Completeness and incompleteness of the Binet–Legendre metric
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Abstract
The goal of this short paper is to give conditions for the completeness of the Binet–Legendre metric in Finsler geometry. The case of the Funk and Hilbert metrics in a convex domain are discussed.
Keywords
Finsler metric Binet–Legendre metric John ellipsoid Hilbert geometry Affine metricMathematics Subject Classification
58A10 58A12 53Cxx1 Introduction and statement of the main result
Given a Finsler manifold \((M,F)\), there are several natural ways to construct a Riemannian metric \(g\) on the manifold \(M\) that is associated to the given Finsler metric. Recently, such constructions were shown to be a useful tool in Finsler geometry, see for example [24, 25, 26, 34, 35].

The construction is pointwise: the associated Riemannian metric \(g\), restricted to any tangent space of the manifold \(M\), depends only on the restriction of the Finsler metric to this tangent space.

The construction is homogenous: If we multiply the Finsler metric by a conformal factor \(\lambda \), the associated Riemannian metric is multiplied by \(\lambda ^2\).
The construction in [26] is called the Binet–Legendre metric ^{1} and has proven to be a flexible and useful tool in Finsler geometry, its definition will be recalled in Sect. 2.2.
Our goal in the present paper is to relate the completeness, or incompleteness, of the Binet–Legendre metric to that of the given Finsler metric. Our main result is in fact the following stronger theorem.
Theorem 1.1
Remarks

Our proof will give explicit (though perhaps not optimal) values for the constants \(C_1,C_2,C_3\). The constants \(C_1\) and \(C_3\) play the same role, but in the reversible case we have a better constant (namely Open image in new window ).

Our theorem implies that if the Binet–Legendre metric associated to a Finsler metric \(F\) is complete, then the Finsler metric is also complete, see Corollary 4.2. The converse statement holds in the case of quasireversible metric but not in general. We illustrate this phenomenon by an example in Sect. 5.3.

The quasireversibility hypothesis in the second statement is necessary. For instance the Funk metric (discussed below) is forward complete but not backward complete, hence it cannot be bilipschitz equivalent to any Riemannian metric. In fact it is quite clear from Main Theorem that a Finsler metric is bilipschitz to a Riemannian metric if and only if it is quasireversible (note the “if” direction follows from Main Theorem, while the “only if” direction is obvious).

The associated Riemannian metrics constructed in the papers [24, 25, 35] involve the second derivatives of the given Finlser metric and they generally do not satisfy (1).
The rest of the paper is organized as follows: In Sect. 2 we recall some basic definitions from Finsler geometry and we recall the definition and some basic properties of the Binet–Legendre metric. In Sect. 3 we discuss another auxiliary Riemannian metric, based on the John ellipsoid from convex geometry, and we use it as a tool to prove Theorem 1.1, in Sect. 4, where we also derive some of its simple but important consequences.
In Sect. 5 we discuss some examples. We first recall in Sect. 5.1 the definition of Zermelo metrics in Euclidean domains and in particular the Funk and reverse Funk metrics. In the next Sect. 5.2, we explicitly compute the Binet–Legendre metric associated to a Zermelo metric and in Sect. 5.3 we construct an example of a complete Finsler metric with incomplete Binet–Legendre metric. In Sect. 5.4 we discuss the Hilbert Finsler metric in a convex domain and we use it to compare the Binet–Legendre metric to the socalled affine metric, which is another important Riemannian metric defined in an arbitrary convex domain. In an appendix we show by an example that the Riemannian metric obtained from the John ellipsoid construction may be nonsmooth, even if the initial Finsler metric is smooth.
2 A brief review of Finsler geometry
2.1 Basic definitions: Finsler manifolds, completeness and quasireversibility

\(F_x(\xi ) >0\) if \(\xi \ne 0\),

\(F_x( \xi +\eta ) \le F_x(\xi )+ F_x(\eta )\),

\(F_x(\lambda \xi )=\lambda F_x(\xi )\) for all \(\lambda \ge 0\).
It is also known, but somewhat delicate to prove, that if \(F\) is a reversible Finsler metric on \(M\), then \(d\mu _F\) coincides with the \(n\)dimensional Hausdorff measure of the metric space associated to the Finsler structure, see [2, 8, 9, 10].
2.2 The Binet–Legendre metric
Definition 2.1
The Binet–Legendre metric \(g_{\mathrm{BL}}\) associated to the Finsler metric \(F\) is the Riemannian metric dual to the the scalar product \(g_{\mathrm{BL}}^*\) defined above on \(T_x^*(M)\).
The Binet–Legendre metric enjoys a number of important properties, let us state in particular the following theorem.
Theorem 2.2
 (a)
If \(F\) is of class \(C^k\) on the complement of the zero section of \(TM\), then \(g_{\mathrm{BL}}\) is a Riemannian metric of class \(C^k\).
 (b)
If \(\varphi \) is an isometry of \((M,F)\), then it is also an isometry of \((M,g_{\mathrm{BL}})\).
 (c)If \(F_1, F_2\) are two Finsler metrics on \(M\) such that Open image in new window for some function \(\lambda :M \rightarrow \mathbb {R}_+\), then the corresponding Binet–Legendre metrics satisfy
 (d)
If the Finsler metric \(F\) is derived from a Riemannian metric \(g\), that is \(F = \sqrt{g}\), then \(g_{\mathrm{BL}}= g\).
We refer to [26], Theorem 2.4] for the first statement, which is in fact proven for the wider class of partially smooth Finsler metrics. The second statement is obvious and the third and fourth statements are proved in [26], Proposition 12.1].
3 The John metric on a Finsler manifold
Proposition 3.1
This Riemannian metric \(g_{\mathrm{John}}\) will be called the John metric associated to the Finsler metric, it is a natural construction and appeared in the papers [31, 32]. Note that the John metric may fail to be \(C^1\), even if the Finsler metric \(F\) is analytic, an example is given in Appendix.
Proof
Let \({\Omega }_x \subset T_xM\) be the Finsler tangent unit ball at \(x\in M\), and let us denote by \(J_0[{\Omega }_x]\subseteq T_xM\) the corresponding centered John ellipsoid. This ellipsoid is the unit ball of a uniquely defined positive symmetric definite bilinear form on \(T_xM\). By continuity of the John ellipsoid, these bilinear forms give us a \(C^0\)Riemannian metric \(g_{\mathrm{John}}\) on \(M\), that is naturally associated to the Finsler metric \(F\).
Inclusion (5) gives us Open image in new window , which immediately implies inequality (6). In the reversible case, \({\Omega }_x \subseteq T_xM\) is symmetric around the origin and from (3) we have the inclusions Open image in new window which are equivalent to (7). The proof of the last assertion is straightforward. \(\square \)
Our next result says that the volume form of the John metric is comparable to the Busemann measure of the Finsler metric \(F\).
Proposition 3.2
Proof
4 Proof of Main Theorem and some consequences
Remark 4.1
Let us now state some simple consequences of Main Theorem.
Corollary 4.2
Let \((M,F)\) be an arbitrary Finsler manifold. If the Binet–Legendre metric \(g_{\mathrm{BL}}\) is complete, then \(F\) is both forward and backward complete.
Proof
Let \(\{x_j\}\) be a forward Cauchy sequence for the metric \(F\), then the first statement from Main Theorem implies that \(\{x_j\}\) is a Cauchy sequence for the Riemannian metric \(g_{\mathrm{BL}}\). It is therefore a convergent sequence by hypothesis. The proof for a backward Cauchy sequence is the same. \(\square \)
Corollary 4.3
 (a)
The Binet–Legendre metric \(g_{\mathrm{BL}}\) is complete if and only if the given Finsler metric \(F\) is complete.
 (b)
The Riemannian volume density of \(g_{\mathrm{BL}}\) is comparable to the Busemann density \(d\mu _F\).
 (c)
Two quasireversible Finsler manifolds are quasiisometric if and only if the associated Riemannian manifold with their respective Binet–Legendre metrics are quasiisometrics.
Recall that the Finsler manifolds \((M_1,F_1)\) and \((M_2,F_2)\) are quasiisometric if there exists a map \(f :M_1 \rightarrow M_2\) and a constant \(A\) such that for any \(p,q \in M_1\) we have Open image in new window and for any \(y \in M_2\) there exists \(x\in M_1\) with \(d_{F_2}(f(x),y) \le A\).
Proof
The property (a) is an immediate consequence of Main Theorem since completeness is a property which is stable under bilipschitz equivalence.
Remark 4.4
The first inequality in (17) can be improved: it is known that the Riemannian volume is in fact always smaller or equal to the Busemann measure, that is \(d\mu _{\mathrm{BL}}\le d\mu _F\), and the equality holds if and only if \(F\) is Riemannian. This fact also holds without the reversibility assumption and follows, e.g. from [22], Theorem 1], see also [12], Theorem 3.2].
5 Examples and applications
5.1 Zermelo metrics in a domain
Examples
(a) If \(u(x) = c \in {\Omega }\) is constant, then the corresponding Zermelo metric is invariant by translation. It is thus the Minkowski metric whose unit ball is given by \({\Omega } c\) (Fig. 1).
(b) If \(\fancyscript{U} = {\Omega }\) and \(u(x) = x\) is the identity map, then the corresponding Zermelo metric is called the Funk metric and denoted by \(F_{\mathrm{Funk}}\). The Finsler unit ball at the point \(x\in {\Omega }\) is the convex domain \({\Omega }\) itself, but with the point \(x\) as its center (this metric is therefore also called the tautological Finsler structure).
Proposition 5.1

The reverse Funk metric satisfies Open image in new window . They are both invariant under affine transformations preserving \({\Omega }\).

The Funk metric in \({\Omega }\) is forward complete but not backward complete. The reverse Funk metric is backward complete and not forward complete.

Both metrics are projective, meaning that the Euclidean straight lines are geodesics.

If the bounded convex domain \({\Omega }\subset \mathbb {R}^n\) has a boundary of class \(C^k\), then \(F_{\mathrm{Funk}}\) and \(F_{\mathrm{RFunk}}\) are also of class \(C^k \;(\)on the complement of the zero section).
The first statement is obvious and second and third statements are proved in [16] for domains with smooth and strongly convex domains and in [30], Chapters 2 and 3]. The last statement follows from the implicit function theorem.
5.2 Computation of the Binet–Legendre metric for a Zermelo metric
Since the Funk metric is not backward complete, it follows from Corollary 4.2 that its associated Binet–Legendre metric is incomplete. In this section we provide another proof for the incompleteness. More generally we compute the Binet–Legendre metric for a general Zermelo metric in a domain \(\fancyscript{U}\) and show that it is never complete unless \(\fancyscript{U}= \mathbb {R}^n\).
Observe in particular that since \(u(x)\) belongs to the bounded domain \({\Omega }\) for any \(x \in \fancyscript{U}\), the tensors (21) and (22) are always bounded. This implies in particular that the Binet–Legendre metric of a Zermelo metric in a domain \(\fancyscript{U}\) is bilipschitz equivalent to the Euclidean metric. In particular it is complete if and only if \(\fancyscript{U} = \mathbb {R}^n\).
Remark 5.2
Remark 5.3
5.3 An example of a complete metric with incomplete Binet–Legendre metric
In this subsection we briefly give an example of a Finsler metric that is both forward and backward complete and whose associated Binet–Legendre metric is incomplete, showing that the converse to Corollary 4.2 fails.
The example is given by a Zermelo metric that interpolates between the Funk metric (which is forward complete) and the reverse Funk metric (which is backward complete). It can be built in any bounded convex domain, but we will only describe it in the standard unit ball \(\mathbb {B}^n \subset \mathbb {R}^n\).
5.4 The Hilbert metric and the “affine metric” in a bounded convex domain
We have the following result about the Binet–Legendre metric associated to the Hilbert metric.
Proposition 5.4
The Binet–Legendre metric associated to the Hilbert metric in a bounded convex domain \({\Omega }\) is a complete Riemannian metric. This metric is invariant under the group of projective transformations preserving the domain and it is bilipschitz equivalent to the Hilbert metric.
Proof
The Hilbert metric is clearly reversible and it is not difficult to check from formula (23) that it is complete. The second statement in Theorem 1.1 implies that its associated Binet–Legendre metric is bilipshitz equivalent to the Hilbert metric, in particular it is also complete.
The Hilbert metric is invariant under projective transformations since the distance is expressed in terms of the cross ratio of four aligned points. Using Theorem 2.2 (b), we deduce that the Binet–Legendre metric is also invariant under projective transformations. \(\square \)
Another important projectively invariant metric in a convex domain can be constructed from the solution to some Monge–Ampère equation. It is based on the following statement.
Theorem 5.5
This theorem was first proved in 1974 by Loewner and Nirenberg for the case of smooth, \(2\)dimensional strictly convex domain [19] and in 1977 by Cheng and Yau for the general case [13, 19, 21].
Definition 5.6
Observe that by the strict concavity of \(u\), the metric \(g_{\mathrm {Aff}}\) is positive definite, hence Riemannian. The name “affine metric” has been proposed in relation to the Blaschke theory of affine hypersurfaces, see [7, 20, 28]. The affine metric enjoys the following properties.
Theorem 5.7

The affine metric \(g_{\mathrm {Aff}}\) is complete and invariant under projective transformations leaving the domain \(\fancyscript{U}\) invariant.
 The affine metric \(g_{\mathrm {Aff}} \) is bilipschitz equivalent to the Hilbert metric: there exists a constant \(c\) such that
A proof of the first statement is given in [19], Sections 6 and 9], see also [14, 15]. The second statement is a recent result by Benoist and Hulin [7], Proposition 3.4]. Observe that the completeness of \(g_{\mathrm {Aff}} \) also follows from the second statement, since the Hilbert metric is complete. From the previous theorem and Theorem 1.1 we then have
Corollary 5.8
The Binet–Legendre metric \(g_{\mathrm{BL}}\) associated to the Hilbert metric in a properly convex domain Open image in new window is bilipschitz equivalent to the affine metric \(g_{\mathrm{Aff}}\).
In conclusion, both the Binet–Legendre and the affine metric in a convex domain are complete, invariant under projective transformation and bilipschitz equivalent to the Hilbert metric. Observe however that the construction of the affine metric is based on hard analysis to solve a nonlinear elliptic partial differential equation, so even the existence of such a metric is a nontrivial fact. On the other hand, the Binet–Legendre metric is based on a direct and quite elementary geometric construction. This metric can be effectively computed, at least for sufficiently simple domains, see e.g. [27].
Footnotes
Notes
Acknowledgments
The authors are thankful to Rolf Schneider for useful discussions.
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