Geometriae Dedicata

, Volume 179, Issue 1, pp 91–138 | Cite as

Riemannian simplices and triangulations

Open Access
Original Paper

Abstract

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the “thickness” or “fatness” of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.

Keywords

Riemannian centre of mass Karcher means Barycentric coordinates Triangulation Riemannian manifold  Sampling conditions Riemannian simplices 

Mathematics Subject Classification

57R05 

1 Introduction

In this work we study a natural definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension. The definition is intrinsic; the simplex is defined by the positions of its vertices in the manifold, which need not be embedded in an ambient space. The standard definition of a Euclidean simplex as the convex hull of its vertices is not useful for defining simplices in general Riemannian manifolds. Besides the problem that convex hulls are difficult to compute (almost nothing is known about the convex hull of three distinct points, for example [4], section 6.1.3]), the resulting objects could not be used as building blocks for triangulations, i.e., they cannot be used to define geoemetric simplicial complexes. This is because if two full dimensional convex simplices share a boundary facet, that facet must itself be convex. This constrains the facet to lie on a totally geodesic submanifold (i.e., minimising geodesics between points on the facet must lie in the facet), and when the curvature is not constant such submanifolds cannot be expected to exist (see [4], Thm 58] or [16], Section 11]).

Given the vertices, a geometric Euclidean simplex can also be defined as the domain on which the barycentric coordinate functions are non-negative. This definition does extend to general Riemannian manifolds in a natural way. The construction is based on the fact that the barycentric coordinate functions can be defined by a “centre of mass” construction. Suppose \(\{v_0, \ldots , v_n\} \subset \mathbb {R}^n\), and \(\{\lambda _{i}\}_{0 \le i \le n}\) is a set of non-negative weights that sum to 1. If u is the point that minimises the function
$$\begin{aligned} y \mapsto \sum _{i=0}^n \lambda _{i}d_{\mathbb {R}^n}(y,v_i)^2, \end{aligned}$$
(1)
where \(d_{\mathbb {R}^n}(x,y) = \left| x-y\right| \) is the Euclidean distance, then \(u = \sum \lambda _{i} v_i\), and the \(\{\lambda _{i}\}\) are the barycentric coordinates of u in the simplex \([v_0, \ldots , v_n]\).

We can view a given set of barycentric coordinates \(\lambda = (\lambda _{0},\ldots ,\lambda _{n})\) as a point in \(\mathbb {R}^{n+1}\). The set \(\varvec{\varDelta }^{n}\) of all points in \(\mathbb {R}^{n+1}\) with non-negative coefficients that sum to 1 is called the standard Euclideann-simplex. Thus the minimisation of the function (1) defines a map from the standard Euclidean simplex to the Euclidean simplex \([v_0,\ldots ,v_n] \subset \mathbb {R}^n\)

If instead the points \(\{v_i\}\) lie in a convex set W in a Riemannian manifold \(M\), then, by using the metric of the manifold instead of \(d_{\mathbb {R}^n}\) in Eq. (1), we obtain a function \(\fancyscript{E}_{\lambda }:W \rightarrow \mathbb {R}\) that has a unique minimum \(x \in W\), provided \(W\) is sufficently small (see Sect. 2.1). In this way we obtain a mapping \(\lambda \mapsto x\) from \(\varvec{\varDelta }^{n}\) to W. We call the image of this map an intrinsic simplex, or a Riemannian simplex.

1.1 Previous work

Equation (1) defines a point with given barycentric coordinates as a weighted centre of mass. Centres of mass were apparently introduced in this context in 1929 by Cartan [13] for a finite number of points in a symmetric setting [4], Section 6.1]. Fréchet also studied such functions in a more general setting, with integrals instead of sums, in 1948 [22]. However, Karcher [26] gave an extensive treatment particular to the Riemannian setting, and averages defined in this way are often referred to as “Karcher means”.

Karcher’s exposition [26] is the standard reference for Karcher means. However, for our purposes a particularly good resource is the work by Buser and Karcher [11], Sections 6, 8]. This work was exploited by Peters [32], where Karcher means are used to interpolate between locally defined diffeomorphisms between manifolds in order to construct a global diffeomorphism in a proof of Cheeger’s finiteness theorem. Chavel [14], Ch. IX] gives a detailed exposition of Peters’s argument. Kendal [28] provides another important reference for Karcher means. Riemannian simplices are not explicitly considered in any of these works.

More recently, Rustamov [33] introduced barycentric coordinates on a surface via Karcher means. Sander [34] used the method in arbitrary dimensions to define Riemannian simplices as described above. He called them geodesic finite elements, reflecting the application setting in numerical solutions to partial differential equations involving functions which take values in a manifold.

Our work is motivated by a desire to develop sampling requirements for representing a compact smooth Riemannian manifold with a simplicial complex. By this we mean that we seek conditions on a finite set \(S\subset M\) that guarantee that \(S\) can be the vertex set of an (abstract) simplicial complex \(\fancyscript{A}\) that is homeomorphic to \(M\). A simplicial complex \(\fancyscript{A}\) defines a topological space, \(\left| \fancyscript{A}\right| \) called the carrier of \(\fancyscript{A}\). We call a homeomorphism \(H :\left| \fancyscript{A}\right| \rightarrow M\) a triangulation of \(M\). In this work we present criteria sufficient to construct a triangulation \(H\), given \(\fancyscript{A}\).

We are particularly interested in manifolds of dimension greater than 2. For 2-dimensional manifolds a triangulation is guaranteed to exist when \(S\) meets density requirements that can be specified either in terms of extrinsic criteria, for surfaces embedded in Euclidean space [1, 10], or in terms of intrinsic criteria [20, 29]. In higher dimensions, although it is well known that a smooth manifold admits a triangulation, to the best of our knowledge well-founded sampling conditions sufficient to guarantee the existence of a triangulation with a given sample points as vertices have yet to be described.

For arbitrary finite dimension, Cairns [12] first demonstrated that a smooth compact manifold admits a triangulation by embedding Euclidean complexes into the manifold via coordinate charts, and showing that if the complexes were sufficiently refined the embedding maps could be perturbed so that they remain embeddings and the images of simplices coincide where patches overlap, thus constructing a global embedding of a complex. Whitehead [36] refined the technique into a general approximation theory which is described in detail by Munkres [30] and is not restricted to compact manifolds. Whitney [37] used his result that a manifold can be embedded into Euclidean space to triangulate the manifold by intersecting it with a fine Cartesian grid in the ambient space. The problem has been revisited more recently in the computational geometry community, where the focus is on the algorithm used to construct a triangulation when a compact submanifold is known only through a finite set of sample points. Cheng et al. [17] used the generic triangulation result of Edelsbrunner and Shah [21] to argue that a weighted Delaunay complex will triangulate a manifold, and Boissonnat and Ghosh [9] adapted Whitney’s argument to demonstrate a triangulation by a Delaunay-based complex whose computation depends only linearly on the ambient dimension.

In every case a metric on the manifold was employed in the computation of the triangulation. A simplicial complex is constructed, some measure of simplex quality is introduced, and a lower bound on this quality measure is an essential component of the construction. In the computational geometry setting that motivated this work, the construction of the simplicial complex starts by introducing sample points on the manifold, which will serve as the vertices of the constructed complex. What becomes interesting in this context is the density of sample points that is required to construct a triangulation. Although density bounds have been made that are sufficient to construct a manifold simplicial complex, there have been no explicit estimates on sampling density sufficient to ensure the existence of a homeomorphism between the complex and the original manifold. In all of the above-mentioned triangulation results, there is only the unquantified assurance that if the maximum distance between adjacent vertices is small enough, a triangulation may be obtained.

In this work we do not address the problem of constructing a simplicial complex. We focus on the problem of establishing sampling density criteria sufficient to ensure that a given complex defines a natural triangulation. Thus our sampling density criteria are dependent on the properties of the given simplicial complex. In fact, the only quantative property of the complex that comes into play is a measure of the quality of the simplices involved. Rather than supplying sampling density estimates for triangulation that are implicitly tied to a specific construction, we provide a flexible framework that allows one to determine the sampling density sufficient to ensure a triangulation by any given construction. We express this density by a scale parameter \(h\) which is effectively an upper bound on the geodesic distance between adjacent vertices.

1.2 Overview

When we speak about the quality of a Euclidean simplex, we are referring to a function that parameterises how close the simplex is to being degenerate. A common quality measure for an n-simplex is the ratio of the volume to the nth power of the longest edge length. Another useful quality measure is the ratio of the smallest altitude to the longest edge length. A Euclidean simplex is degenerate if and only if its quality measure vanishes.

In this work we shed light on the relationship between the local curvature in the manifold, and the size and quality of the simplices involved in a triangulation. We articulate explicit criteria that are sufficient to guarantee that a simplicial complex with vertices on the manifold is homeomorphic to the manifold. The intrinsic simplices defined by the centre of mass construction provide a convenient tool for this purpose.

Although the idea of Riemannian simplices defined in this way has been in the mathematical community for some time (see Berger [4], 6.1.5]), we are not aware of any published work exploiting the notion (of simplices in particular) prior to that of Rustamov [33] and Sander [34]. For our purposes we need to establish a property that Sander did not consider. We need to ensure that the map from the Euclidean simplex to the manifold is a smooth embedding (i.e., the map extends to a smooth map from an open neighbourhood of the Euclidean simplex). This ensures that the barycentric coordinates mapped to the manifold do in fact provide a local system of coordinates. If the map is not a smooth embedding, we call the Riemannian simplex degenerate. Independently, von Deylen [19] has also treated the question of degeneracy of Riemannian simplices. His work includes a detailed analysis of the geometry of the barycentric coordinate map, and several applications. He does not address the problem of sampling criteria for triangulation.

A Euclidean simplex is non-degenerate if and only if its vertices are affinely independent. We show that a Riemannian simplex is non-degenerate if and only if for every point in the simplex the vertices are affinely independent when they are lifted by the inverse of the exponential map to the tangent space of that point.

In a two dimensional manifold this condition is satisfied for a triangle as long as the vertices do not lie on a common geodesic. Similar to the Euclidean case, such a configuration can be avoided by applying an arbitrarily small perturbation to the vertices. However, when the dimension is greater than two, a non-trivial constraint on simplex quality is required. In dimension 2 a sampling density for triangulation can be specified in terms of the convexity radius [20, 29] (maximal radius for which a geodesic ball is convex, see Sect. 2.1), and depends only on an upper bound on the sectional curvatures (Lemma 1). In dimension higher than 2, we require the simplex size (maximum edge length) to also be constrained by a lower bound on the sectional curvatures (the upper bound on the edge lengths is inversely proportional to the square root of an upper bound on the absolute value of the sectional curvatures), so we cannot express the sampling density requirements in terms of a convexity radius alone.

We may define a quality measure for a Riemannian simplex by considering the quality of the Euclidean simplex obtained by lifting the vertices to the tangent space at one of the vertices. For our purposes we require a lower bound on the smallest such quality measure when each of the vertices is considered.

The quality of the Riemannian simplex that is required to ensure that it is non-degenerate depends on the maximum edge length, as well as the magnitude of the sectional curvatures in the neighbourhood. We establish this relationship with the aid of the Rauch comparison theorem, which provides an estimate on the differences in edge lengths of Euclidean simplices obtained by lifting the vertices of the Riemannian simplex to different tangent spaces. By exploiting previously established bounds on the degradation of the quality of a Euclidean simplex under perturbations of the edge lengths [7], we establish conditions that guarantee that the Riemannian simplex is non-degenerate.

We use this result to establish conditions that guarantee that a simplicial complex is homeomorphic to the manifold. This is the primary motivation for this work. Given an abstract simplicial complex whose vertex set is identified with points on the manifold, we are ensured that it triangulates the manifold if certain conditions are met, the principle one being a relationship between the size and quality of the Riemannian simplices.

1.3 Outline and main results

In Sect. 2 we present the framework for centre of mass constructions, and introduce the barycentric coordinate map and Riemannian simplices. Riemannian simplices are defined (Definition 4) as the image of the barycentric coordinate map, so they are “filled in” geometric simplices. A Riemannian simplex \(\varvec{\sigma }_M\) is defined by its vertices \(\varvec{\sigma }= \{p_0, \ldots , p_n\} \subset M\), which are constrained to lie in a convex neighbourhood \(B_{\rho }\subset M\). For any \(x \in B_{\rho }\) we define a Euclidean simplex \(\varvec{\sigma }(x) \subset T_{x}{M}\) by \(\varvec{\sigma }(x) = \{v_0(x), \ldots , v_n(x)\}\), where \(v_i(x) = \exp _x^{-1}(p_i)\). We use a subscript indicating the ambient space when we are referring to a simplex as a set of non-negative barycentric coordinates, otherwise we mean a finite vertex set; the convex hull of \(\varvec{\sigma }(x)\) is \(\varvec{\sigma }_{\mathbb {E}}(x)\).

In Sect. 2.3 we give a characterisation of non-degenerate Riemannian simplices in terms of affine independence. We show that \(\varvec{\sigma }_M\) is non-degenerate if and only if \(\varvec{\sigma }(x)\) is non-degenerate for every \(x \in \varvec{\sigma }_M\).

In Sect. 3 we establish criteria to ensure that a Riemannian simplex is non-degenerate. We first review properties of Euclidean simplices, including thickness, the quality measure we employ. The thickness is essentially the ratio of the smallest altitude to the longest edge length of the simplex. If the edge lengths in a Euclidean simplex change by a small amount, we can quantify the change in the thickness. In particular, if \(F:\mathbb {R}^n\rightarrow \mathbb {R}^n\) is a bi-Lipschitz map, we can quantify a bound on the thickness, \(t\left( \varvec{\sigma }\right) \), of a simplex \(\varvec{\sigma }\) relative to the metric distortion (i.e., the bi-Lipschitz constant) that establishes when the Euclidean simplex \(F(\varvec{\sigma })\) is non-degenerate.

The Rauch theorem establishes bounds on the norm of the differential of the exponential map, relative to the sectional curvatures. Using this we obtain a bound on the metric distortion of the transition function
$$\begin{aligned} \exp _x^{-1} \circ \exp _p:T_pM\rightarrow T_xM\end{aligned}$$
(2)
which maps \(\varvec{\sigma }(p)\) to \(\varvec{\sigma }(x)\), and so we are able to establish conditions ensuring that \(\varvec{\sigma }(x)\) is non-degenerate, based on quality assumptions on \(\varvec{\sigma }(p)\).

An open geodesic ball of radius r centred at \(x \in M\) is the set \(B_{M}(x;r)\) of all points in \(M\) whose geodesic distance from x is less than r. The injectivity radius atx, denoted \(\iota (x)\), is the supremum of the radii r for which \(\exp _x\) restricts to a diffeomorphism between the Euclidean ball of radius r centred at \(0\in T_xM\), and \(B_{M}(x;r)\). The injectivity radius of \(M\) is the infimum of \(\iota (x)\) over all \(x\in M\), and is denoted \(\iota _{M}\).

Theorem 1

(Non-degeneracy criteria) Suppose \(M\) is a Riemannian manifold with sectional curvatures \(K\) bounded by \(\left| K\right| \le \varLambda \), and \(\varvec{\sigma }_M\) is a Riemannian simplex, with \(\varvec{\sigma }_M\subset B_{\rho }\subset M\), where \(B_{\rho }\) is an open geodesic ball of radius \(\rho \) with
$$\begin{aligned} \rho < \rho _0= \min \left\{ \frac{\iota _{M}}{2}, \frac{\pi }{4\sqrt{\varLambda }} \right\} . \end{aligned}$$
(3)
Then \(\varvec{\sigma }_M\) is non-degenerate if there is a point \(p \in B_{\rho }\) such that the lifted Euclidean simplex \(\varvec{\sigma }(p)\) has thickness satisfying
$$\begin{aligned} t\left( \varvec{\sigma }(p)\right) > 10 \sqrt{\varLambda }L\left( \varvec{\sigma }_M\right) , \end{aligned}$$
(4)
where \(L\left( \varvec{\sigma }_M\right) \) is the geodesic length of the longest edge in \(\varvec{\sigma }_M\).

If p is a vertex in an abstract simplicial complex \(\fancyscript{A}\), we define the star ofp to be the subcomplex \(\underline{{{\mathrm{St}}}}(p)\) of \(\fancyscript{A}\) consisting of all simplices that contain p, together with the faces of these simplices. The underlying topological space (or carrier) of a complex \(\fancyscript{A}\) is denoted \(\left| \fancyscript{A}\right| \). We say that \(\underline{{{\mathrm{St}}}}(p)\) is a full star if \(\left| \underline{{{\mathrm{St}}}}(p)\right| \) is a closed topological ball of dimension n with p in its interior, and \(\fancyscript{A}\) contains no simplices of dimension greater than n.

In Sect. 4 we develop our sampling criteria for triangulating manifolds. We establish properties of maps whose differentials are bounded close to a fixed linear isometry, and use these properties to reveal conditions under which a full star will be embedded into a manifold. This allows us to express generic conditions that ensure that a simplicial complex is homeomorphic to the manifold. We then exploit a refinement of the Rauch theorem, and other estimates established by Buser and Karcher [11], to bound the differential of the barycentric coordinate map as required, and thus arrive at:

Theorem 2

(Triangulation criteria) Suppose \(M\) is a compact n-dimensional Riemannian manifold with sectional curvatures \(K\) bounded by \(\left| K\right| \le \varLambda \), and \(\fancyscript{A}\) is an abstract simplicial complex with finite vertex set \(S\subset M\). Define a quality parameter \(t_0> 0\), and let
$$\begin{aligned} h= \min \left\{ \frac{\iota _{M}}{4}, \frac{\sqrt{n}t_0}{6 \sqrt{\varLambda }} \right\} . \end{aligned}$$
(5)
If
  1. 1.

    For every \(p \in S\), the vertices of \(\underline{{{\mathrm{St}}}}(p)\) are contained in \(B_{M}(p;h)\), and the balls \(\{B_{M}(p;h)\}_{p \in S}\) cover \(M\).

     
  2. 2.

    For every \(p \in S\), the restriction of the inverse of the exponential map \(\exp _p^{-1}\) to the vertices of \(\underline{{{\mathrm{St}}}}(p) \subset \fancyscript{A}\) defines a piecewise linear embedding of \(\left| \underline{{{\mathrm{St}}}}(p)\right| \) into \(T_{p}{M}\), realising \(\underline{{{\mathrm{St}}}}(p)\) as a full star such that every simplex \(\varvec{\sigma }(p)\) has thickness \(t\left( \varvec{\sigma }(p)\right) \ge t_0\).

     
then \(\fancyscript{A}\) triangulates \(M\), and the triangulation is given by the barycentric coordinate map on each simplex.

The techniques employed to obtain Theorem 2 exploit stronger bounds on the differential of the exponential map, and provide a slightly better bound for non-degeneracy than the one stated in Theorem 1, but at the expense of a stronger constraint on the allowed diameter of the simplex. This is the reason Eq. (4) appears as a stronger constraint on the thickness than the curvature controlled part of Eq. (5).

We refer to the criteria of Theorem 2 as sampling criteria, even though they require a simplicial complex for their definition. Although there is no explicit constraint on the minimal distance between points of \(S\), one is implicitly imposed by the quality constraint on the Riemannian simplices. The required sampling density depends on the quality of the Riemannian simplices, which leaves open the question of what kind of quality of simplices can we hope to attain. Recent work [7] constructs a Delaunay complex conforming to the requirements of Theorem 2 with the thickness \(t_0\) bounded by \(1/2^{\fancyscript{O}(n^3)}\). It would be interesting to see this improved.

The complex \(\fancyscript{A}\) in Theorem 2 naturally admits a piecewise linear metric by assigning edge lengths to the simplices given by the geodesic distance in \(M\) between the endpoints. In Sect. 5 we observe that in order to ensure that this does in fact define a piecewise-flat metric, we need to employ slightly stronger constraints on the scale parameter h. In this case, the complex \(\fancyscript{A}\) becomes a good geometric approximation of the original manifold, and we find:

Theorem 3

(Metric distortion) If the requirements of Theorem 2, are satisfied with the scale parameter (5) replaced by
$$\begin{aligned} h= \min \left\{ \frac{\iota _{M}}{4}, \frac{t_0}{6 \sqrt{\varLambda }} \right\} , \end{aligned}$$
then \(\fancyscript{A}\) is naturally equipped with a piecewise flat metric \(d_{\fancyscript{A}}\) defined by assigning to each edge the geodesic distance in \(M\) between its endpoints.
If \(H :\left| \fancyscript{A}\right| \rightarrow M\) is the triangulation defined by the barycentric coordinate map in this case, then the metric distortion induced by \(H\) is quantified as
$$\begin{aligned} \left| d_{M}(H(x),H(y)) - d_{\fancyscript{A}}(x,y) \right| \le \frac{50 \varLambda h^2}{t_0^2} d_{\fancyscript{A}}(x,y), \end{aligned}$$
for all \(x,y \in \left| \fancyscript{A}\right| \).

The criteria of these three theorems can also be formulated in terms of the thickness of the Euclidean simplices defined by the geodesic edge lengths of the Riemannian simplices, rather than the Euclidean simplices we find in the tangent spaces. In “Appendix 1” we briefly mention this alternative formulation of our results. We also compare the thickness quality measure for simplices with a commonly used volumetric quality measure which we call fatness.

In “Appendix 1” an alternate approach to non-degenerate Riemannian simplices is presented. This approach is based on bounding angles and edge lengths in geodesic triangles via the Toponogov comparison theorem.

2 Riemannian simplices

In this section we summarise the results of the theory of Riemannian centres of mass that we need in order to define Riemannian simplices. We then give an explicit description of the barycentric coordinate map that is used to define these simplices. We take the view that if the barycentric coordinate map is well defined, then the simplex is well defined, but it may be degenerate. The geodesic finite elements employed by Sander [34] are Riemannian simplices without a requirement of non-degeneracy. In Sect. 2.3 we demonstrate that non-degeneracy of a Riemannian simplex \(\varvec{\sigma }_M\) is characterised by the affine independence of the vertices when lifted to the tangent space of any point in \(\varvec{\sigma }_M\).

2.1 Riemannian centre of mass

We work with an n-dimensional Riemannian manifold \(M\). The centre of mass construction developed by Karcher [26] hinges on the notion of convexity in a Riemannian manifold. A set \(B \subseteq M\) is convex if any two points \(x,y \in B\) are connected by a minimising geodesic \(\gamma _{xy}\) that is unique in \(M\), and contained in \(B\). For \(c \in M\), the geodesic ball of radius r is the set \(B_{M}(c;r)\) of points in \(M\) whose distance from c is less than r, and we denote its closure by \(\overline{B}_{M}(c;r)\). If r is small enough, \(\overline{B}_{M}(c;r)\) will be convex; the following lemma quantifies “small enough”.

In order to obtain non-degeneracy criteria for Riemannian simplices we require both an upper and a lower bound on the sectional curvatures, so it is convenient to work with a bound \(\varLambda \) on the absolute value of the sectional curvatures, \(\left| K\right| \le \varLambda \). However, the definition of Riemannian simplices only requires an upper bound on the sectional curvatures. In order to emphasise this we introduce distinct symbols for the upper and lower bounds on the sectional curvatures. Thus \(\varLambda _{-}\le K\le \varLambda _{+}\), and \(\varLambda = \max \{\varLambda _{+}, - \varLambda _{-}\}\).

We have [14], Thm. IX.6.1]:

Lemma 1

Suppose the sectional curvatures of \(M\) are bounded by \(K\le \varLambda _{+}\), and \(\iota _{M}\) is the injectivity radius. If
$$\begin{aligned} r < \min \left\{ \frac{\iota _{M}}{2}, \frac{\pi }{2\sqrt{\varLambda _{+}}} \right\} , \end{aligned}$$
then \(\overline{B}_{M}(x;r)\) is convex. (If \(\varLambda _{+}\le 0\), we take \(1 / \sqrt{\varLambda _{+}}\) to be infinite.)

Remark 2

Lemma 1 is stated in terms of global bounds on the injectivity radii and sectional curvatures (on a non-compact manifold, these may be useless), but really we only need these bounds in a neighbourhood of x. Let \(K(x)\) be an upper bound on the sectional curvatures at x, and denote the injectivity radius at x by \(\iota (x)\). Now define \(I(x)\) and \(\varLambda _{+}(x)\) to be the infimum and supremum respectively of \(\iota (y)\) and \(K(y)\), where \(y\) ranges over the ball \(B_{M}(x;R)\) of radius
$$\begin{aligned} R = \min \left\{ \frac{\iota (x)}{2}, \frac{\pi }{2\sqrt{K(x)}} \right\} . \end{aligned}$$
Then Lemma 1 holds if \(\iota _{M}\) and \(\varLambda _{+}\) are replaced by \(I(x)\) and \(\varLambda _{+}(x)\) respectively in the bound on r. For simplicity, we will continue to refer to global bounds, but everywhere they occur a similar remark applies.

Also, in all cases where an upper bound on the sectional curvatures is employed, this bound is only relevant when it is positive. If \(M\) has non-positive curvature, then \(1 / \sqrt{\varLambda _{+}}\) may be assumed to be infinite.

In our context, we are interested in finding a weighted centre of mass of a finite set \(\{p_0, \ldots , p_j\} \subset B \subset M\), where the containing set \(B\) is open, and its closure \(\overline{B}\) is convex. The centre of mass construction is based on minimising the function \(\fancyscript{E}_{\lambda }:\overline{B} \rightarrow \mathbb {R}\) defined by
$$\begin{aligned} \fancyscript{E}_{\lambda }(x) = \frac{1}{2} \sum _i \lambda _{i} d_{M}(x,p_i)^2, \end{aligned}$$
(6)
where the \(\lambda _{i} \ge 0\) are non-negative weights that sum to \(1\), and \(d_{M}\) is the geodesic distance function on \(M\). Karcher’s first simple observation is that the minima of \(\fancyscript{E}_{\lambda }\) must lie in the interior of \(\overline{B}\), i.e., in \(B\) itself. This follows from considering the gradient of \(\fancyscript{E}_{\lambda }\):
$$\begin{aligned} {{\mathrm{grad}}}\fancyscript{E}_{\lambda }(x) = - \sum _i \lambda _{i} \exp ^{-1}_x(p_i). \end{aligned}$$
(7)
At any point x on the boundary of \(\overline{B}\), the gradient vector lies in a cone of outward pointing vectors. It follows that the minima of \(\fancyscript{E}_{\lambda }\) lie in \(B\). The more difficult result that the minimum is unique, Karcher showed by demonstrating that \(\fancyscript{E}_{\lambda }\) is convex. If \(B \subseteq M\) is a convex set, a function \(f :B \rightarrow \mathbb {R}\) is convex if for any geodesic \(\gamma :I \rightarrow B\), the function \(f \circ \gamma \) is convex (here \(I \subseteq \mathbb {R}\) is an open interval). If \(f\) has a minimum in \(B\), it must be unique. By Eq. (7), it is the point x where
$$\begin{aligned} \sum _i \lambda _{i} \exp ^{-1}_x(p_i) = 0. \end{aligned}$$
We have the following result [26], Thm 1.2]:

Lemma 3

(Unique centre of mass) If \(\{p_0, \ldots , p_j\} \subset B_{\rho } \subset M\), and \(B_{\rho }\) is an open ball of radius \(\rho \) with
$$\begin{aligned} \rho < \min \left\{ \frac{\iota _{M}}{2}, \frac{\pi }{4\sqrt{\varLambda _{+}}} \right\} , \end{aligned}$$
then on any geodesic \(\gamma :I \rightarrow B_{\rho }\), we have
$$\begin{aligned} \frac{\mathrm {d}^2}{\mathrm {d}t^2}\fancyscript{E}_{\lambda }(\gamma (t)) \ge C(\varLambda _{+}, \rho ) > 0, \end{aligned}$$
(8)
where \(C(\varLambda _{+}, \rho )\) is a positive constant depending only on \(\varLambda _{+}\) and \(\rho \). In particular, \(\fancyscript{E}_{\lambda }\) is convex and has a unique minimum in \(B_{\rho }\).

Karcher gives an explicit expression for \(C(\varLambda _{+}, \rho )\), but we will not need to refer to it here. Also, Karcher expresses the centre of mass concept in more generality by using an integral over a set whose measure is 1, rather than a weighted sum over a finite set as we have used.

2.2 The barycentric coordinate map

Let \(\varvec{\varDelta }^{j}\) denote the standard Euclidean j-simplex. This can be realised as the set of points \(\lambda \in \mathbb {R}^{j+1}\) whose components are non-negative, \(\lambda _{i} \ge 0\), and sum to one: \(\sum _i \lambda _{i} = 1\). We index the coordinates starting from zero: these are the barycentric coordinates on the standard simplex.

Definition 4

(Riemannian simplex) If a finite set \(\varvec{\sigma }^j = \{p_0, \ldots , p_j\} \subset M\) in an n-manifold is contained in an open geodesic ball \(B_{\rho }\) whose radius, \(\rho \), satisfies Eq. (3), then \(\varvec{\sigma }^j\) is the set of vertices of a geometric Riemannian simplex, denoted \(\varvec{\sigma }_M^j\), and defined to be the image of the map
$$\begin{aligned} \begin{aligned}&\quad \fancyscript{B}_{\varvec{\sigma }^j} :\varvec{\varDelta }^{j} \rightarrow M\\&\lambda \mapsto \mathop {\mathrm{argmin}}\limits _{x \in \overline{B}_{\rho }} \fancyscript{E}_{\lambda }(x). \end{aligned} \end{aligned}$$
We say that \(\varvec{\sigma }_M^j\) is non-degenerate if \(\fancyscript{B}_{\varvec{\sigma }^j}\) is a smooth embedding; otherwise it is degenerate.

Define an i-face of \(\varvec{\sigma }_M^j\) to be the image of an i-face of \(\varvec{\varDelta }^{j}\). Since an i-face of \(\varvec{\varDelta }^{j}\) may be identified with \(\varvec{\varDelta }^{i}\) (e.g., by an order preserving map of the vertex indices), the i-faces of \(\varvec{\sigma }_M^j\) are themselves Riemannian i-simplices. In particular, if \(\tau \) and \(\mu \) are the vertices of Riemannian simplices \(\varvec{\tau }_M\) and \(\varvec{\mu }_M\), and \(\varvec{\sigma }^i = \tau \cap \mu \), then the Riemannian i-simplex \(\varvec{\sigma }_M^i\) is a face of both \(\varvec{\tau }_M\) and \(\varvec{\mu }_M\). The edges of a Riemannian simplex are the Riemannian 1-faces. We observe that these are geodesic segments. We will focus on full dimensional simplices, i.e., unless otherwise specified, \(\varvec{\sigma }_M\) will refer to a Riemannian simplex defined by a set \(\varvec{\sigma }\) of \(n+1\) vertices in our n-dimensional manifold \(M\).

Remarks The barycentric coordinate map \(\fancyscript{B}_{\varvec{\sigma }}\) is differentiable. This follows from the implicit function theorem, as is shown by Buser and Karcher [11], Section 8.3.3], for example. They work in local coordinates on the tangent bundle, and use the connection to split the derivative of \({{\mathrm{grad}}}\fancyscript{E}_{\lambda }:M\rightarrow TM\) into horizontal and vertical components. The strict convexity condition (8) implies that the vertical component of the derivative is full rank, and permits the use of the implicit function theorem.

The argument of Buser and Karcher assumes that the map is defined on an open domain. We observe that \(\fancyscript{B}_{\varvec{\sigma }}\) is well defined if we allow negative barycentric coordinates of small magnitude. For a sufficiently small \(\epsilon > 0\), Lemma 3 holds if the barycentric coordinates \(\lambda _{i}\) satisfy \(\sum \lambda _{i} = 1\) and \(\lambda _{i} > - \epsilon \) for all \(i \in \{0, \ldots , n\}\), albeit with \(C(\varLambda _{+}, \rho )\) replaced with a smaller positive constant. This follows from the observation that \(\frac{d^2}{dt^2}\fancyscript{E}_{\lambda }\) is continuous in the barycentric coordinates, thus since it is strictly positive on the boundary of \(\varvec{\varDelta }^{n}\), it can be extended to an open neighbourhood. This means that \(\fancyscript{B}_{\varvec{\sigma }}\) is smooth on the closed domain \(\varvec{\varDelta }^{n}\), as defined in Sect. 4.1.

Karcher himself mentioned that his result can accommodate signed measures [26], Remark 1.8], and Sander has demonstrated this in some detail [35]. However, for our current purposes we are only claiming that we can accommodate arbitrarily small negative barycentric coordinates assuming the stated bound on \(\rho _0\) (Eq. 3).

A Riemannian simplex is not convex in general, but as Karcher observed [26], being the image of the barycentric coordinate map, it will be contained in any convex set that contains the vertices of the simplex. Thus the Riemannian simplex is contained in the intersection of such sets.

Equation (3) gives an upper bound on the size of a Riemannian simplex that depends only on the injectivity radius and an upper bound on the sectional curvature. For example, in a non-positively curved manifold, the size of a well defined Riemannian simplex is constrained only by the injectivity radius. However, if the dimension n of the manifold is greater than \(2\), we will require also a lower bound on the sectional curvatures in order to ensure that the simplex is non-degenerate.

Lemma 3 demands that a Riemannian simplex be contained in a ball whose radius is constrained by \(\rho _0\). Thus Riemannian simplices always have edge lengths less than \(2\rho _0\). If the longest edge length, \(L\left( \varvec{\sigma }_M\right) \), of \(\varvec{\sigma }_M\) is less than \(\rho _0\), then \(\varvec{\sigma }_M\) must be contained in the closed ball of radius \(L\left( \varvec{\sigma }_M\right) \) centred at a vertex. Indeed, any open ball centred at a vertex whose radius is larger than \(L\left( \varvec{\sigma }_M\right) \), but smaller than \(\rho _0\), must contain the vertices and have a convex closure. The simplex is thus contained in the intersection of these balls. If \(L\left( \varvec{\sigma }_M\right) \ge \rho _0\), then a ball of radius \(L\left( \varvec{\sigma }_M\right) \) need not be convex. In this case we claim only that \(\varvec{\sigma }_M\) is contained in a ball of radius \(2\rho _0\) centred at any vertex.

2.3 The affine independence criterion for non-degeneracy

In this subsection we show that a Riemannian simplex \(\varvec{\sigma }_M\) is non-degenerate if, and only if, for any \(x \in \varvec{\sigma }_M\), the lift of the vertices by the inverse exponential map yields a non-degenerate Euclidean simplex. We first introduce some notation and terminology to better articulate this statement.

Notation A Euclidean simplex \(\varvec{\sigma }\) of dimension \(k\) is defined by a set of \(k+1\) points in Euclidean space \(\varvec{\sigma }= \{v_0, \ldots , v_k \} \subset \mathbb {R}^n\). In general we work with abstract simplices, even though we attribute geometric properties to the simplex, inherited from the embedding of the vertices in the ambient space (see Sect. 3.1). When we wish to make the dimension explicit, we write it as a superscript, thus \(\varvec{\sigma }^k\) is a \(k\)-simplex. Traditional “filled in” geometric simplices are denoted by boldface symbols; \(\varvec{\sigma }_{\mathbb {E}}= {{\mathrm{conv}}}(\varvec{\sigma })\) is the convex hull of \(\varvec{\sigma }\). If such a simplex is specified by a vertex list, we employ square brackets: \(\varvec{\sigma }_{\mathbb {E}}= [v_0, \ldots , v_k]\).

The barycentric coordinate functions\(\{ \lambda _{i} \}\) associated to \(\varvec{\sigma }\) are affine functions \(\mathbb {R}^n \rightarrow \mathbb {R}\) that satisfy \(\lambda _{i}(v_j) = \delta _{ij}\) and \(\sum _{i=0}^n \lambda _{i} = 1\). It is often convenient to choose one of the vertices, \(v_0\) say, of \(\varvec{\sigma }\) to be the origin. We let P be the \(n \times k\) matrix whose \(i^{\text {th}}\) column is \(v_i - v_0\). Then the barycentric coordinate functions \(\{\lambda _{i}\}\) are linear functions for \(i>0\), and they are dual to the basis defined by the columns of P. This means that if we represent the function \(\lambda _{i}\) as a row vector, then the matrix Q whose ith row is \(\lambda _{i}\) satisfies \(QP = I_{k \times k}\).

A full dimensional Euclidean simplex \(\varvec{\sigma }\) is non-degenerate, if and only if the corresponding matrix P is non-degenerate. In particular, if \(\varvec{\sigma }\) is full dimensional (i.e., \(k=n\)), then \(Q = {P}^{-1}\). Suppose \(\varvec{\sigma }\subset \mathbb {R}^n\) is an n-simplex. If \(\xi \in \mathbb {R}^n\), let \(\lambda (\xi ) = {(\lambda _{1}(\xi ), \ldots , \lambda _{n}(\xi ))}^\mathsf {T}\). Then \(\lambda (\xi )\) is the vector of coefficients of \(\xi - v_0\) in the basis defined by the columns of P. I.e., \(\xi - v_0 = P\lambda (\xi )\).

We will be interested in Euclidean simplices that are defined by the vertices of a Riemannian simplex: If \(\varvec{\sigma }= \{p_0, \ldots , p_n\} \subset B_{\rho }\subset M\) is the set of vertices of \(\varvec{\sigma }_M\), it is convenient to introduce the notation \(v_i(x) = \exp _x^{-1}(p_i)\), and \(\varvec{\sigma }(x) = \exp _x^{-1}(\varvec{\sigma })\). Thus \(\varvec{\sigma }(x) = \{v_0(x), \ldots , v_n(x)\}\) is a Euclidean simplex in \(T_{x}{M}\).

The norm of a vector \(v\) in a Euclidean space is denoted \(\left| v\right| \). For example, if \(v \in T_pM\), then \(\left| v\right| = \left\langle v,v\right\rangle ^{\frac{1}{2}}\), where \(\left\langle \cdot ,\cdot \right\rangle \) is inner product defined by the Riemannian metric tensor on \(M\). The differential of a map \(F :M\rightarrow \bar{M}\) is denoted by \(dF\); so \(dF_x :T_xM\rightarrow T_{F(x)}\bar{M}\) is a linear map whose operator norm is \(\left\| dF_x\right\| \). All differentiable maps, operators, and manifolds are assumed to be \(C^\infty \).

An expression for the differential The expression for the differential obtained in Equation (10) below is obtained as a particular case of an argument presented by Buser and Karcher [11], Section 8.3]. The argument was later exploited by Peters [32] to sharpen bounds on Cheeger’s finiteness theorem [15]. A thorough exposition appears also in Chavel [14], IX.8].

We work in a domain \(U \subset \mathbb {R}^n\) defined by a chart \(\phi :M\supset W \rightarrow U\) such that \(B_{\rho }\subset W\). Let \(\tilde{\varvec{\sigma }}= \phi (\varvec{\sigma })\) be the image of the vertices of a Riemannian n-simplex \(\varvec{\sigma }_M\subset B_{\rho }\). Label the vertices of \(\tilde{\varvec{\sigma }}= \{\tilde{v}_0, \ldots , \tilde{v}_n\}\) such that \(\tilde{v}_i = \phi (p_i)\), and assume \(\tilde{v}_0\) is at the origin. The affine functions \(\lambda _{i} :u \mapsto \lambda _{i}(u)\) are the barycentric coordinate functions of \(\tilde{\varvec{\sigma }}\). We consider \({{\mathrm{grad}}}\fancyscript{E}_{\lambda }\), introduced in Eq. (7), now to be a vector field that depends on both \(u \in U\) and \(x \in B_{\rho }\). Specifically, we consider the vector field \(\nu :U \times B_{\rho }\rightarrow T_{}{M}\) defined by
$$\begin{aligned} \nu (u,x) = - \sum _{i=0}^n \lambda _{i}(u)v_i(x). \end{aligned}$$
(9)
Let \(b:\varvec{\tilde{\sigma }}_{\mathbb {E}}\rightarrow \varvec{\sigma }_M\) be defined by \(b= \fancyscript{B}_{\varvec{\sigma }}\circ \fancyscript{L}\), where \(\fancyscript{L}\) is the canonical linear isomorphism that takes the vertices of \(\tilde{\varvec{\sigma }}\) to those of \(\varvec{\varDelta }^{n}\), and \(\fancyscript{B}_{\varvec{\sigma }}\) is the barycentric coordinate map introduced in Definition 4. This map is differentiable, by the arguments presented by Buser and Karcher, and \(\nu (u, b(u)) = 0\) for all \(u \in \varvec{\tilde{\sigma }}_{\mathbb {E}}\). Regarding \(\nu \) as a vector field along \(b\), its derivative may be expanded as
$$\begin{aligned} \partial _u\nu + \left( \nabla \nu \right) db= 0, \end{aligned}$$
where \(\partial _u\nu \) denotes the differential of \(\nu (u,x)\) with x fixed, i.e.,
$$\begin{aligned}&\partial _u\nu _{(u,x)} :T_{u}{\mathbb {R}^n} \rightarrow T_{x}{M}\\&\left( \partial _u \nu _{(u,x)}\right) \dot{u}(0) = \frac{d}{d t} \nu (u(t),x) \big |_{t=0}, \end{aligned}$$
with \(\dot{u}(0)\) denoting the tangent vector at \(0\) to some curve \(t \mapsto u(t)\) in \(U\subset \mathbb {R}^n\). Similarly \(\nabla \nu \) is the covariant differential when \(u\) is fixed:
$$\begin{aligned}&\nabla \nu _{(u,x)} :T_{x}{M} \rightarrow T_{x}{M}\\&\left( \nabla \nu _{(u,x)}\right) \dot{x}(0) = D_t \nu (u, x(t)) \big |_{t=0}, \end{aligned}$$
where \(D_t\nu \) is the covariant derivative along the curve \(x(t)\). Finally \(db:T_{u}{\mathbb {R}^n} \rightarrow T_{x}{M}\) is the differential of \(b\), our barycentric coordinate map onto the Riemannian simplex \(\varvec{\sigma }_M\).
Our objective is to exhibit conditions that ensure that \(db\) is non-degenerate. It follows from the strict convexity conditon (8) of Lemma 3 that the map \(\nabla \nu :w \mapsto \nabla _w\nu \) is non-degenerate. Indeed, if \(w \in T_xM\) for some \(x \in B_{\rho }\), there is a geodesic \(\gamma :I \rightarrow B_{\rho }\) with \(\gamma '(0)=w\), and \(\frac{d^2}{dt^2}\fancyscript{E}_{\lambda }(\gamma (t))\big |_{t=0} = \left\langle \nabla _{w}\nu ,w\right\rangle > 0\). Therefore, we have that
$$\begin{aligned} db= - \left( \nabla \nu \right) ^{-1} \partial _u \nu , \end{aligned}$$
(10)
and thus \(db\) has full rank if and only if \(\partial _u\nu \) has full rank.
The differential as a matrix Recalling Eq. (9), notice that when x is fixed, \(\nu \) is an affine map \(\mathbb {R}^n\supset U \rightarrow T_xM\), and so \(\left( \partial _u \nu \right) _v = \left( \partial _u \nu \right) _w\) for all \(v,w \in U\). We see that
$$\begin{aligned} \partial _u \nu = - \sum _{i=0}^n v_i(x)\, d\lambda _{i}. \end{aligned}$$
Since \(\sum _{i=0}^n \lambda _{i} = 1\), we have that \(\sum _{i=0}^n \, d \lambda _{i} = 0\). We may thus write \(d\lambda _{0} = - \sum _{i=1}^n d\lambda _{i}\), and so for \(\xi \in T_{u}{U}\), we have
$$\begin{aligned} \left( \partial _u \nu \right) \xi = - \sum _{i=1}^n \big ( v_i(x) - v_0(x) \big ) d\lambda _{i}(\xi )\,. \end{aligned}$$
(11)
Now, since the domain of the barycentric coordinates is \(U\), and the origin of \(U \subset \mathbb {R}^n\) coincides with \(\tilde{v}_0\), the functions \(\lambda _{i}\) for \(i \in \{1,\ldots ,n \}\) are linear functions, and we use the canonical identification of tangent spaces in \(\mathbb {R}^n\) to conclude that \(d\lambda _{i}(\xi ) = \lambda _{i}(\xi )\), where in the right hand side we view \(\xi \) as an element of \(\mathbb {R}^n\), rather than an element of \(T_{u}{\mathbb {R}^n}\). As discussed above, we have \(\lambda (\xi ) = \tilde{P}^{-1}\xi \), where \(\lambda (\xi ) = {(\lambda _{1}(\xi ), \ldots , \lambda _{n}(\xi ))}^\mathsf {T}\), and \(\tilde{P}\) is the matrix whose \(i^{\text {th}}\) column is \(\tilde{v}_i\). Thus, using an arbitrary linear isometry to get a coordinate system for \(T_{x}{M}\), and letting P be the matrix whose \(i^{\text {th}}\) column is \((v_i(x) - v_0(x))\), we may rewrite Eq. (11) as
$$\begin{aligned} \left( \partial _u \nu \right) \xi = - P\lambda (\xi ) = - P\tilde{P}^{-1}\xi . \end{aligned}$$
(12)
From Eq. (12) we conclude that \(\partial _u \nu \) is full rank if and only if P is of full rank, and this is the case if and only if \(\varvec{\sigma }(x)\) is a non-degenerate Euclidean simplex, i.e., its vertices \(\{v_i(x)\}\) are affinely independent.

We observe that if \(db\) is non-degenerate on \(\varvec{\sigma }_M\), then \(b\) must be injective. Indeed, if \(x = b(u)\), then \(\{ \lambda _{i}(u) \}\), the barycentric coordinates of u with respect to \(\tilde{\varvec{\sigma }}\), are also the barycentric coordinates of the origin in \(T_{x}{M}\), with respect to the simplex \(\varvec{\sigma }(x)\). Thus if \(b(u) = x = b(\tilde{u})\), then \(\lambda _{i}(u) = \lambda _{i}(\tilde{u})\), and we must have \(\tilde{u} = u\) by the uniqueness of the barycentric coordinates.

In summary, we have

Proposition 5

A Riemannian simplex \(\varvec{\sigma }_M\subset M\) is non-degenerate if and only if \(\varvec{\sigma }(x) \subset T_xM\) is non-degenerate for every \(x \in \varvec{\sigma }_M\).

3 Non-degeneracy criteria

In this section we exploit Proposition 5 to establish geometric criteria that ensure that a Riemannian simplex is non-degenerate. In Sect. 2.3 we worked in an arbitrary coordinate chart \(\phi :M\supset W \rightarrow \mathbb {R}^n\), where the convex ball \(B_{\rho }\) containing \(\varvec{\sigma }_M\) is contained in \(W\). Now we will choose \(\phi \) to be the inverse of the exponential map at some fixed point \(p \in B_{\rho }\). Specifically, we set \(\phi = \mathbf {u}\circ \exp _p^{-1} :W \rightarrow \mathbb {R}^m\), where \(\mathbf {u}:T_pM\rightarrow \mathbb {R}^n\) is an arbitrary linear isometry that defines the \(u\)-coordinate functions in \(U = \phi (W)\). The Euclidean simplex \(\tilde{\varvec{\sigma }}\) in the coordinate domain can now be identified with \(\varvec{\sigma }(p)\), and we observe that \(\exp _x^{-1} \circ \exp _p\) maps \(\varvec{\sigma }(p)\) to \(\varvec{\sigma }(x)\).

In Sect. 3.1 we review some properties of Euclidean simplices, including thickness, the quality measure that we use, and recall a lemma that bounds the difference in thickness between two simplices whose corresponding edge lengths are almost the same. Thus given an assumed thickness of \(\varvec{\sigma }(p)\), the question of whether or not \(\varvec{\sigma }_M\) is degenerate becomes a question of how much the exponential transition function (2) distorts distances. In order to address this question, we exploit the Rauch comparison theorem, which we discuss in Sect. 3.2. We put these observations together in Sect. 3.3 to obtain explicit bounds on the required quality of \(\varvec{\sigma }(p)\), relative to its size (longest edge length) and the sectional curvatures in \(B_{\rho }\).

3.1 The stability of Euclidean simplex quality

A Euclidean simplex \(\varvec{\sigma }= \{v_0, \ldots , v_k\} \subset \mathbb {R}^n\) has a number of geometric attributes. An i-face of \(\varvec{\sigma }\) is a subset of \(i+1\) vertices, and a \((k-1)\) face of a \(k\)-simplex is a facet. The facet of \(\varvec{\sigma }\) that does not have \(v_i\) as a vertex is denoted \({\varvec{\sigma }}_{v_i}\). The altitude of \(v_i \in \varvec{\sigma }\) is the distance from \(v_i\) to the affine hull of \({\varvec{\sigma }}_{v_i}\), denoted \(a_{v_i}(\varvec{\sigma })\), and the longest edge length is denoted \(L\left( \varvec{\sigma }\right) \). When there is no risk of confusion, we will omit explicit reference to the simplex, and ignore the distinction between the vertices and their labels. Thus we write \(L\), and \(a_i\) instead of \(L\left( \varvec{\sigma }\right) \) and \(a_{v_i}(\varvec{\sigma })\).

The simplex quality measure that we will use is the thickness of a \(k\)-simplex \(\varvec{\sigma }\), defined as
$$\begin{aligned} t\left( \varvec{\sigma }\right) = {\left\{ \begin{array}{ll} 1&{} \text {if }k=0 \\ \min _{v \in \varvec{\sigma }} \dfrac{a_{v}}{k L}&{} \text {otherwise.} \end{array}\right. } \end{aligned}$$
(13)
If \(t\left( \varvec{\sigma }\right) = 0\), then \(\varvec{\sigma }\) is degenerate. We say that \(\varvec{\sigma }\) is \(t_0\)-thick, if \(t\left( \varvec{\sigma }\right) \ge t_0\). If \(\varvec{\sigma }\) is \(t_0\)-thick, then so are all of its faces. We write \(t\) for the thickness if the simplex in question is clear.

As discussed in Sect. 2.3, we can associate a matrix P to a Euclidean simplex. The quality of a simplex \(\varvec{\sigma }\) is closely related to the quality of P, which can be quantified by means of its singular values. In fact, we are only interested in the smallest and largest singular values. The smallest singular value, \(s_{k}(P) = \inf _{\left| x\right| =1}\left| Px\right| \), vanishes if and only if the matrix P does not have full rank. The largest singular value is the same as the operator norm of P, i.e., \(s_{1}(P) = \left\| P\right\| = \sup _{\left| x\right| =1}\left| Px\right| \). We have the following result [8], Lem. 2.4] relating the thickness of \(\varvec{\sigma }\) to the smallest singular value of P:

Lemma 6

(Thickness and singular value) Let \(\varvec{\sigma }= \{v_0, \ldots , v_k\}\) be a non-degenerate \(k\)-simplex in \(\mathbb {R}^n\), with \(k>0\), and let P be the \(n \times k\) matrix whose \(i\text {th}\) column is \(v_i - v_0\). Then the \(ith\) row of the pseudo-inverse \({P}^{-1}_{\text {left}} = ({P}^\mathsf {T}P)^{-1}{P}^\mathsf {T}\) is given by \({w}^\mathsf {T}_i\), where \(w_i\) is orthogonal to \(aff({\varvec{\sigma }}_{v_i})\), and
$$\begin{aligned} \left| w_i\right| = a_{i}^{-1}. \end{aligned}$$
We have the following bound on the smallest singular value of P:
$$\begin{aligned} s_{k}(P) \ge \sqrt{k} tL. \end{aligned}$$

The appearance of the dimension k in the denominator in the definition of thickness is a convention introduced so that \(t\) provides a dimension-independent bound on the condition number of P: Since the columns of P have norm bounded by \(L\), we have that \(s_{1}(P) \le \sqrt{k}L\), and thus Lemma 6 implies \(\frac{s_{1}(P)}{s_{k}(P)} \le t^{-1}\). Although we adhere to definition (13) in this work, we acknowledge that this normalisation convention may obscure the relationship between simplex quality and dimension. We frequently make use of the fact that for a k-simplex \(\varvec{\sigma }\), we have \(kt\left( \varvec{\sigma }\right) \le 1\).

The crucial property of thickness for our purposes is its stability. If two Euclidean simplicies with corresponding vertices have edge lengths that are almost the same, then their thicknesses will be almost the same. This allows us to quantify a bound on the smallest singular value of the matrix associated with one of the simplices, given a bound on the other. To be precise, we have the following consequence of the more general Lemma 25 demonstrated in Sect. 5.1:

Lemma 7

(Thickness under distortion) Suppose that \(\varvec{\sigma }= \{v_0, \ldots , v_k\}\) and \(\tilde{\varvec{\sigma }}= \{\tilde{v}_0, \ldots , \tilde{v}_k\}\) are two \(k\)-simplices in \(\mathbb {R}^n\) such that
$$\begin{aligned} \left| \left| v_i - v_j\right| - \left| \tilde{v}_i - \tilde{v}_j\right| \right| \le C_0L\left( \varvec{\sigma }\right) \end{aligned}$$
for all \(0 \le i < j \le k\). Let P be the matrix whose \(i{\text {th}}\) column is \(v_i - v_0\), and define \(\tilde{P}\) similarly.
If
$$\begin{aligned} C_0= \frac{\eta t\left( \varvec{\sigma }\right) ^2}{4} \qquad \text { with } \quad 0 \le \eta \le 1, \end{aligned}$$
then
$$\begin{aligned} s_{k}(\tilde{P}) \ge (1 - \eta ) s_{k}(P). \end{aligned}$$
and
$$\begin{aligned} t\left( \tilde{\varvec{\sigma }}\right) \ge \frac{4}{5\sqrt{k}}(1 - \eta ) t\left( \varvec{\sigma }\right) . \end{aligned}$$

3.2 The Rauch comparison theorem

The Rauch comparison theorem gives us bounds on the norm of the differential of the exponential map. This in turn implies a bound on how much the exponential map can distort distances. It is called a comparison theorem because it is implicitly comparing the exponential map on the given manifold to that on a space of constant sectional curvatures. In this context we encounter the functions
$$\begin{aligned} \mathbf {S}_{\kappa }(r) = {\left\{ \begin{array}{ll} (1/ \sqrt{\kappa })\sin \sqrt{\kappa }r &{}\qquad \kappa >0\\ r &{}\qquad \kappa =0\\ (1/ \sqrt{-\kappa })\sinh \sqrt{-\kappa }r &{}\qquad \kappa <0, \end{array}\right. } \end{aligned}$$
parameterised by \(\kappa \), which can be thought of as representing a constant sectional curvature.

The Rauch theorem can be found in Buser and Karcher [11], Section 6.4] or in Chavel [14], Thm. IX.2.3], for example. In the statement of the theorem we implicitly use the identification between the tangent spaces of a tangent space and the tangent space itself.

Lemma 8

(Rauch theorem) Radially the exponential map \(\exp _p :T_{p}{M} \rightarrow M\) is an isometry:
$$\begin{aligned} \left| (d\exp _p)_{v} v\right| = \left| v\right| . \end{aligned}$$
Assume the sectional curvatures, \(K\), are bounded by \(\varLambda _{-}\le K\le \varLambda _{+}\). Taking \(\left| v\right| =1\), one has for any \(w\) perpendicular to \(v\)
$$\begin{aligned} \frac{\mathbf {S}_{\varLambda _{+}}(r)}{r} \left| w\right| \le \left| (d\exp _p)_{rv}w\right| \le \frac{\mathbf {S}_{\varLambda _{-}}(r)}{r} \left| w\right| . \end{aligned}$$
The inequalities hold when \(r < 2\rho _0\) (defined in Eq. (3)). Also, if \(\varLambda _{-}< 0\), then the right inequality is valid for all r, and if \(\varLambda _{+}> 0\), then the left inequality is valid for all r.
For convenience, we will use a bound on the absolute value of the sectional curvatures, rather than separate upper and lower bounds. Thus \(\left| K\right| \le \varLambda \), where \(\varLambda = \max \{ \varLambda _{+}, - \varLambda _{-}\}\). We use Taylor’s theorem to obtain
$$\begin{aligned} \mathbf {S}_{-\varLambda }(r)&\le r + \frac{\varLambda r^3}{2} \qquad \text {when } 0 \le r < \frac{\pi }{2 \sqrt{\varLambda }}\\ \mathbf {S}_{\varLambda }(r)&\ge r - \frac{\varLambda r^3}{6} \qquad \text {for all } r \ge 0. \end{aligned}$$
Since \(d\exp _p\) preserves the lengths of any radial vector \(v\), the bound on the distortion of the length of a vector orthogonal to \(v\) implies a bound for all vectors. We can restate the Rauch theorem in a weaker, but more convenient form:

Lemma 9

Suppose the sectional curvatures in \(M\) are bounded by \(\left| K\right| \le \varLambda \). If \(v \in T_pM\) satisfies \(\left| v\right| = r < \frac{\pi }{2\sqrt{\varLambda }}\), then for any vector \(w \in T_v(T_pM) \cong T_pM\), we have
$$\begin{aligned} \left( 1 - \frac{\varLambda r^2}{6} \right) \left| w\right| \le \left| (d\exp _p)_{v}w\right| \le \left( 1 + \frac{\varLambda r^2}{2} \right) \left| w\right| . \end{aligned}$$

3.3 Non-degenerate Riemannian simplices

Our goal now is to estimate the metric distortion incurred when we map a simplex from one tangent space to another via the exponential maps
$$\begin{aligned} \exp _x^{-1} \circ \exp _{p} :T_{p}M\rightarrow T_xM, \end{aligned}$$
and this is accomplished by the bounds on the differential. Specifically, if \(F:\mathbb {R}^n \rightarrow \mathbb {R}^n\) satisfies \(\left\| dF\right\| \le \eta \), then the length of the image of the line segment between x and \(y\) provides an upper bound on the distance between \(F(x)\) and \(F(y)\):
$$\begin{aligned} \left| F(y) - F(x)\right| \le \int _0^1 \left| dF_{x +s(y-x)}(y-x)\right| \, ds \le \eta \left| y-x\right| . \end{aligned}$$
(14)
If \(x,p, y \in B_{\rho }\), with \(y = \exp _{p}(v)\), then \(\left| v\right| < 2\rho \), and \(\left| \exp ^{-1}_x(y)\right| < 2\rho \). Then, if \(\rho < \rho _0\) given in Eq. (3), Lemma 9 tells us that
$$\begin{aligned} \left\| d\left( \exp _x^{-1} \circ \exp _{p} \right) _{v}\right\|\le & {} \left\| \left( d\exp _x^{-1}\right) _y\right\| \left\| \left( d\exp _{p} \right) _{v}\right\| \le \left( 1 + \frac{\varLambda (2\rho )^2}{3} \right) \left( 1 + \frac{\varLambda (2\rho )^2}{2} \right) \\\le & {} 1 + 5\varLambda \rho ^2. \end{aligned}$$
Therefore (14) yields
$$\begin{aligned} \left| v_i(x) - v_j(x)\right| \le (1 + 5\varLambda \rho ^2) \left| v_i(p) - v_j(p)\right| . \end{aligned}$$
We can do the same argument the other way, so
$$\begin{aligned} \left| v_i(p) - v_j(p)\right| \le (1 + 5\varLambda \rho ^2) \left| v_i(x) - v_j(x)\right| , \end{aligned}$$
and we find
$$\begin{aligned} \begin{aligned} \big | \left| v_i(x) - v_j(x)\right| - \left| v_i(p) - v_j(p)\right| \big |&\le 5\varLambda \rho ^2(1 + 5\varLambda \rho ^2) \left| v_i(p) - v_j(p)\right| \\&\le 21\varLambda \rho ^2 \left| v_i(p) - v_j(p)\right| \qquad \text {when } \rho < \rho _0. \end{aligned} \end{aligned}$$
(15)
Letting P be the matrix associated with \(\varvec{\sigma }(p)\), and using \(C_0= 21\varLambda \rho ^2\), in Lemma 7, we find that the matrix \(\tilde{P}\) associated with \(\varvec{\sigma }(x)\) in Proposition 5 is non-degenerate if \(\varvec{\sigma }(p)\) satisfies a thickness bound of \(t_0> 10 \sqrt{\varLambda }\rho \), and we have

Theorem 1

(Non-degeneracy criteria) Suppose \(M\) is a Riemannian manifold with sectional curvatures \(K\) bounded by \(\left| K\right| \le \varLambda \), and \(\varvec{\sigma }_M\) is a Riemannian simplex, with \(\varvec{\sigma }_M\subset B_{\rho }\subset M\), where \(B_{\rho }\) is an open geodesic ball of radius \(\rho \) with
$$\begin{aligned} \rho < \rho _0= \min \left\{ \frac{\iota _{M}}{2}, \frac{\pi }{4\sqrt{\varLambda }} \right\} . \end{aligned}$$
Then \(\varvec{\sigma }_M\) is non-degenerate if there is a point \(p \in B_{\rho }\) such that the lifted Euclidean simplex \(\varvec{\sigma }(p)\) has thickness satisfying
$$\begin{aligned} t\left( \varvec{\sigma }(p)\right) > 10 \sqrt{\varLambda }\rho . \end{aligned}$$
The ball \(B_{\rho }\) may be chosen so that this inequality is necessarily satisfied if
$$\begin{aligned} t\left( \varvec{\sigma }(p)\right) > 10 \sqrt{\varLambda }L\left( \varvec{\sigma }_M\right) , \end{aligned}$$
where \(L\left( \varvec{\sigma }_M\right) \) is the geodesic length of the longest edge in \(\varvec{\sigma }_M\).

The last assertion follows from the remark at the end of Sect. 2.2: If \(L\left( \varvec{\sigma }_M\right) < \rho _0\), then \(\varvec{\sigma }_M\) is contained in a closed ball of radius \(L\left( \varvec{\sigma }_M\right) \) centred at one of the vertices.

Remark 10

Using Proposition 17 and Lemma 11 of Sect. 4, we can replace Eq. (15) with
$$\begin{aligned} \big | \left| v_i(x) - v_j(x)\right| - \left| v_i(p) - v_j(p)\right| \big | \le 6\varLambda \rho ^2 \left| v_i(p) - v_j(p)\right| \qquad \text {when } \rho < \frac{1}{2}\rho _0, \end{aligned}$$
and we find, that the Riemannian simplex \(\varvec{\sigma }_M\) of Theorem 1 is non-degenerate if
$$\begin{aligned} t\left( \varvec{\sigma }(p)\right) > 5 \sqrt{\varLambda }\rho , \end{aligned}$$
but with the caveat that \(\rho \) must now satisfy \(\rho \le \frac{1}{2}\rho _0\).

Orientation In Euclidean space \(\mathbb {E}^n\) we can define an orientation as an equivalence class of frames, two frames being equivalent if the linear transformation between them has positive determinant. We can likewise associate an orientation to a (non-degenerate) Euclidean n-simplex \(\varvec{\sigma }= \{v_0, \ldots , v_n\}\): it is the orientation associated with the basis \(\{(v_i - v_0)\}_{i \in \{1,\ldots ,n\}}\). The orientation depends on how we have indexed the points. Any even permutation of the indices yields the same orientation.

In a manifold, we can assign an orientation locally, in a neighbourhood \(U \subset M\) on which the tangent bundle admits a local trivialisation, for example. Then we can define an orientation by defining an orientation on \(T_{p}{M}\) for some \(p \in U\). If \(\varvec{\sigma }= \{p_0, \ldots , p_n\} \subset U\) defines a non-degenerate Riemannian simplex, then we can associate an orientation to that simplex: it is the orientation of \(\varvec{\sigma }(p_0) \subset T_{p_0}{M}\). Again, we will get agreement on the orientation if we perform any even permutation of the vertex indices. The reason is that our non-degeneracy assumption implies that the orientation of \(\varvec{\sigma }(p_i)\) will agree with the orientation of \(\varvec{\sigma }(p_j)\) for any \(i,j \in \{0, \ldots , n\}\).

In the particular case discussed in this section, where \(\phi = u \circ \exp _{p}^{-1}\), for \(p \in B_{\rho }\), the barycentric map \(b:\varvec{\sigma }_{\mathbb {E}}(p) \rightarrow \varvec{\sigma }_M\) is orientation preserving. Since \(\exp _x^{-1}\circ \exp _y\) is orientation preserving for any \(x,y \in B_{\rho }\), it is enough to consider the case where \(p=p_0 \in \varvec{\sigma }\). Consider Eq. (10):
$$\begin{aligned} (db)_{v_0(p_0)} = - \left( \nabla \nu \right) ^{-1} \partial _u \nu _{v_0(p_0)}. \end{aligned}$$
By Eq. (12), we have \((\partial _u \nu )_{v_0(p_0)} = - {{\mathrm{Id}}}\). Also, it follows from Lemma 3 that \(\nabla \nu \) has positive determinant. (Buser and Karcher [11], p. 132] show that \(\nabla \nu \) is bounded near the identity), and thus so must db everywhere, since it does not vanish on its domain.

4 Triangulation criteria

We are interested in the following scenario. Suppose we have a finite set of points \(S\subset M\) in a compact Riemannian manifold, and an (abstract) simplicial complex \(\fancyscript{A}\) whose vertex set is \(S\), and such that every simplex in \(\fancyscript{A}\) defines a non-degenerate Riemannian simplex. When can we be sure that \(\fancyscript{A}\) triangulates \(M\)? Consider a convex ball \(B_{\rho }\) centred at \(p \in S\). We require that, when lifted to \(T_pM\), the simplices near p triangulate a neighbourhood of the origin. If we require that the simplices be small relative to \(\rho \), and triangulate a region extending to near the boundary of the lifted ball, then Riemannian simplices outside of \(B_{\rho }\) cannot have points in common with the simplices near the centre of the ball, and it is relatively easy to establish a triangulation.

Instead, we aim for finer local control of the geometry. We establish geometric conditions (Lemma 14) that ensure that the complex consisting of simplices incident to p, (i.e., the star of p) is embedded by a given map into the manifold. In order to achieve this result we require a strong constraint on the differential of the map in question. Since we work locally, in a coordinate chart, we consider maps \(F:\mathbb {R}^n \supseteq U \rightarrow \mathbb {R}^n\). We demand that for some linear isometry \(T :\mathbb {R}^n \rightarrow \mathbb {R}^n\) we have
$$\begin{aligned} \left\| dF_u - T\right\| \le \eta , \end{aligned}$$
(16)
for some \(0 \le \eta \le 1\), and all \(u \in U\). This is stronger than the kind of bounds found, for example, in the Rauch theorem (Lemma 9), which have the form
$$\begin{aligned} (1-\eta )\left| w\right| \le \left| dF_uw\right| \le (1 + \eta )\left| w\right| . \end{aligned}$$
(17)
Whereas (17) implies that \(dF_u\) is close to a linear isometry at every \(u\in U\), Eq. (16) means that \(dF_u\) is close to the same linear isometry for all \(u \in U\).

Using a local constraint of the form (16) to establish the embedding of vertex stars, we demonstrate, in Sect. 4.1, generic criteria which ensure that a map from a simplicial complex to a Riemannian manifold is a homeomorphism. We then turn our attention to the specific case where the map in question is the barycentric coordinate map on each simplex.

Using a refinement of the Rauch theorem established by Buser and Karcher [11] we show, in Sect. 4.2, that transition functions arising from the exponential map (Eq. (2)) are subject to bounds of the form (16). Then in Sect. 4.3 we observe that the barycentric coordinate map is also subjected to such bounds, and thus yield Theorem 2 as a particular case of the generic triangulation criteria.

4.1 Generic triangulation criteria

We say that a map \(F :\mathbb {R}^m\rightarrow \mathbb {R}^n\) is smooth if it is of class \(C^\infty \). If \(A \subset \mathbb {R}^m\), then \(F :A \rightarrow \mathbb {R}^n\) is smooth on \(A\) if F can be extended to a function that is smooth in an open neighbourhood of A, i.e., there exists an open neighbourhood \(U \in \mathbb {R}^m\) and a smooth map \(\tilde{F} :U \rightarrow \mathbb {R}^n\) such that \(A \subseteq U\), and \(\tilde{F}|_A = F\). This definition is independent of the ambient space \(\mathbb {R}^m\) containing \(A\). In particular, if \(A \subseteq \mathbb {R}^k \subseteq \mathbb {R}^m\), then the smoothness of \(F\) does not depend on whether we consider A to be a subset of \(\mathbb {R}^k\) or of \(\mathbb {R}^m\). In the case that A is the closure of a non-empty open set, continuity of the partial derivatives implies that they are well defined on all of A and independent of the chosen extension. See Munkres [30], Section 1] for more details.

For our purposes, we are interested in smooth maps from non-degenerate closed Euclidean simplices of dimension n into an n-dimensional manifold \(M\). We will work within coordinate charts, so our primary focus will be on maps of the form
$$\begin{aligned} F :\varvec{\sigma }_{\mathbb {E}}^n \rightarrow \mathbb {R}^n, \end{aligned}$$
such that Eq. (16) is satisfied for all \(u \in \varvec{\sigma }_{\mathbb {E}}^n\). As an example of how we can exploit this bound, we observe that a map satisfying Eq. (16) is necessarily an embedding with bounded metric distortion if its domain is convex:

Lemma 11

Suppose \(A \subset \mathbb {R}^n\) is convex, and \(F :A \rightarrow \mathbb {R}^n\) is a smooth map such that, for some non-negative \(\eta <1\),
$$\begin{aligned} \left\| dF_u - T\right\| \le \eta , \end{aligned}$$
for all \(u \in A\), and some linear isometry \(T :\mathbb {R}^n\rightarrow \mathbb {R}^n\). Then
$$\begin{aligned} \left| \, \left| F(u) - F(v)\right| - \left| u - v\right| \,\right| \le \eta \left| u - v\right| \quad \text {for all } u,v \in A. \end{aligned}$$

Proof

We observe that it is sufficient to consider the case \(T = {{\mathrm{Id}}}\), because if \(\tilde{F} = T^{-1} \circ F\), then \(\left\| d\tilde{F} - {{\mathrm{Id}}}\right\| = \left\| dF - T\right\| \), and \(\left| \tilde{F}(u) - \tilde{F}(v)\right| = \left| F(u) - F(v)\right| \).

Assume \(u\ne v\). For the lower bound we consider the unit vector \(\hat{u} = \frac{u-v}{\left| u-v\right| }\), and observe that
$$\begin{aligned} dF(u-v)\varvec{\cdot }\hat{u} \ge (1-\eta )\left| u-v\right| > 0, \end{aligned}$$
so, by integrating along the segment \([u,v]\), we find
$$\begin{aligned} \left| F(u)- F(v)\right| \ge \big (F(u) - F(v) \big )\varvec{\cdot }\hat{u} \ge (1-\eta ) \left| u-v\right| . \end{aligned}$$
For the upper bound we employ the unit vector \(\hat{w} = \frac{F(u) - F(v)}{\left| F(u) - F(v)\right| }\):
$$\begin{aligned} \big ( F(u)-F(v) \big )\varvec{\cdot }\hat{w} = \left| F(u) - F(v)\right| \le (1 + \eta ) (u-v)\varvec{\cdot }\hat{w} \le (1 + \eta ) \left| u-v\right| . \end{aligned}$$
\(\square \)

For our purposes we will be free to choose a coordinate system so that \(F\) keeps a vertex fixed. We will have use for the following observation, which can be demonstrated with an argument similar to the proof of Lemma 11.

Lemma 12

If \(A \subseteq \mathbb {R}^n\) is a convex set and \(F :A \rightarrow \mathbb {R}^n\) is a smooth map with a fixed point \(p \in A\) and
$$\begin{aligned} \left\| dF_u - {{\mathrm{Id}}}\right\| \le \eta , \quad \text {for all } u \in A, \end{aligned}$$
then
$$\begin{aligned} \left| F(u) - u \right| \le \eta {\left| u-p\right| } \quad \text {for all } u \in A. \end{aligned}$$

Embedding complexes In preparation for considering triangulations we first consider the problem of mappings of complexes into \(\mathbb {R}^n\).

A simplicial complex \(\fancyscript{C}\) is a set of abstract simplices such that if \(\varvec{\sigma }\in \fancyscript{C}\), then \(\tau \in \fancyscript{C}\) for every face \(\tau \subset \varvec{\sigma }\). We will only consider finite simplicial complexes. A subcomplex of \(\fancyscript{C}\) is a subset that is also a simplicial complex. The star of a simplex \(\varvec{\sigma }\in \fancyscript{C}\) is the smallest subcomplex of \(\fancyscript{C}\) consisting of all simplices that have \(\varvec{\sigma }\) as a face, and is denoted \(\underline{{{\mathrm{St}}}}(\varvec{\sigma })\). In particular, if p is a vertex of \(\fancyscript{C}\), then \(\underline{{{\mathrm{St}}}}(p)\) is the set of simplices that contain p, together with the faces of these simplices.

The carrier (“geometric realisation”) of \(\fancyscript{C}\) is denoted \(\left| \fancyscript{C}\right| \). We are interested in complexes endowed with a piecewise flat metric. This is a metric on \(\left| \fancyscript{C}\right| \) that can be realised by assigning lengths to the edges in \(\fancyscript{C}\) such that each simplex \(\varvec{\sigma }\in \fancyscript{C}\) is associated with a Euclidean simplex \(\varvec{\sigma }_{\mathbb {E}}\subset \left| \fancyscript{C}\right| \) that has the prescribed edge lengths. Certain constraints on the edge lengths must be met in order to define a valid piecewise flat metric, but for our current purposes we will have a metric inherited from an embedding in Euclidean space.

We say that \(\fancyscript{C}\) is embedded in \(\mathbb {R}^n\) if the vertices lie in \(\mathbb {R}^n\) and the convex hulls of any two simplices in \(\fancyscript{C}\) either do not interesect, or their intersection is the convex hull of a simplex in \(\fancyscript{C}\). In other words, to each \(\varvec{\sigma },\tau \in \fancyscript{C}\) we associate \(\varvec{\sigma }_{\mathbb {E}}= {{\mathrm{conv}}}(\varvec{\sigma }), \varvec{\tau }_{\mathbb {E}}= {{\mathrm{conv}}}(\tau )\), and we have \(\varvec{\sigma }_{\mathbb {E}}\cap \varvec{\tau }_{\mathbb {E}}= {{\mathrm{conv}}}(\varvec{\sigma }\cap \tau )\). The topological boundary of a set \(B \subset \mathbb {R}^n\) is denoted by \(\partial {B}\), and its topological interior by \({{\mathrm{int}}}(B)\). If \(\fancyscript{C}\) is embedded in \(\mathbb {R}^n\), and p is a vertex of \(\fancyscript{C}\), we say that \(\underline{{{\mathrm{St}}}}(p)\) is a full star if \(p \in {{\mathrm{int}}}(\left| \underline{{{\mathrm{St}}}}(p)\right| )\).

The scale of \(\fancyscript{C}\) is an upper bound on the length of the longest edge in \(\fancyscript{C}\), and is denoted by \(h\). We say that \(\fancyscript{C}\) is \(t_0\)-thick if each simplex in \(\fancyscript{C}\) has thickness greater than \(t_0\). The dimension of \(\fancyscript{C}\) is the largest dimension of the simplices in \(\fancyscript{C}\). We call a complex of dimension n an n-complex. If every simplex in \(\fancyscript{C}\) is the face of an n-simplex, then \(\fancyscript{C}\) is a puren-complex.

A map \(F:\left| \fancyscript{C}\right| \rightarrow \mathbb {R}^n\) is smooth on\(\fancyscript{C}\) if for each \(\varvec{\sigma }\in \fancyscript{C}\) the restriction \(F \big |_{\varvec{\sigma }_{\mathbb {E}}}\) is smooth. This means that \(d(F\big |_{\varvec{\sigma }_{\mathbb {E}}})\) is well defined, and even though \(dF\) is not well defined, we will use this symbol when the particular restriction employed is either evident or unimportant. When the underlying complex on which \(F\) is smooth is unimportant, we simply say that \(F\) is piecewise smooth.

F is piecewise linear if its restriction to each simplex is an affine map. The secant map of \(F\) is the piecewise linear map defined by the restriction of \(F\) to the vertices of \(\fancyscript{C}\).

We are interested in conditions that ensure that \(F :\left| \fancyscript{C}\right| \rightarrow \mathbb {R}^n\) is a topological embedding. Our primary concern is with the behaviour of the boundary. The reason for this is captured by the following variation of a lemma by Whitney [37], Lem AII.15a]:

Lemma 13

(Whitney) Let \(\fancyscript{C}\) be a (finite) simplicial complex embedded in \(\mathbb {R}^n\) such that \({{\mathrm{int}}}(\left| \fancyscript{C}\right| )\) is non-empty and connected, and \(\partial {\left| \fancyscript{C}\right| }\) is a compact \((n-1)\)-manifold. Suppose \(F :\left| \fancyscript{C}\right| \rightarrow \mathbb {R}^n\) is smooth on \(\fancyscript{C}\) and such that \(\det d(F\big |_{\varvec{\sigma }_{\mathbb {E}}}) > 0\) for each n-simplex \(\varvec{\sigma }\in \fancyscript{C}\). If the restriction of \(F\) to \(\partial {\left| \fancyscript{C}\right| }\) is an embedding, then \(F\) is a topological embedding.

Proof

The assumptions on \({{\mathrm{int}}}(\left| \fancyscript{C}\right| )\) and \(\partial {\left| \fancyscript{C}\right| }\) imply that \(\fancyscript{C}\) is a pure n-complex, and that each \((n-1)\) simplex is either a boundary simplex, or the face of exactly two n-simplices. Whitney showed [37], Lem AII.15a] that any \(x \in {{\mathrm{int}}}(\left| \fancyscript{C}\right| )\) admits an open neighbourhood \(U \subset {{\mathrm{int}}}(\left| \fancyscript{C}\right| )\) such that the restriction of \(F\) to \(U\) is a homeomorphism. In particular, \(F({{\mathrm{int}}}(\left| \fancyscript{C}\right| ))\) is open.

By the Jordan-Brouwer separation theorem [31], Section IV.7], \(\mathbb {R}^n \setminus F(\partial {\left| \fancyscript{C}\right| })\) consists of two open components, one of which is bounded. Since \(F(\left| \fancyscript{C}\right| )\) is compact, \(F({{\mathrm{int}}}(\left| \fancyscript{C}\right| ))\) must coincide with the bounded component, and in particular \(F({{\mathrm{int}}}(\left| \fancyscript{C}\right| )) \cap F(\partial {\left| \fancyscript{C}\right| }) = \emptyset \), so \(F({{\mathrm{int}}}(\left| \fancyscript{C}\right| ))\) is a single connected component.

We need to show that \(F\) is injective. First we observe that the set of points in \(F({{\mathrm{int}}}(\left| \fancyscript{C}\right| ))\) that have exaclty one point in the preimage is non-empty. It suffices to look in a neigbhourhood of a point \(y \in F(\partial {\left| \fancyscript{C}\right| })\). Choose \(y = F(x)\), where x is in the relative interior of \(\varvec{\sigma }_{\mathbb {E}}^{n-1} \subset \partial {\left| \fancyscript{C}\right| }\). Then there is a neighbourhood \(V\) of \(y\) such that \(V\) does not intersect the image of any other simplex of dimension less than or equal to \(n-1\). Let \(\varvec{\sigma }^n\) be the unique n-simplex that has \(\varvec{\sigma }^{n-1}\) as a face. Then \(F^{-1}(V \cap F(\left| \fancyscript{C}\right| ) \subset \varvec{\sigma }_{\mathbb {E}}^n\), and it follows that every point in \(V \cap {{\mathrm{int}}}(\left| \fancyscript{C}\right| )\) has a unique point in its image.

Now the injectivity of \(F\) follows from the fact that the number of points in the preimage is locally constant on \(F({{\mathrm{int}}}(\left| \fancyscript{C}\right| )) \setminus F(\partial {\left| \fancyscript{C}\right| })\), which in our case is connected. This is a standard argument in degree theory [31], Prop. IV.1.2]: A point \(a \in F({{\mathrm{int}}}(\left| \fancyscript{C}\right| )) \setminus F(\partial {\left| \fancyscript{C}\right| })\) has \(k\) points \(\{x_1, \ldots , x_k\}\) in its preimage. There is a neighbourood \(V\) of \(a\) and disjoint neighbourhoods \(U_i\) of \(x_i\) such that \(F|_{U_i} :U_i \rightarrow V\) is a homeomorphism for each \(i \in \{1,\ldots ,k\}\). It follows that the number of points in the preimage is \(k\) for every point in the open neighbourhood of \(a\) defined as
$$\begin{aligned} W = V \setminus F\left( \left| \fancyscript{C}\right| \setminus \bigcup _i U_i\right) . \end{aligned}$$
\(\square \)

Lemma 14

(Embedding a star) Suppose \(\fancyscript{C}= \underline{{{\mathrm{St}}}}(p)\) is a \(t_0\)-thick, pure n-complex embedded in \(\mathbb {R}^n\) such that all of the n-simplices are incident to a single vertex, p, and \(p \in {{\mathrm{int}}}(\left| \fancyscript{C}\right| )\) (i.e., \(\underline{{{\mathrm{St}}}}(p)\) is a full star). If \(F :\left| \fancyscript{C}\right| \rightarrow \mathbb {R}^n\) is smooth on \(\fancyscript{C}\), and satisfies
$$\begin{aligned} \left\| dF - {{\mathrm{Id}}}\right\| < nt_0\end{aligned}$$
(18)
on each n-simplex of \(\fancyscript{C}\), then \(F\) is an embedding.

Proof

If \(\left| \fancyscript{C}\right| \) is convex, then the claim follows immediately from Lemma 11.

From the definition of thickness, we observe that \(nt_0\le 1\), and therefore \(\left\| dF\right\| > 0\). By Lemma 13, it suffices to consider points \(x,y \in \partial {\left| \fancyscript{C}\right| }\). Rather than integrating the differential of a direction, as we did implicitly in the proof of Lemma 11, we will integrate the differential of an angle.

Let Q be the 2-dimensional plane defined by \(p, x\), and \(y\). We define the angle function \(\phi :\mathbb {R}^n\rightarrow \mathbb {R}\) as follows: For \(z \in \mathbb {R}^n\), let \(\check{z}\) be the orthogonal projection of \(z\) into \(Q\). Then \(\phi (z)\) is the angle that \(\check{z}-p\) makes with \(x-p\), where the orientation is chosen so that \(\phi (y) < \pi \) (We can assume that x and \(y\) are not colinear with p, since in that case \([x,y]\) must be contained in \(\left| \fancyscript{C}\right| \), and the arguments of Lemma 11 ensure that we would not have \(F(x)=F(y)\)).

Let \(\alpha \) be the piecewise linear curve obtained by projecting the segment \([x,y] \subset Q\) onto \(\partial {\left| \fancyscript{C}\right| }\) via the radial rays emanating from p. Parameterise \(\alpha \) by the angle \(\phi \), i.e., by the arc between \(\frac{x-p}{\left| x-p\right| }\) and \(\frac{y-p}{\left| y-p\right| }\) on the unit circle in \(Q\). Then \(d\phi (\alpha ') = 1\), and we have
$$\begin{aligned} \phi (F(y)) - \phi (F(x))&= \int _{F(\alpha )}d\phi = \int _0^{\phi (y)} d\phi (dF\alpha '(s)) \, ds. \end{aligned}$$
We will show that \(d\phi (dF\alpha '(s)) > 0\); it follows that \(\phi (F(y)) > \phi (F(x))\) and hence \(F(y) \ne F(x)\).
We need an observation about thick simplices: Suppose that p is a vertex of a \(t_0\)-thick n-simplex \(\varvec{\sigma }_{\mathbb {E}}\), and \(\varvec{\tau }_{\mathbb {E}}\) is the facet opposite p. Then a line r through p and \(\varvec{\tau }_{\mathbb {E}}\) makes an angle \(\theta \) with \(\varvec{\tau }_{\mathbb {E}}\) that is bounded by
$$\begin{aligned} \sin \theta \ge nt_0. \end{aligned}$$
Indeed, the altitude of p satisfies \(a_{p} \ge nt_0L\) by the definition of thickness, and the distance between p and the point of intersection of r with \(\varvec{\tau }_{\mathbb {E}}\) is less than \(L\).
If \(\alpha (s)\) is the point of intersection of r with \(\varvec{\tau }_{\mathbb {E}}\), we observe that \(d\phi (\alpha '(s)) = \left| \alpha '(s)\right| \sin \theta \), i.e., the magnitude of the component of \(\alpha '(s)\) orthogonal to r. By the hypothesis (18), the angle \(\beta \) between \(dF\alpha '(s)\) and \(\alpha '(s)\) satisfies \(\sin \beta < nt_0\). Therefore
$$\begin{aligned} d\phi (dF\alpha '(s))&\ge \left| dF\alpha '(s)\right| (\sin \theta - \sin \beta )\\&> 0. \end{aligned}$$
\(\square \)

Triangulations For our purposes a manifold is always compact and without boundary. A simplicial complex \(\fancyscript{A}\) is a manifold simplicial complex if \(\left| \fancyscript{A}\right| \) is a topological manifold. A triangulation of a manifold \(M\) is a homeomorphism \(H :\left| \fancyscript{A}\right| \rightarrow M\), where \(\fancyscript{A}\) is a simplicial complex. If \(M\) is a differentiable manifold, then \(H\) is a smooth triangulation if it is smooth on \(\fancyscript{A}\), i.e., the restriction of \(H\) to any simplex in \(\fancyscript{A}\) is smooth. We are concerned with smooth triangulations of compact Riemannian manifolds.

Our homeomorphism argument is based on the following observation:

Lemma 15

Let \(\fancyscript{A}\) be a manifold simplicial complex of dimension n with finite vertex set \(S\), and let \(M\) be a compact n-manifold. Suppose \(H :\left| \fancyscript{A}\right| \rightarrow M\) is such that for each \(p \in S, H \big |_{\left| \underline{{{\mathrm{St}}}}(p)\right| } :\left| \underline{{{\mathrm{St}}}}(p)\right| \rightarrow M\) is an embedding. If for each connected component \(M_i\) of \(M\) there is a point \(y \in M_i\) such that \(h^{-1}(y)\) contains exactly one point in \(\left| \fancyscript{A}\right| \), then \(H\) is a homeomorphism.

Proof

The requirement that the star of each vertex be embedded means that \(H\) is locally a homeomorphism, so it suffices to observe that it is bijective. It is surjective by Brouwer’s invariance of domain; thus \(H\) is a covering map. The requirement that each component of \(M\) has a point with a single point in its pre-image implies that \(H :\left| \fancyscript{A}\right| \rightarrow M\) is a single-sheeted covering, and therefore a homeomorphism. \(\square \)

The following proposition generically models the situation we will work with when we describe a triangulation by Riemannian simplices:

Proposition 16

(Triangulation) Let \(\fancyscript{A}\) be a manifold simplicial n-complex with finite vertex set \(S\), and \(M\) a compact Riemannian manifold with an atlas \(\{(W_p,\phi _p)\}_{p \in S}\) indexed by \(S\). Suppose
$$\begin{aligned} H :\left| \fancyscript{A}\right| \rightarrow M\end{aligned}$$
satisfies:
  1. 1.

    For each \(p \in S\) the secant map of \(\phi _p \circ H\) restricted to \(\left| \underline{{{\mathrm{St}}}}(p)\right| \) is a piecewise linear embedding \(\fancyscript{L}_p :\left| \underline{{{\mathrm{St}}}}(p)\right| \rightarrow \mathbb {R}^n\) such that each simplex \(\varvec{\sigma }\in \fancyscript{C}_p = \fancyscript{L}_p(\underline{{{\mathrm{St}}}}(p))\) is \(t_0\)-thick, and \(\left| \fancyscript{C}_p\right| \subset B_{\mathbb {R}^n}(\fancyscript{L}_p(p);h)\), with \(\fancyscript{L}_p(p) \in {{\mathrm{int}}}(\left| \fancyscript{C}_p\right| )\). The scale parameter \(h\) must satisfy \(h < \frac{\iota _{M}}{4}\), where \(\iota _{M}\) is the injectivity radius of \(M\).

     
  2. 2.

    For each \(p \in S, \phi _p :W_p \mathop {\longrightarrow }\limits ^{\cong } U_p \subset \mathbb {R}^n\) is such that \(\overline{B} = \overline{B}_{\mathbb {R}^n}(\fancyscript{L}_p(p);\frac{3}{2}h) \subseteq U_p\), and \(\left\| (d\phi _p^{-1})_u\right\| \le \frac{4}{3}\), for every \(u \in \overline{B}\).

     
  3. 3.
    The map
    $$\begin{aligned} F_p = \phi _p \circ H \circ \fancyscript{L}_p^{-1} :\left| \fancyscript{C}_p\right| \rightarrow \mathbb {R}^n\end{aligned}$$
    satisfies
    $$\begin{aligned} \left\| (dF_p)_u-{{\mathrm{Id}}}\right\| \le \frac{nt_0}{2} \end{aligned}$$
    on each n-simplex \(\varvec{\sigma }\in \fancyscript{C}_p\), and every \(u \in \varvec{\sigma }_{\mathbb {E}}\).
     
Then H is a smooth triangulation of \(M\).

Proof

By Lemma 14, \(F_p\) is a homeomorphism onto its image. It follows then that \(H\big |_{\left| \underline{{{\mathrm{St}}}}(p)\right| }\) is an embedding for every \(p\in S\). Therefore, since \(\left| \fancyscript{A}\right| \) is compact, \(H :\left| \fancyscript{A}\right| \rightarrow M\) is a covering map.

Given \(x \in \left| \fancyscript{A}\right| \), with \(x \in \varvec{\sigma }_{\mathbb {E}}\), and p a vertex of \(\varvec{\sigma }_{\mathbb {E}}\), let \(\tilde{x} = \fancyscript{L}_p(x) \in \left| \fancyscript{C}_p\right| \). Then the bound on \(dF\) implies that \(\left| F_p(\tilde{x}) - \fancyscript{L}_p(p)\right| \le \left( 1 + \frac{nt_0}{2} \right) h \le \frac{3}{2}h\), so \(F_p(\tilde{x}) \in \overline{B}\). Since \(\phi _p^{-1} \circ F_p(\tilde{x}) = H(x)\), and
$$\begin{aligned} \left| \left( d\phi _p^{-1}\right) _{F(u)}(dF_p)_u\right| \le \frac{4}{3} \left( 1 + \frac{nt_0}{2} \right) \le 2 \end{aligned}$$
for any \(u \in \varvec{\sigma }_{\mathbb {E}}\subset \left| \fancyscript{C}_p\right| \), we have that \(d_{M}(H(p),H(x)) \le 2h\).

Suppose \(y \in \left| \fancyscript{A}\right| \) with \(H(y) = H(x)\). Let \(\tau \in \fancyscript{A}\) with \(y \in \varvec{\tau }_{\mathbb {E}}\), and \(q \in \tau \) a vertex. Then \(d_{M}(H(p),H(q)) \le 4h < \iota _{M}\). Thus there is a path \(\gamma \) from \(H(x)\) to \(H(p)\) to \(H(q)\) to \(H(y) = H(x)\) that is contained in the topological ball \(B_{M}(H(p);\iota _{M})\), and is therefore null-homotopic. Since \(H\) is a covering map, this implies that \(x = y\). Thus \(H\) is injective, and therefore defines a smooth triangulation. \(\square \)

4.2 The differential of exponential transitions

If there is a unique minimising geodesic from x to \(y\), we denote the parallel translation along this geodesic by \(T_{yx}\). As a preliminary step towards exploiting Proposition 16 in the context of Riemannian simplices, we show here that the estimates of Buser and Karcher [11], Section 6] imply

Proposition 17

(Strong exponential transition bound) Suppose the sectional curvatures on \(M\) satisfy \(\left| K\right| \le \varLambda \). Let \(v \in T_pM\), with \(y = \exp _p(v)\). If \(x,y \in B_{M}(p;\rho )\), with
$$\begin{aligned} \rho < \frac{1}{2}\rho _0 = \frac{1}{2}\min \left\{ \frac{\iota _{M}}{2}, \frac{\pi }{4\sqrt{\varLambda }} \right\} , \end{aligned}$$
then
$$\begin{aligned} \left\| d\left( \exp ^{-1}_x \circ \exp _p\right) _v - T_{xp}\right\| \le 6\varLambda \rho ^2. \end{aligned}$$

The primary technical result that we use in the demonstration of Proposition 17 is a refinement of the Rauch theorem demonstrated by Buser and Karcher [11], Section 6.4]. We make use of a simplified particular case of their general result:

Lemma 18

(Strong Rauch theorem) Assume the sectional curvatures on \(M\) satisfy \(\left| K\right| \le \varLambda \), and suppose there is a unique minimising geodesic between x and p. If \(v = \exp _p^{-1}(x)\), and
$$\begin{aligned} \left| v\right| = d_{M}(p,x) = r \le \frac{\pi }{2\sqrt{\varLambda }}, \end{aligned}$$
then
$$\begin{aligned} \left\| (d\exp _p)_v - T_{xp}\right\| \le \frac{\varLambda r^2}{2}. \end{aligned}$$

Proof

Given distinct upper and lower bounds on the sectional curvatures, \(\varLambda _{-}\le K\le \varLambda _{+}\), the result of Buser and Karcher [11], Section 6.4.2] is stated as
$$\begin{aligned} \left| (d\exp _p)_{v}w - T_{xp} \left( \frac{\mathbf {S}_{\kappa }(r)w}{r} \right) \right| \le \left| w\right| \left( \frac{\mathbf {S}_{\kappa -\lambda }(r) - \mathbf {S}_{\kappa }(r)}{r} \right) , \end{aligned}$$
for any vector w perpendicular to v, and as long as \(\mathbf {S}_{\kappa }\) is nonnegative. Here \(\kappa \) is arbitrary, and \(\lambda = \max \{\varLambda _{+}- \kappa , \kappa - \varLambda _{-}\}\).
We take \(\varLambda = \max \{ \varLambda _{+}, - \varLambda _{-}\}\), and \(\kappa = 0\). The stated bound results since now \(\mathbf {S}_{\kappa }(r) = \mathbf {S}_0(r) = r\), and the constraint \(r \le \frac{\pi }{2\sqrt{\varLambda }}\) ensures that
$$\begin{aligned} \frac{\mathbf {S}_{-\varLambda }(r) - r}{r} \le \frac{\varLambda r^2}{2}, \end{aligned}$$
as observed in Sect. 3.2. The result applies to all vectors since the exponential preserves lengths in the radial direction. \(\square \)

We obtain a bound on the differential of the inverse of the exponential map from Lemma 18 and the following observation:

Lemma 19

Suppose \(A :\mathbb {R}^n\rightarrow \mathbb {R}^n\) is a linear operator that satisfies
$$\begin{aligned} \left\| A-T\right\| \le \eta , \end{aligned}$$
for some linear isometry \(T :\mathbb {R}^n\rightarrow \mathbb {R}^n\). If \(\eta \le \frac{1}{2}\), then
$$\begin{aligned} \left\| A^{-1} - T^{-1}\right\| \le 2\eta . \end{aligned}$$

Proof

We first bound \(\left\| A^{-1}\right\| = s_{n}(A)^{-1}\), the inverse of the smallest singular value. Since \(s_{n}(A) = s_{n}(T^{-1}A)\), and \(\left\| T^{-1}A - {{\mathrm{Id}}}\right\| \le \eta \), we have \(\left| s_{n}(A) - 1\right| \le \eta \). Thus \(s_{n}(A)^{-1} \le (1-\eta )^{-1} \le 1 + 2\eta \).

Now write \(A = T + \eta E\), where \(\left\| E\right\| \le 1\). The trick [24], p. 50] is to observe that
$$\begin{aligned} A^{-1}&= T^{-1} - A^{-1}(A-T)T^{-1}\\&= T^{-1} - \eta A^{-1}ET^{-1}, \end{aligned}$$
and the stated bound follows. \(\square \)

Lemma 20

Suppose the sectional curvatures on \(M\) satisfy \(\left| K\right| \le \varLambda \). Let \(v \in T_pM\), with \(y = \exp _p(v)\). If \(x,y \in B_{M}(p;\rho )\), with
$$\begin{aligned} \rho \le \min \left\{ \frac{\iota _{M}}{2},\frac{1}{2\sqrt{\varLambda }} \right\} , \end{aligned}$$
then
$$\begin{aligned} \left\| d(\exp ^{-1}_x \circ \exp _p)_v - T_{xy}T_{yp}\right\| \le 5\varLambda \rho ^2. \end{aligned}$$

Proof

By Lemma 1, \(B_{M}(p;\rho )\) is convex. Since \(d_{M}(x,y) < 2\rho \), Lemma 18 yields \(\left\| (d\exp _x)_w - T_{yx}\right\| < 2\varLambda \rho ^2 \le \frac{1}{2}\), where \(w = \exp _x^{-1}(y)\). Since \(T_{xy} = T_{yx}^{-1}\), we may use Lemma 19 to write
$$\begin{aligned} (d\exp _x^{-1})_y = T_{xy} + \left( 4\varLambda \rho ^2 \right) E, \end{aligned}$$
where \(E\) satisfies \(\left\| E\right\| \le 1\). We obtain the result by composing this with
$$\begin{aligned} (d\exp _p)_v = T_{yp} + \left( \frac{\varLambda \rho ^2}{2} \right) \tilde{E}, \end{aligned}$$
where \(\left\| \tilde{E}\right\| \le 1\). \(\square \)

In order to compare \(T_{xp}\) with \(T_{xy}T_{yp}\) we exploit further estimates demonstrated by Buser and Karcher. If \(\alpha :[0,1] \rightarrow M\) is a curve, let \(T_{\alpha (t)} :T_{\alpha (0)}M\rightarrow T_{\alpha (t)}M\) denote the parallel translation operator (we do not require that \(\alpha \) be a minimising geodesic). Buser and Karcher [11], Sections 6.1, 6.2] bound the difference in the parallel translation operators between two homotopic curves:

Lemma 21

(Parallel translation comparison) Let \(c_i :[0,1] \rightarrow M\) be piecewise smooth curves from p to \(q\), and let
$$\begin{aligned} c :[1,2] \times [0,1] \rightarrow M\end{aligned}$$
be a piecewise smooth homotopy between \(c_1\) and \(c_2\), i.e., \(c(1,t) = c_1(t)\), and \(c(2,t) = c_2(t)\). Let \(\mathfrak {a} = \int \det dc_{(s,t)} \,dsdt\) be the area of the homotopy. If the sectional curvatures are bounded by \(\left| K\right| \le \varLambda \), then
$$\begin{aligned} \left\| T_{c_2(1)} - T_{c_1(1)}\right\| \le \frac{4}{3}\varLambda \mathfrak {a}. \end{aligned}$$

In our case the two curves of interest form the edges of a geodesic triangle. A geodesic triangle in \(M\) is a set of three points (vertices) such that each pair is connected by a unique minimising geodesic, together with these three minimising geodesics (edges). Any three points in a convex set are the vertices of a geodesic triangle. Buser and Karcher [11], Section 6.7] demonstrate an estimate of A. D. Aleksandrow that says that the edges of a small geodesic triangle are the boundary of a topological disk whose area admits a natural bound:

Lemma 22

(Small triangle area) Let \(p,x,y \in M\) be the edges of a geodesic triangle whose edge lengths, \(\ell _{px},\ell _{xy},\ell _{yp}\) satisfy
$$\begin{aligned} \ell _{px} + \ell _{xy} + \ell _{yp} \le \min \left\{ \iota _{M}, \frac{2\pi }{\sqrt{\varLambda _{+}}} \right\} , \end{aligned}$$
where \(\varLambda _{+}\) is an upper bound on the sectional curvatures of \(M\), and \(\iota _{M}\) is the injectivity radius (if \(\varLambda _{+}\le 0\), then \(\varLambda _{+}^{-\frac{1}{2}}\) is considered infinite). Then the edges of triangle pxy form the boundary of an immersed topological disk whose area \(\mathfrak {a}\) satisfies
$$\begin{aligned} \mathfrak {a} \le \mathfrak {a}_{\varLambda _{+}}, \end{aligned}$$
where \(\mathfrak {a}_{\varLambda _{+}}\) is the area of a triangle with the same edge lengths in the sphere of radius \(\frac{1}{\sqrt{\varLambda _{+}}}\).
Consider \(x,y \in B_{M}(p;\rho )\), where \(\rho < \frac{1}{2}\rho _0\) and as usual \(\varLambda \) is a bound on the absolute values of the sectional curvatures. In this case, Buser and Karcher [11], Section 6.7.1] observe that the area of the triangle in the sphere of radius \(\frac{1}{\sqrt{\varLambda }}\) that has the same edge lengths as pxy satisfies
$$\begin{aligned} \mathfrak {a}_{\varLambda } \le \frac{5}{8}\rho ^2. \end{aligned}$$
It follows then, from Lemma 21, that
$$\begin{aligned} \left\| T_{xp} - T_{xy}T_{yp}\right\| \le \frac{5}{6}\varLambda \rho ^2. \end{aligned}$$
This, together with Lemma 20, yields
$$\begin{aligned} \left\| d(\exp ^{-1}_x \circ \exp _p)_v - T_{xp}\right\| \le 5\varLambda \rho ^2 + \frac{5}{6}\varLambda \rho ^2 \le 6\varLambda \rho ^2, \end{aligned}$$
and we obtain Proposition 17.

4.3 Triangulations with Riemannian simplices

We now exploit Proposition 17 to demonstrate that a bound of the form (16) is satisfied by the differential (10) of the barycentric coordinate map defining a Riemannian simplex \(\varvec{\sigma }_M\)
$$\begin{aligned} db= - \left( \nabla \nu \right) ^{-1} \partial _u \nu , \end{aligned}$$
and find a bound on the scale that allows us to exploit Proposition 16.

Choose a vertex \(p_0\) of \(\varvec{\sigma }_M\), and an arbitrary linear isometry \(u :T_{p_0}M\rightarrow \mathbb {R}^n\) to establish a coordinate system on \(T_{p_0}{M}\), i.e., \(v_0(p_0)\) remains the origin. Let P be the matrix whose ith column is \(v_i(p_0)\). For \(x \in B_{\rho }\), rather than placing an arbitrary coordinate system on \(T_xM\), we identify \(T_{p_0}M\) and \(T_xM\) by the parallel translation operator \(T_{p_0x}\), i.e., use \(u \circ T_{p_0x}\) for coordinates. Let \(\tilde{P}\) be the matrix whose \(i^{\text {th}}\) column is \(v_i(x) - v_0(x)\).

Now the map
$$\begin{aligned} F :v \mapsto \exp _x^{-1} \circ \exp _{p_0}(v) - v_0(x) \end{aligned}$$
can be considered as a map \(\mathbb {R}^n\supset U \rightarrow \mathbb {R}^n\), and the matrix whose \(i^{\text {th}}\) column is \(F(v_i(p_0))\) is \(\tilde{P}\). It follows from Proposition 17 that if \(h< \frac{1}{2}\rho _0\), then for any \(u \in B_{\mathbb {R}^n}(0;h)\), we have \(\left\| (dF)_u - {{\mathrm{Id}}}\right\| \le \eta \), with \(\eta = 6\varLambda h^2\).
Lemma 12 implies a bound on the difference of the column vectors of P and \(\tilde{P}\):
$$\begin{aligned} \left| v_i(p_0) - (v_i(x) - v_0(x))\right| \le \eta \left| v_i(p_0)\right| \le \eta L\left( \varvec{\sigma }(p_0)\right) . \end{aligned}$$
It follows that \(\left\| P - \tilde{P}\right\| \le \sqrt{n}\eta L\left( \varvec{\sigma }(p_0)\right) \). Assume also that \(t\left( \varvec{\sigma }(p_0)\right) \ge t_0\). Then, recalling Eq. (12), and recognising that \(T_{xp_0}\) is represented by the identity matrix in our coordinate systems, we have
$$\begin{aligned} \left\| - \left( \partial _u \nu \right) - T_{xp_0}\right\|&= \left\| \tilde{P} P ^{-1} - PP ^{-1} \right\| \\&= \left\| \left( \tilde{P} - P \right) P^{-1} \right\| \\&\le \sqrt{n}6 \varLambda h^2 L\left( \varvec{\sigma }(p_0)\right) \left\| P^{-1}\right\| \\&\le \frac{\sqrt{n}6\varLambda h^2 L\left( \varvec{\sigma }(p_0)\right) }{\sqrt{n} t_0L\left( \varvec{\sigma }(p_0)\right) } \qquad \text {by Lemma 6}\\&\le \frac{6 \varLambda h^2}{t_0}. \end{aligned}$$
Buser and Karcher show [11], Section 8.1.3] that for any \(x \in B_{M}(p;h)\), with \(h < \rho _0\), we have
$$\begin{aligned} \left\| (\nabla \nu )_x - {{\mathrm{Id}}}\right\| \le 2\varLambda h^2. \end{aligned}$$
When \(h < \frac{1}{2}\rho _0\), we have \(2\varLambda h^2 < \frac{1}{2}\), and Lemma 19 yields
$$\begin{aligned} \left\| (\nabla \nu )^{-1} - {{\mathrm{Id}}}\right\| \le 4\varLambda h^2. \end{aligned}$$
Therefore we have, when \(b(u) = x\)
$$\begin{aligned} \begin{aligned} \left\| db_u - T_{xp_0}\right\|&= \left\| - \left( \nabla \nu \right) ^{-1} \partial _u \nu - T_{xp_0}\right\| \\&\le 4\varLambda h^2 + \frac{6 \varLambda h^2}{t_0} + 4\varLambda h^2 \left( \frac{6 \varLambda h^2}{t_0} \right) \\&\le \frac{14 \varLambda h^2}{t_0}, \end{aligned} \end{aligned}$$
(19)
using \(h < \frac{\pi }{8\sqrt{\varLambda }}\).
Finally, in order to employ Proposition 16 we consider the composition \(\exp ^{-1}_{p_0} \circ b\). From Lemmas 18 and 19 we have that
$$\begin{aligned} \left\| (d \exp ^{-1}_{p_0})_x - T_{p_0x}\right\| \le \varLambda h^2. \end{aligned}$$
Therefore, since \(T_{p_0x} = T_{xp_0}^{-1}\) we have
$$\begin{aligned} \left\| d(\exp _{p_0}^{-1} \circ b)_u - {{\mathrm{Id}}}\right\|&\le \varLambda h^2 + \frac{14 \varLambda h^2}{t_0} + \varLambda h^2 \left( \frac{14 \varLambda h^2}{t_0} \right) \\&\le \frac{18 \varLambda h^2}{t_0}. \end{aligned}$$
In order to meet the conditions of Proposition 16, we require
$$\begin{aligned} \frac{18 \varLambda h^2}{t_0} \le \frac{1}{2}nt_0, \end{aligned}$$
or
$$\begin{aligned} h\le \frac{\sqrt{n}t_0}{6 \sqrt{\varLambda }}. \end{aligned}$$
We obtain

Theorem 2

(Triangulation criteria) Suppose \(M\) is a compact n-dimensional Riemannian manifold with sectional curvatures \(K\) bounded by \(\left| K\right| \le \varLambda \), and \(\fancyscript{A}\) is an abstract simplicial complex with finite vertex set \(S\subset M\). Define a quality parameter \(t_0> 0\), and let
$$\begin{aligned} h= \min \left\{ \frac{\iota _{M}}{4}, \frac{\sqrt{n}t_0}{6 \sqrt{\varLambda }} \right\} . \end{aligned}$$
If
  1. 1.

    For every \(p \in S\), the vertices of \(\underline{{{\mathrm{St}}}}(p)\) are contained in \(B_{M}(p;h)\), and the balls \(\{B_{M}(p;h)\}_{p \in S}\) cover \(M\).

     
  2. 2.

    For every \(p \in S\), the restriction of the inverse of the exponential map \(\exp _p^{-1}\) to the vertices of \(\underline{{{\mathrm{St}}}}(p) \subset \fancyscript{A}\) defines a piecewise linear embedding of \(\left| \underline{{{\mathrm{St}}}}(p)\right| \) into \(T_{p}{M}\), realising \(\underline{{{\mathrm{St}}}}(p)\) as a full star such that every simplex \(\varvec{\sigma }(p)\) has thickness \(t\left( \varvec{\sigma }(p)\right) \ge t_0\).

     
then \(\fancyscript{A}\) triangulates \(M\), and the triangulation is given by the barycentric coordinate map on each simplex.

5 The piecewise flat metric

The complex \(\fancyscript{A}\) described in Theorem 2 naturally inherits a piecewise flat metric from the construction. The length assigned to an edge \(\{p,q\} \in \fancyscript{A}\) is the geodesic distance in \(M\) between its endpoints: \(\ell _{pq} = d_{M}(p,q)\). We first examine, in Sect. 5.1, conditions which ensure that this assignment of edge lengths does indeed make each \(\varvec{\sigma }\in \fancyscript{A}\) isometric to a Euclidean simplex. With this piecewise flat metric on \(\fancyscript{A}\), the barycentric coordinate map is a bi-Lipschitz map between metric spaces \(H :\left| \fancyscript{A}\right| \rightarrow M\). In Sect. 5.2 we estimate the metric distortion of this map.

Several of the lemmas in this section are generalisations of lemmas that appeared in [7], Section A.1]. The arguments are essentially the same, but we have included the proofs here for convenience.

5.1 Euclidean simplices defined by edge lengths

If G is a symmetric positive definite \(n \times n\) matrix, then it can be written as a Gram matrix, \(G = {P}^\mathsf {T}P\) for some \(n \times n\) matrix P. Then P describes a Euclidean simplex with one vertex at the origin, and the other vertices defined by the column vectors. The matrix P is not unique, but if \(G= {Q}^\mathsf {T}Q\), then \(Q = OP\) for some linear isometry O. Thus a symmetric positive definite matrix defines a Euclidean simplex, up to isometry.

If \(\varvec{\sigma }= \{p_0, \ldots , p_n\} \subset B_{\rho }\), is the vertex set of a Riemannian simplex \(\varvec{\sigma }_M\), we define the numbers \(\ell _{ij} = d_{M}(p_i,p_j)\). These are the edge lengths of a Euclidean simplex \(\varvec{\sigma }_{\mathbb {E}}\) if and only if the matrix G defined by
$$\begin{aligned} G_{ij} = \frac{1}{2}\left( \ell _{0i}^2 + \ell _{0j}^2 - \ell _{ij}^2\right) \end{aligned}$$
(20)
is positive definite.

We would like to use the smallest eigenvalue of G to estimate the thickness of \(\varvec{\sigma }_{\mathbb {E}}\), however, an unfortunate choice of vertex labels can prevent us from doing this easily. We make use of the following observation:

Lemma 23

Suppose \(\varvec{\sigma }= \{v_0, \ldots , v_k\} \subset \mathbb {R}^n\) is a Euclidean \(k\)-simplex, and let P be the \(n\times k\) matrix whose ith column is \(v_i - v_0\). If for some \(i\ne 0\), an altitude at least as small as \(a_0\) is realised, i.e, \(a_i \le a_0\), then
$$\begin{aligned} t\left( \varvec{\sigma }\right) \ge \frac{s_{k}(P)}{k L\left( \varvec{\sigma }\right) }. \end{aligned}$$

Proof

We assume that \(\varvec{\sigma }\) is non-degenerate, since otherwise the bound is trivial. If \(v_i\) is a vertex of minimal altitude, then by Lemma 6, the \(i{\text {th}}\) row of the pseudo-inverse \({P}^{-1}_{\text {left}}\) is given by \({w_i}^\mathsf {T}\), where
$$\begin{aligned} \left| w_i\right| = a_i^{-1} = (ktL)^{-1}. \end{aligned}$$
It follows then that \(s_{1}({P}^{-1}_{\text {left}}) \ge (k tL)^{-1}\), and therefore \(s_{k}(P) \le k tL\), yielding the stated bound. \(\square \)
If G is positive definite, then we may write \(G= {P}^\mathsf {T}P\), where P is a matrix describing \(\varvec{\sigma }_{\mathbb {E}}= [v_0, \ldots , v_k]\), with the edge lengths \(\{\ell _{ij}\}\) dictating the vertex labelling. If \(\mu _k(G) = s_{k}(P)^2\) is the smallest eigenvalue of G, then provided some vertex other than \(v_0\)realises the smallest altitude in \(\varvec{\sigma }_{\mathbb {E}}\), Lemma 23 yields
$$\begin{aligned} t\left( \varvec{\sigma }_{\mathbb {E}}\right) \ge \frac{\sqrt{\mu _k(G)}}{k L\left( \varvec{\sigma }\right) }. \end{aligned}$$
(21)
For our current purposes, we can ensure the existence, and bound the thickness of \(\varvec{\sigma }_{\mathbb {E}}\) by comparing it with a related simplex such as \(\varvec{\sigma }(p_0)\). To this end we employ the following observation (where \(\tilde{\varvec{\sigma }}\) plays the role of \(\varvec{\sigma }(p_0)\)):

Lemma 24

Suppose \(\tilde{\varvec{\sigma }} = \{\tilde{v}_0, \ldots , \tilde{v}_k\}\) is a Euclidean \(k\)-simplex, and \(\{{\ell }_{ij}\}\) is a set of positive numbers defined for all \(0 \le i\ne j \le k\) such that \({\ell }_{ij} = {\ell }_{ji}\), and
$$\begin{aligned} \left| \left| \tilde{v}_i - \tilde{v}_j\right| - {\ell }_{ij}\right| \le C_0L\left( \tilde{\varvec{\sigma }}\right) . \end{aligned}$$
Let \(\tilde{P}\) be the matrix whose ith column is \(\tilde{v}_i - \tilde{v}_0\), and define the matrix G by
$$\begin{aligned} G_{ij} = \frac{1}{2}\left( {\ell }_{0i}^2 + {\ell }_{0j}^2 - {\ell }_{ij}^2\right) . \end{aligned}$$
Let E be the matrix that records the difference between G and the Gram matrix \({\tilde{P}}^\mathsf {T}\tilde{P}\):
$$\begin{aligned} G = {\tilde{P}}^\mathsf {T} \tilde{P} + E. \end{aligned}$$
If \(C_0\le \frac{2}{3}\), then the entries of \(E\) are bounded by \(\left| E_{ij}\right| \le 4 C_0L\left( \tilde{\varvec{\sigma }}\right) ^2\), and in particular
$$\begin{aligned} \left\| E\right\| \le 4 k C_0L\left( \tilde{\varvec{\sigma }}\right) ^2. \end{aligned}$$
(22)

Proof

Let \(\tilde{\ell }_{ij} = \left| \tilde{v}_i- \tilde{v}_j\right| \). By the cosine rule we have
$$\begin{aligned} \left[ {\tilde{P}}^\mathsf {T} \tilde{P} \right] _{ij} = \frac{1}{2}\left( \tilde{\ell }_{0i}^2 + \tilde{\ell }_{0j}^2 - \tilde{\ell }_{ij}^2\right) , \end{aligned}$$
and we obtain a bound on the magnitude of the coefficients of \(E\):
$$\begin{aligned} \begin{aligned} \left| G_{ij} - \left[ {\tilde{P}}^\mathsf {T} \tilde{P} \right] _{ij} \right|&\le \frac{1}{2} \left( \left| \ell _{0i}^2 - \tilde{\ell }_{0i}^2 \right| + \left| \ell _{0j}^2 - \tilde{\ell }_{0j}^2 \right| + \left| \ell _{ij}^2 - \tilde{\ell }_{ij}^2 \right| \right) \\&\le \frac{3}{2}(2 + C_0)C_0L\left( \tilde{\varvec{\sigma }}\right) ^2\\&\le 4 C_0L\left( \tilde{\varvec{\sigma }}\right) ^2. \end{aligned} \end{aligned}$$
This leads us to a bound on \(s_{1}(E) = \left| E\right| \). Indeed, the magnitude of the column vectors of \(E\) is bounded by \(\sqrt{k}\) times a bound on the magnitude of their coefficients, and the magnitude of \(s_{1}(E)\) is bounded by \(\sqrt{k}\) times a bound on the magnitude of the column vectors. We obtain Eq. (22). \(\square \)

We have the following extension of the “Thickness under distortion” Lemma 7 ([7], Section 4.2]):

Lemma 25

(Abstract Euclidean simplex) Suppose \(\tilde{\varvec{\sigma }} = \{\tilde{v}_0, \ldots , \tilde{v}_k\} \subset \mathbb {R}^n\), and \(\{\ell _{ij}\}_{0\le i,j \le k}\) is a set of positive numbers defined for all \(0 \le i\ne j \le k\) such that \({\ell }_{ij} = {\ell }_{ji}\), and such that
$$\begin{aligned} \left| \left| \tilde{v}_i - \tilde{v}_j\right| - \ell _{ij}\right| \le C_0L\left( \tilde{\varvec{\sigma }}\right) \end{aligned}$$
for all \(0 \le i < j \le k\).
If
$$\begin{aligned} C_0= \frac{\eta t\left( \tilde{\varvec{\sigma }}\right) ^2}{4} \qquad \text { with } \quad 0 \le \eta \le 1, \end{aligned}$$
(23)
then there exists a Euclidean simplex \(\varvec{\sigma }= \{{v}_0, \ldots , {v}_k\}\) whose edge lengths are described by the numbers \(\ell _{ij}\). Let \(\tilde{P}\) and P be matrices whose \(i^{\text {th}}\) column is given by \(\tilde{v}_i - \tilde{v}_0\), and \(v_i-v_0\) respectively. Then
$$\begin{aligned} s_{k}({P}) \ge (1 - \eta )s_{k}(\tilde{P}), \end{aligned}$$
and the thickness of \(\varvec{\sigma }\) satisfies
$$\begin{aligned} t\left( \varvec{\sigma }\right) \ge \frac{4}{5\sqrt{k}}(1 - \eta ) t\left( \tilde{\varvec{\sigma }}\right) . \end{aligned}$$

Proof

If \(\tilde{\varvec{\sigma }}\) is degenerate, then by (23), \(\{\ell _{ij}\}\) is the set of edge lengths of \(\tilde{\varvec{\sigma }}\) and there is nothing to prove. Therefore, assume \(t\left( \tilde{\varvec{\sigma }}\right) >0\).

Let G be the matrix defined by Eq. (20), and define the matrix E by \(G= {\tilde{P}}^\mathsf {T}\tilde{P} + E\), and let x be a unit eigenvector of G associated with the smallest eigenvalue \(\mu _k\). Then
$$\begin{aligned} \begin{aligned} \mu _k = {x}^\mathsf {T}Gx&= {x}^\mathsf {T}{\tilde{P}}^\mathsf {T}\tilde{P}x + {x}^\mathsf {T}Ex\\&\ge s_{k}(\tilde{P})^2 - s_{1}(E)\\&= \left( 1 - \frac{s_{1}(E)}{s_{k}(\tilde{P})^2} \right) s_{k}(\tilde{P})^2. \end{aligned} \end{aligned}$$
From Lemma 6 we have \(s_{k}(\tilde{P})^2 \ge k t\left( \tilde{\varvec{\sigma }}\right) ^2 L\left( \tilde{\varvec{\sigma }}\right) ^2\), and by Lemma 24\(s_{1}(E) \le 4kC_0L\left( \tilde{\varvec{\sigma }}\right) ^2\), so by the definition (23) of \(C_0\), we have that
$$\begin{aligned} \mu _k \ge (1 - \eta )s_{k}(\tilde{P})^2, \end{aligned}$$
and thus \(G\) is positive semi-definite, and the first inequality is satisfied because \(\mu _k = s_{k}(P)^2\) and \(\sqrt{1-\eta } \ge 1-\eta \).
In order to obtain the thickness bound, we employ Lemma 23. Since thickness is independent of the vertex labelling, we may assume that some vertex other than \({v}_0\) realises the minimal altitude in \({\varvec{\sigma }}\) (if necessary, we relabel the vertices of \(\tilde{\varvec{\sigma }}\) and \({\varvec{\sigma }}\), maintaining the correspondence). Then using Lemma 23 and Lemma 6 we have
$$\begin{aligned} k t\left( {\varvec{\sigma }}\right) L\left( {\varvec{\sigma }}\right) \ge s_{k}({P}) \ge (1 - \eta )s_{k}(\tilde{P}) \ge (1- \eta )\sqrt{k}t\left( \tilde{\varvec{\sigma }}\right) L\left( \tilde{\varvec{\sigma }}\right) . \end{aligned}$$
The stated thickness bound follows since \(\frac{L\left( \tilde{\varvec{\sigma }}\right) }{L\left( \varvec{\sigma }\right) } \ge \frac{1}{1 + C_0} \ge \frac{4}{5}\). \(\square \)
Now we examine whether the simplices of the complex \(\fancyscript{A}\) of Theorem 2 meet the requirements of Lemma 25. If \(\varvec{\sigma }\in \fancyscript{A}\), with \(p \in \varvec{\sigma }\), then we can use the Rauch theorem 9 to compare \(\varvec{\sigma }\) with the Euclidean simplex \(\varvec{\sigma }(p) \in T_pM\). Under the assumptions of Theorem 2, we have \(\left\| d\exp _p\right\| \le 1 +\frac{\varLambda h^2}{2}\), and \(\left\| d\exp _p^{-1}\right\| \le 1 + \frac{\varLambda h^2}{3}\). Thus
$$\begin{aligned} \ell _{ij} - \left| v_i(p) - v_j(p)\right| \le \frac{\varLambda h^2}{2} \left| v_i(p) - v_j(p)\right| , \end{aligned}$$
and
$$\begin{aligned} \left| v_i(p) - v_j(p)\right| - \ell _{ij}&\le \frac{\varLambda h^2}{3} \ell _{ij} \le \frac{\varLambda h^2}{3} \left( 1 + \frac{\varLambda h^2}{2} \right) \left| v_i(p) - v_j(p)\right| \\&\le \frac{\varLambda h^2}{2} \left| v_i(p) - v_j(p)\right| , \end{aligned}$$
and we can use
$$\begin{aligned} C_0= \frac{\varLambda h^2}{2} \end{aligned}$$
(24)
in Lemma 25. Thus in order to guarantee that the \(\ell _{ij}\) describe a non-degenerate Euclidean simplex, we require that
$$\begin{aligned} \varLambda h^2 = \frac{\eta t_0^2}{2}, \end{aligned}$$
for some non-negative \(\eta < 1\).

Under the conditions of Theorem 2 we may have \(h^2 \varLambda = \frac{n t_0^2}{36}\), which gives us \(\eta = \frac{n}{18}\). Thus when \(n \ge 18\) we require stronger bounds on the scale than those imposed by Theorem 2 if we wish to ensure the existence of a piecewise flat metric on \(\fancyscript{A}\). Reducing the curvature controlled constraint on \(h\) in Theorem 2 by a factor of \(1/\sqrt{n}\) gives us \(\eta = \frac{1}{18}\), and Lemma 25 yields:

Proposition 26

If the requirements of Theorem 2, are satisfied when the scale parameter (5) is replaced with
$$\begin{aligned} h= \min \left\{ \frac{\iota _{M}}{4}, \frac{t_0}{6 \sqrt{\varLambda }} \right\} , \end{aligned}$$
then the geodesic distances between the endpoints of the edges in \(\fancyscript{A}\) defines a piecewise flat metric on \(\fancyscript{A}\) such that each simplex \(\varvec{\sigma }\in \fancyscript{A}\) satisfies
$$\begin{aligned} t\left( \varvec{\sigma }\right) > \frac{3}{4\sqrt{n}}t_0. \end{aligned}$$

5.2 Metric distortion of the barycentric coordinate map

In the context of Theorem 2 the barycentric coordinate map on each simplex defines a piecewise smooth homeomorphism \(H :\left| \fancyscript{A}\right| \rightarrow M\). If the condition of Proposition 26 is also met, then \(\fancyscript{A}\) is naturally endowed with a piecewise flat metric. We wish to compare this metric with the Riemannian metric on \(M\). It suffices to consider an n-simplex \(\varvec{\sigma }\in \fancyscript{A}\), and establish bounds on the singular values of the differential \(dH\). If \(p \in \varvec{\sigma }\), then we can write \(H \big |_{\varvec{\sigma }_{\mathbb {E}}} = b\circ \fancyscript{L}_p\), where \(\fancyscript{L}_p :\varvec{\sigma }_{\mathbb {E}}\rightarrow \varvec{\sigma }_{\mathbb {E}}(p)\) is the linear map that sends \(\varvec{\sigma }\in \fancyscript{A}\) to \(\varvec{\sigma }(p) \in T_pM\).

A bound on the metric distortion of a linear map that sends one Euclidean simplex to another is a consequence of the following (reformulation of [7], Lem A.4]):

Lemma 27

(Linear distortion bound) Suppose that P and \(\tilde{P}\) are non-degenerate \(k \times k\) matrices such that
$$\begin{aligned} {\tilde{P}}^\mathsf {T}\tilde{P} = {P}^\mathsf {T} P + E. \end{aligned}$$
(25)
Then there exists a linear isometry \(\varPhi :\mathbb {R}^k \rightarrow \mathbb {R}^k\) such that
$$\begin{aligned} \left\| \tilde{P} P ^{-1} - \varPhi \right\| \le \frac{s_{1}(E)}{s_{k}(P)^2}. \end{aligned}$$

Proof

Multiplying by \({P}^{-T} := { ( {P}^\mathsf {T})}^{-1}\) on the left, and by \({P}^{-1}\) on the right, we rewrite Eq. (25) as
$$\begin{aligned} {A}^\mathsf {T}A = I + F, \end{aligned}$$
(26)
where \(A = \tilde{P}{P}^{-1}\), and \(F = {P}^{-T}E{P}^{-1}\). Using the singular value decomposition \(A = U_A \varSigma _A {V}^\mathsf {T}_A\), we let \(\varPhi = U_A {V}^\mathsf {T}_A\) so that
$$\begin{aligned} (A - \varPhi ) = U_A( \varSigma _A - I ) {V}^\mathsf {T}_A . \end{aligned}$$
(27)
From Eq. (26) we deduce that \(s_{1}(A)^2 \le 1 + s_{1}(F)\), and also that \(s_{k}(A)^2 \ge 1 - s_{1}(F)\). Using these two inequalities we find
$$\begin{aligned} \max _i \left| s_{i}(A) - 1\right| \le \frac{s_{1}(F)}{1 + s_{i}(A)} \le s_{1}(F), \end{aligned}$$
and thus
$$\begin{aligned} \left\| \varSigma _A - I\right\| \le s_{1}(F) \le s_{1}({P}^{-1})^2 s_{1}(E) = s_{k}(P)^{-2} s_{1}(E). \end{aligned}$$
The result now follows from Eq. (27). \(\square \)

Lemma 27 implies:

Lemma 28

Suppose \(\varvec{\sigma }= \{v_0, \ldots , v_n\}\) and \(\tilde{\varvec{\sigma }}= \{\tilde{v}_0, \ldots , \tilde{v}_n\}\) are two Euclidean simplices in \(\mathbb {R}^n\) such that
$$\begin{aligned} \left| \, \left| \tilde{v}_i - \tilde{v}_j\right| - \left| v_i - v_j\right| \,\right| \le C_0L\left( \varvec{\sigma }\right) . \end{aligned}$$
If \(A :\mathbb {R}^n \rightarrow \mathbb {R}^n\) is the affine map such that \(A(v_i) = \tilde{v}_i\) for all i, and \(C_0\le \frac{2}{3}\), then for all \(x,y \in \mathbb {R}^n\),
$$\begin{aligned} \left| \, \left| A(x) - A(y)\right| - \left| x-y\right| \,\right| \le \eta \left| x-y\right| , \end{aligned}$$
where
$$\begin{aligned} \eta =\frac{4 C_0}{t\left( \varvec{\sigma }\right) ^2}. \end{aligned}$$

Proof

Let P be the matrix whose ith column is \(v_i - v_0\), and let \(\tilde{P}\) be the matrix whose \(i^{\text {th}}\) column is \(\tilde{v}_i - \tilde{v}_0\). Then we have the matrix form \(A(x) = \tilde{P}P^{-1}x +(\tilde{v}_0 - \tilde{P}P^{-1}v_0)\). It follows then from Lemma 27 that \(\eta \le s_{n}(P)^{-2}s_{1}(E)\), where \(E = {\tilde{P}}^\mathsf {T}\tilde{P} - {P}^\mathsf {T}P\).

By Lemma 24, \(s_{1}(E) \le 4n C_0L\left( \varvec{\sigma }\right) ^2\), and by Lemma 6, \(s_{n}(P)^2 \ge n t\left( \varvec{\sigma }\right) ^2L\left( \varvec{\sigma }\right) ^2\), and the result follows. \(\square \)

Observe that if A in Lemma 28 is a linear map, then the lemma states that \(s_{1}(A) \le 1+\eta \) and \(s_{n}(A) \ge 1 - \eta \). We use this to estimate the metric distortion of \(H|_{\varvec{\sigma }_{\mathbb {E}}} = b\circ \fancyscript{L}_p\). Under the assumption of Proposition 26, specifically, given that \(h \le \frac{t_0}{6\sqrt{\varLambda }}\), we again exploit Eq. (24), so
$$\begin{aligned} \left\| \fancyscript{L}_p^{-1}\right\| \le 1 + \frac{2\varLambda h^2}{t_0^2} \end{aligned}$$
and, since \(\left\| \fancyscript{L}_p\right\| ^{-1} = s_{n}(\fancyscript{L}_p^{-1}) \ge 1 - \frac{2\varLambda h^2}{t_0^2}\), and the second term is less than \(\frac{1}{2}\), we also have
$$\begin{aligned} \left\| \fancyscript{L}_p\right\| \le 1 + \frac{4\varLambda h^2}{t_0^2}. \end{aligned}$$
Using Eq. (19) we have
$$\begin{aligned} \left\| db\right\| \le 1 + \frac{14\varLambda h^2}{t_0}, \end{aligned}$$
and
$$\begin{aligned} \left\| db^{-1}\right\| \le 1 + \frac{28\varLambda h^2}{t_0}. \end{aligned}$$
Recalling that \(dH \big |_{\varvec{\sigma }_{\mathbb {E}}} = (db) \fancyscript{L}_p\), and \(h^2 \le \frac{t_0^2}{36\varLambda }\), we obtain
$$\begin{aligned} \left\| dH\right\| \le 1 + \frac{20\varLambda h^2}{t_0^2}, \end{aligned}$$
and
$$\begin{aligned} \left\| dH^{-1}\right\| \le 1 + \frac{32\varLambda h^2}{t_0^2}. \end{aligned}$$
The bound on the differential of \(H\) and its inverse enables us to estimate the Riemannian metric on \(M\) using the piecewise flat metric on \(\fancyscript{A}\). The metric distortion bound on \(H\) is found with the same kind of calculation as exhibited in Eq. (15), for example. We find:

Theorem 3

(Metric distortion) If the requirements of Theorem 2, are satisfied with the scale parameter (5) replaced by
$$\begin{aligned} h= \min \left\{ \frac{\iota _{M}}{4}, \frac{t_0}{6 \sqrt{\varLambda }} \right\} , \end{aligned}$$
then \(\fancyscript{A}\) is naturally equipped with a piecewise flat metric \(d_{\fancyscript{A}}\) defined by assigning to each edge the geodesic distance in \(M\) between its endpoints.
If \(H :\left| \fancyscript{A}\right| \rightarrow M\) is the triangulation defined by the barycentric coordinate map in this case, then the metric distortion induced by \(H\) is quantified as
$$\begin{aligned} \left| d_{M}(H(x),H(y)) - d_{\fancyscript{A}}(x,y) \right| \le \frac{50 \varLambda h^2}{t_0^2} d_{\fancyscript{A}}(x,y), \end{aligned}$$
for all \(x,y \in \left| \fancyscript{A}\right| \).

Footnotes

  1. 1.

    We have included the factor \(1/k\) to shorten the calculation below.

  2. 2.

    By which we mean that following inequality implies the previous.

Notes

Acknowledgments

We thank Stefan von Deylen for pointing out the work of Sander [34], and for stimulating discussions. We have also benefited from discussions with Arijit Ghosh. This research has been partially supported by the 7th Framework Programme for Research of the European Commission, under FET-Open Grant Number 255827 (CGL Computational Geometry Learning).

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Authors and Affiliations

  1. 1.Johann Bernoulli InstituteRijksuniversiteit GroningenGroningenThe Netherlands

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