1 Introduction

Let M be a smooth connected manifold of dimension \(n\ge 3\) equipped with a sub-Riemannian structure \((\Delta ,g)\) which consists of a totally nonholonomic distribution \(\Delta \) of rank \(m<n\) on M, that is, a smooth subbundle of TM of dimension m generated locally by m smooth vector fields \(X^1, \ldots ,X^m\) satisfying the Hörmander condition

$$\begin{aligned} \text{ Lie } \Bigl \{ X^1, \ldots ,X^m \Bigr \} (x) = T_xM \qquad \forall \,x \in {\mathcal {V}}, \end{aligned}$$

and a smoothly varying inner product g on \(\Delta \). By the Chow-Rashevsky Theorem, M is horizontally path-connected with respect to \(\Delta \); in other words, for every pair of points \(x,y \in M\) there is a horizontal path \(\gamma : [0,1] \rightarrow M\) connecting them, i.e., an absolutely continuous curve \(\gamma : [0,1] \rightarrow M\) with derivative in \(L^2\) satisfying

$$\begin{aligned} {\dot{\gamma }}(t) \in \Delta _{\gamma (t)} \qquad \text{ for } \text{ a.e. } t \in [0,1],\qquad \gamma (0)=x,\quad \gamma (1)=y. \end{aligned}$$

As a consequence, thanks to the metric g on \(\Delta \) that allows to measure the length of horizontal paths, we can define a distance \(d_{SR}\) on M, called the sub-Riemannian distance, which reflects the metric properties of \((\Delta ,g)\). The sub-Riemannian geometry is precisely concerned with the study of the metric and geometric properties of the metric space \((M,d_{SR})\) and its related objects, such as for instance the minimizing horizontal paths between two points. The presence of the so-called singular horizontal paths, which is specific to the sub-Riemannian world (such phenomenon does not occur in Riemannian geometry), is the source of many troubles that include two outstanding problems within sub-Riemannian geometry: the Sard conjecture and the problem of regularity of minimizing geodesics. The purpose of the present paper is to settle these two problems in the case of analytic sub-Riemannian structures in dimension 3. In order to define the notion of singular horizontal paths and to state our results, in the next paragraph we first identify the horizontal paths with the trajectories of a control system. For further details on the notions and results of sub-Riemannian geometry given in the introduction, we refer the reader to Bellaïche’s monograph [7], or to the books by Montgomery [28], by Agrachev, Barilari and Boscain [1], or by the fourth author [36].

Given a sub-Riemannian structure \((\Delta ,g)\) as above, it can be shown that there exist an integer \(k \in [m,m(n+1)]\) and a family of k smooth vector fields \({\mathcal {F}}=\{X^1,\ldots ,X^k\}\) such that

$$\begin{aligned} \Delta _x = \text{ Span } \Bigl \{ X^1(x), \ldots , X^k(x) \Bigr \}\qquad \forall \,x \in M. \end{aligned}$$

For every \(x\in M\), the set of controls \(u=(u_1,\ldots ,u_k) \in L^2([0,1],{\mathbb {R}}^k)\) for which the solution \({\mathbf {x}}(\cdot )={\mathbf {x}}(\cdot \,;x,u)\) to the Cauchy problem

$$\begin{aligned} \dot{{\mathbf {x}}}(t) = \sum _{i=1}^k u_i(t) X^i({\mathbf {x}}(t)) \quad \text{ for } \text{ a.e. } t \in [0,1] \quad \text{ and } \quad {\mathbf {x}}(0)=x \end{aligned}$$

exists over [0, 1] is a nonempty open set \({\mathcal {U}}^x\subset L^2([0,1],{\mathbb {R}}^k)\). By construction, any solution \({\mathbf {x}}(\cdot \,; x,u) : [0,1] \rightarrow M\) with \(u\in {\mathcal {U}}^x\) is a horizontal path in \(\Omega _{\Delta }^{x}\), the set of horizontal paths on [0, 1] starting from x. Moreover, by definition, any path \(\gamma \in \Omega _{\Delta }^{x}\) is equal to \({\mathbf {x}}(\cdot \,;x,u)\) for some \(u\in {\mathcal {U}}^x\) (which is not necessarily unique). Given a point \(x\in M\), the End-Point Mapping from x (associated with \({\mathcal {F}}\) in time 1) is defined as

$$\begin{aligned} \begin{array}{rcl} \text{ E}^{x} : {\mathcal {U}}^{x} &{} \longrightarrow &{} M \\ u &{} \longmapsto &{} {\mathbf {x}}(1;x,u), \end{array} \end{aligned}$$

and it is of class \(C^{\infty }\) on \({\mathcal {U}}^x\) equipped with the \(L^2\)-topology. A control \(u \in {\mathcal {U}}^{x}\subset L^2([0,1], {\mathbb {R}}^k)\) is called singular (with respect to \(E^u\)) if the linear mapping

$$\begin{aligned} D_u\text{ E}^{x} \, : \, L^2 \left( [0,1],{\mathbb {R}}^k\right) \, \longrightarrow T_{\text{ E}^{x}(u)} M \end{aligned}$$

is not onto, that is, if \(\text{ E}^{x}\) is not a submersion at u. Then, a horizontal path \(\gamma \in \Omega _{\Delta }^{x}\) is called singular if it coincides with \({\mathbf {x}}(\cdot \,; x,u)\) for some singular control \(u \in {\mathcal {U}}^x\). It is worth noticing that the property of singularity of a horizontal path is independent of the choice of the family \(X^1, \ldots , X^k\) and of the control u which is chosen to parametrize the path. Moreover, by the Cauchy-Schwarz inequality, all singular horizontal paths have finite length with respect to g. For every \(x\in M\), we denote by \({\mathcal {S}}^{x}_{\Delta }\) the set of singular horizontal paths starting at x and we set

$$\begin{aligned} {\mathcal {X}}^x_{\Delta } := \Bigl \{ \gamma (1) \, \vert \, \gamma \in {\mathcal {S}}^{x}_{\Delta }\Bigr \} \subset M. \end{aligned}$$

By construction, the set \({\mathcal {X}}^x_{\Delta }\) coincides with the set of critical values of a smooth mapping over a space of infinite dimension. In analogy with the classical Sard Theorem in finite dimension (see e.g. [13]), the Sard conjecture in sub-Riemannian geometry asserts the following:

Sard Conjecture: For every \(x\in M\), the set \({\mathcal {X}}^x_{\Delta }\) has zero Lebesgue measure inside M.

The Sard Conjecture cannot be obtained as a straightforward consequence of a general Sard Theorem in infinite dimension, as the latter fails to exist (see for example [6]). This conjecture remains still largely open, except for some particular cases in dimension \(n\ge 4\) (see [24, 28, 37]) and for the 3-dimensional case (where actually a stronger conjecture is expected).

Whenever M has dimension 3, the singular horizontal paths can be shown to be contained inside the so-called Martinet surface \(\Sigma \) (see Sect. 2.1 below for the definition of the Martinet surface) which happens to be a 2-dimensional set. So, in this case, the Sard Conjecture is trivially satisfied. For this reason, in the 3-dimensional case, the meaningful version of the Sard conjecture becomes the following (here and in the sequel, \({\mathcal {H}}^s\) denotes the s-dimensional Hausdorff measure):

Sard Conjecture in dimension 3: For every \(x\in M\), \({\mathcal {H}}^2({\mathcal {X}}^x_{\Delta })=0\).

In fact, whenever M has dimension 3, it can be shown that singular horizontal paths correspond exactly to finite length concatenations of integral curves of the characteristic line foliation, also called the Martinet line field, which is given by the trace of the distribution on the Martinet surface (see Sect. 2.2). In [47], Zelenko and Zhitomirskii proved that, for generic rank-two distributions in dimension 3, a stronger version of the Sard conjecture holds. More precisely they showed that, in the generic setting, the Martinet surface is smooth and the sets \({\mathcal {X}}^x_{\Delta }\) are locally unions of finitely many smooth curves which correspond to stable manifold of singularities of the characteristic line foliation of saddle type. In particular, this implies the generic validity of the Strong Sard Conjecture in dimension 3 (we refer the interested reader to [28] for a statement of Strong Sard Conjectures in higher dimensions):

Strong Sard Conjecture in dimension 3: For every \(x\in M\), the set \({\mathcal {X}}^x_{\Delta }\) has Hausdorff dimension at most 1.

We note that such a result is the best one can hope for. Indeed, if \(y \in {\mathcal {X}}^x_{\Delta }\) with \(y=\gamma (1) \ne x\) for some singular curve \(\gamma \), then \(\gamma (t) \in {\mathcal {X}}^x_{\Delta }\) for any \(t \in [0,1]\) (this follows by reparameterizing \(\gamma \)). Thus, whenever \({\mathcal {X}}^x_{\Delta }\) contains a point \(y \ne x\) then automatically it has at least Hausdorff dimension 1.

As mentioned above, the results in [47] are concerned with generic distributions. Hence, it is natural to investigate what one can say without a genericity assumption, both for the Sard Conjecture and for its Strong version. Recently, in [9] the first and fourth authors proved that the Sard Conjecture in dimension 3 holds whenever at least one of these two conditions holds: either \(\Sigma \) is smooth; or the distribution satisfies some suitable tangency assumption over the set of singularities of \(\Sigma \). Unfortunately, the approach used in [9] strongly relies on the assumptions listed above and cannot be adapted to study the general case, nor to attack the Strong Sard Conjecture. Indeed, a possible pathological case (see Example 1.2 below) illustrates a phenomenon that cannot be handled via the methods in [9].

The aim of the present paper is to show that the description of singular curves given in [47] holds in fact for any analytic distribution in dimension 3. In particular, we shall prove that the Strong Sard Conjecture holds for any analytic distribution.

To state precisely our result, we assume that g is given by the restriction of a complete smooth Riemannian metric to \(\Delta \) and, for every \(x\in M\) and every \(L>0\), we denote by \({\mathcal {S}}^{x,L}_{\Delta ,g}\) the set of \(\gamma \) in \({\mathcal {S}}^{x}_{\Delta }\) with length bounded by L (the length being computed with respect to g). Then we define

$$\begin{aligned} {\mathcal {X}}^{x,L}_{\Delta ,g} := \Bigl \{ \gamma (1) \, \vert \, \gamma \in {\mathcal {S}}^{x,L}_{\Delta ,g}\Bigr \} \subset M. \end{aligned}$$

We observe that since g is complete, the sets \({\mathcal {X}}^{x,L}_{\Delta ,g}\) are compact. Moreover we note that, independently of the metric g, there holds

$$\begin{aligned} {\mathcal {X}}^{x}_{\Delta }=\bigcup _{L\in {\mathbb {N}}}{\mathcal {X}}^{x,L}_{\Delta ,g}. \end{aligned}$$

Our first result settles the Strong Sard Conjecture in the analytic case. Here and in the sequel, we call singular horizontal curve any set of the form \(\gamma ([0,1])\), where \(\gamma :[0,1] \rightarrow M\) is a singular horizontal path. Furthermore, we call semianalytic curve in M any semianalytic compact connected subset of M of Hausdorff dimension at most 1 (see “Appendix B”). It is well-known that semianalytic curves admit a nice stratification into 0-dimensional and 1-dimensional pieces, see Lemma B.3.

Theorem 1.1

Let M be an analytic manifold of dimension 3 and \(\Delta \) a rank-two totally nonholonomic analytic distribution on M. Then any singular horizontal curve is a semianalytic curve in M. Moreover, if g is a complete smooth Riemannian metric on M then, for every \(x\in M\) and every \(L>0\), the set \({\mathcal {X}}^{x,L}_{\Delta ,g}\) is a finite union of singular horizontal curves, so it is a semianalytic curve. In particular, for every \(x\in M\), the set \({\mathcal {X}}^{x}_{\Delta }\) is a countable union of semianalytic curves and it has Hausdorff dimension at most 1.

The proof of Theorem 1.1 uses techniques from resolution of singularities, an analytic argument based on the Stokes’ Theorem, and a regularity result for transition maps of hyperbolic singularities. Some of these techniques can be generalized to higher dimensions and some seem, at the moment, restricted to the three dimensional case. Although the first two were already present in [9], what was done there is not sufficient to attack the general problem considered in this paper. Hence, here we have to further develop them, as explained below.

The techniques from resolution of singularities can be applied thanks to a compactness argument which is also well-adapted to higher dimensions. A crucial step in the proof of Theorem 1.1 is to show that the so-called monodromic convergent transverse-singular trajectories (see Definitions 2.7 and 2.9 ) necessarily have infinite length and therefore cannot correspond to singular horizontal paths. This type of trajectories is a generalization of the singular curves that were investigated by the first and fourth authors at the end of the Introduction of [9], but there is an extra transcendental behavior illustrated in the following:

Example 1.2

Consider the case when the Martinet surface \(\Sigma \) is stratified by a singleton \(\{x\}\), a stratum \(\Gamma \) of dimension 1, and two strata of dimension 2 as in Fig. 1. This occurs, for example, for

$$\begin{aligned} X^1= \partial _{x_1}, \qquad X^2 = \partial _{x_2} + \bigl (x_1^3 - x_1 x_2^2(x_3^2-x_2^2)\bigr )\partial _{x_3}, \end{aligned}$$

in which case we get

$$\begin{aligned}&\Sigma = \Bigl \{ 3x_1^2-x_2^2(x_3^2-x_2^2)=0 \Bigr \}, \qquad \Gamma = \Bigl \{x_1=x_2=0, x_3\ne 0\Bigr \}, \\&x = (0,0,0). \end{aligned}$$

Then, from each z in \(\Gamma \) there exist uncountably many distinct integral curves of the characteristic line foliation between x and z. This is in sharp contrast with the study in [9], where the authors only needed to consider a unique such integral curve between x and z.

Following the situation illustrated in the previous example, each z in \(\Gamma \) gives rise to an uncountable number of monodromic convergent transverse-singular trajectories. Fix one of them, and call it \(\gamma ^z\). We note that, if \(\gamma ^z\) had finite length, then it would correspond to a singular horizontal path from x to z. In particular, since the area swept out by the curves \(\gamma ^z\) (as z varies transversally) is 2-dimensional, if the curves \(\gamma ^z\) had finite length then the set \({\mathcal {X}}^x_{\Delta }\) would have dimension 2 and the example of Fig. 1 would contradict the Sard Conjecture. As we shall see this is not the case since all the curves \(\gamma ^z\) have infinite length, so they do not correspond to singular horizontal curves starting from x. In [9], the authors had to handle a similar problem and in that case the lengths of the singular trajectories under consideration were controlled “by hand” under the assumption that \(\Sigma \) was smooth. Here we combine the resolution of singularities with a recent (highly nontrivial) regularity result for transition maps of hyperbolic singularities of planar vector fields due to Speissegger [38] (used in Lemma 4.14), following previous works by Ilyashenko [20], and an analytic argument (based on the Stokes Theorem) in the cotangent bundle. The latter argument, valid in any dimension, generalizes the divergence-type argument used in [9], that is insufficient here because of the phenomenon illustrated in Example 1.2.

Fig. 1
figure 1

Monodromic convergent transverse-singular trajectories

Another important step in the proof of Theorem 1.1 consists in describing the remaining possible singular horizontal paths. We show that the sets \({\mathcal {X}}^{x,L}_{\Delta ,g}\) consist of a finite union of semianalytic curves, which are projections of either characteristic or dicritical orbits of analytic vector fields by an analytic resolution map of the Martinet surface. A key point in the proof is the fact that the singularities of those vector fields are of saddle type, which holds because of a divergence-type restriction, in the same spirit as the one studied in [9].

Theorem 1.1 allows us to address one of the main open problems in sub-Riemannian geometry, namely the regularity of length-minimizing curves. Given a sub-Riemannian structure \((\Delta ,g)\) on M, we recall that a minimizing geodesic from x to y in M is a horizontal path \(\gamma :[0,1] \rightarrow M\) which minimizes the energyFootnote 1 (and so the length) among all horizontal paths joining x to y. It is well-known that minimizing geodesics may be of two types, namely:

- either normal, which means that they are the projection of a trajectory (called normal extremal) of the so-called sub-Riemannian Hamiltonian vector fieldFootnote 2 in \(T^*M\);

- or singular, in which case they are given by the projection of an abnormal extremal (cf. Proposition A.1).

Note that a geodesic can be both normal and singular. In addition, as shown by Montgomery [27] in the 1990s, there exist minimizing geodesics which are singular but not normal. While a normal minimizing geodesic is smooth (being the projection of a trajectory of a smooth dynamical system), a singular minimizing geodesic which is not normal might be nonsmooth. In particular it is widely open whether all singular geodesics (which are always Lipschitz) are of class \(C^1\). We refer the reader to [30, 37, 42] for a general overview on this problem, to [4, 22, 23, 25, 29, 41] for some regularity results on singular minimizing geodesics for specific types of sub-Riemannian structures, and to [14, 31, 39] for partial regularity results for general (possibly analytic) SR structures.

In our setting, the main result of [14] can be combined with our previous theorem to obtain the first \(C^1\) regularity result for singular minimizing geodesics in arbitrary analytic 3-dimensional sub-Riemannian structures. More precisely, we can prove the following result:

Theorem 1.3

Let M be an analytic manifold of dimension 3, \(\Delta \) a rank-two totally nonholonomic analytic distribution on M, and g a complete smooth sub-Riemannian metric on \(\Delta \). Let \(\gamma :[0,1] \rightarrow M\) be a singular minimizing geodesic. Then \(\gamma \) is of class \(C^1\) on [0, 1]. Furthermore \(\gamma ([0,1])\) is semianalytic, and therefore it consists of finitely many points and finitely many analytic arcs.

Theorem 1.3 follows readily from Theorem 1.1, the regularity properties of semianalytic curves recalled in Appendix B, and a breakthrough result of Hakavuori and Le Donne [14] on the absence of corner-type singularities of minimizing geodesics. Their theoremFootnote 3 asserts that if \(\gamma :[0,1] \rightarrow M\) is a minimizing geodesic which is differentiable from the left and from the right at \(t=1/2\), then it is differentiable at \(t=1/2\). By Theorem 1.1 and Lemma B.3, if \(\gamma :[0,1]\rightarrow M\) is a singular minimizing geodesic, then it is piecewise \(C^1\) and so left and right differentiable everywhereFootnote 4 (see Remark B.4 (ii)). Then the main result of [14] implies our Theorem 1.3.

The paper is organized as follows: In Sect. 2, we introduce some preliminary notions (such as the ones of Martinet surface and characteristic line foliation), and introduce the concepts of characteristic and monodromic transverse-singular trajectories. Section 3 is devoted to the proof of Theorem 1.1, which relies on two fundamental results: first Proposition 3.1, which provides a clear description of characteristic orbits, and second Proposition  3.3, which asserts that convergent monodromic transverse-singular trajectories have infinite length and so allow us to rule out monodromic horizontal singular paths. The proofs of Proposition 3.1 and of a part of Proposition 3.3 (namely, Proposition 3.7) are postponed to Sect. 4. That section contains results on the divergence of vector fields and their singularities, a major theorem on resolution of singularities (Theorem 4.7), and the proofs mentioned before. Finally, the four appendices collect some basic results on singular horizontal paths, semianalytic sets, Hardy fields, and resolution of singularities of analytic surfaces and reduction of singularities of planar vector fields.

In the rest of the paper, M is an analytic manifold of dimension 3, \(\Delta \) a rank-two totally nonholonomic analytic distribution on M, and g a complete smooth sub-Riemannian metric on \(\Delta \).

2 Characteristic line foliation and singular trajectories

2.1 The Martinet surface

The Martinet surface \(\Sigma \) associated to \(\Delta \) is defined as

$$\begin{aligned} \Sigma := \Bigl \{ x\in M \, \vert \, \Delta _x + [\Delta ,\Delta ]_x\ne T_xM \Bigr \}, \end{aligned}$$

where \([\Delta ,\Delta ]\) is the (possibly singularFootnote 5) distribution given by

$$\begin{aligned} {[}\Delta ,\Delta ]_x := \Bigl \{ [X,Y](x) \, \vert \, X,Y \text{ smooth } \text{ local } \text{ sections } \text{ of } \Delta \Bigr \}. \end{aligned}$$

We recall that the singular curves for \(\Delta \) are those horizontal paths which are contained in the Martinet surface \(\Sigma \) (see e.g. [36, Example 1.17 p. 27]).

Remark 2.1

(Local Model) Locally, we can always suppose that M coincides with a connected open subset \({\mathcal {V}}\subset {\mathbb {R}}^3\), and that \(\Delta \) is everywhere generated by global analytic sections. More precisely, we can choose one of the following equivalent formulations:

  1. (i)

    \(\Delta \) is a totally nonholonomic distribution generated by an analytic 1-form \(\mu \) (that is, a section in \( \Omega ^1(M)\)) and

    $$\begin{aligned} \mu \wedge d\mu = h \cdot \omega _M, \end{aligned}$$

    where h is an analytic function defined in M whose zero locus defines the Martinet surface (that is, \(\Sigma = \{p\in M\,\vert \, h(p)=0\}\)) and \(\omega _M\) is a local volume form.

  2. (ii)

    \(\Delta \) is generated by two global analytic vector fields \(X^1\) and \(X^2\) which satisfy the Hörmander condition, and \(\Delta +[\Delta ,\Delta ]\) is generated by \(X^1\), \(X^2\), and \([X^1,X^2]\). Also, up to using the Flow-box Theorem and taking a linear combination of \(X^1\) and \(X^2\), we can suppose that

    $$\begin{aligned} X^1 = \partial _{x_1}, \quad X^2 = \partial _{x_2} + A(x) \, \partial _{x_3}, \quad [X^1,X^2] = A_1(x) \, \partial _{x_3}, \end{aligned}$$

    where \((x_1,x_2,x_3)\) is a coordinate system on M, and \(A_1(x) := \partial _{x_1} A(x)\). In this case, the zero locus of \(A_1(x)\) defines the Martinet surface (that is, \(\Sigma = \{p\in M\,\vert \, A_1(p)=0\}\)).

Since M and \(\Delta \) are both analytic, the Martinet surface is an analytic set (see e.g. [16, 33, 40]), and moreover the fact that \(\Delta \) is totally nonholonomic implies that \(\Sigma \) is a proper subset of M of Hausdorff dimension at most 2. Furthermore, we recall that \(\Sigma \) admits a global structure of reduced and coherent real-analytic space,Footnote 6 which we denote by \({\mathscr {M}}\) (see [9, Lemma C.1]). We refer the reader to [9, Appendix C] for the precise definitions.

2.2 Characteristic line foliation

The local models given in Remark 2.1 have been explored, for example, in [47] and later in [9, eqs. (2.2) and (3.1)] in order to construct a locally defined vector field whose dynamics characterizes singular horizontal paths at almost every point (cf. Lemma 2.2(ii) below). Since \(\Sigma \) admits a global structure of coherent analytic space, these local constructions yield a globally defined (singularFootnote 7) line foliation \({\mathscr {L}}\) (in the sense of Baum and Bott [5, p. 281]), which we call characteristic line foliation (following Zelenko and Zhitomirskii [47, Section 1.4]). More precisely, we have:

Lemma 2.2

(Characteristic line foliation) The set

$$\begin{aligned} S := \Bigl \{ p\in \Sigma \,\vert \, p \in Sing({\mathscr {M}}) \text { or }T_p\Sigma \subset \Delta _p\Bigr \} \end{aligned}$$

is analytic of dimension less than or equal to 1, and there exists a line foliation \({\mathscr {L}}\) defined over \(\Sigma \) such that:

  1. (i)

    The line foliation \({\mathscr {L}}\) is regular everywhere in \(\Sigma \setminus S\).

  2. (ii)

    If a horizontal path \(\gamma :[0,1] \rightarrow M\) is singular with respect to \(\Delta \), then its image \(\gamma ([0,1])\) is contained in \(\Sigma \) and it is tangent to \({\mathscr {L}}\) over \(\Sigma \setminus S\), that is

    $$\begin{aligned} \gamma (t) \in \Sigma \setminus S\quad \Longrightarrow \quad {\dot{\gamma }}(t) \in {\mathscr {L}}_{\gamma (t)} \qquad \text{ for } \text{ a.e. } t \in [0,1]. \end{aligned}$$

Proof of Lemma 2.2

S is analytic because \(\Delta \) and \(\Sigma \) are both analytic. The total nonholonomicity of \(\Delta \) implies that S has dimension smaller than or equal to 1, see Lemma 2 of [9].

Let \(i: {\mathscr {M}} \rightarrow M\) be the inclusion. Since \(\Delta \) is a coherent sub-sheaf of \(\Omega ^1(M)\) (cf. Remark 2.1), the pull-back \({\mathscr {L}} := i^{*}(\Delta )\) is also a coherent sub-sheaf of \(\Omega ^1({\mathscr {M}})\). Furthermore, since \(\Delta \) is everywhere locally generated by one section, so is \({\mathscr {L}}\). It is thus enough to study \({\mathscr {L}}\) locally.

Fix a point \(p \in \Sigma \). If \(\Sigma \) has dimension smaller than or equal to 1 at p, then \(\Sigma = S\) (because, in contrast to the definition of singularities of analytic sets, the singular set of the analytic space \({\mathscr {M}}\) contains all the points where the differential of its generator is zero) in a neighborhood of p and the claim of lemma holds trivially. If \(\Sigma \) has dimension 2 at p, then \({\mathscr {L}}\) generates a line foliation over a neighborhood of p in \(\Sigma \).

To prove (i) fix a point p where \({\mathscr {M}}\) is smooth (in particular, \(\Sigma \) is smooth as a subset of M) and \(\Delta _p + T_p\Sigma = T_pM\). Then there exists a local coordinate system \((x_1,x_2,x_3)\) centered at p so that \(\Sigma = \{x_3=0\}\) and \(\mu = dx_1 + A(x) dx_2\), therefore \({\mathscr {L}}\) is regular at p.

Finally, assertion (ii) follows from the above formulae in local coordinates and the characterization of singular horizontal paths given in Proposition A.2. \(\square \)

Remark 2.3

(Characteristic vector-field) We follow [9, eq. (3.1)]. In the notation of Remark 2.1(ii), let h be a reduced analytic function whose zero set is equal to the Martinet surface \(\Sigma \). Consider the vector-field

$$\begin{aligned} {\mathcal {Z}} := X^1(h) X^2 - X^2(h)X^1. \end{aligned}$$

Then the restriction of \({\mathcal {Z}}\) over \(\Sigma \) is a generator of the line foliation \({\mathscr {L}}\).

2.3 Stratification of the Martinet surface

Every real analytic set X (or in general every real analytic space) admits a semianalytic stratification into non-singular analytic strata, i.e., a locally finite partition by locally closed analytic submanifolds of X that are, at the same time, semianalytic subsets of X, see for instance [26, Lemme 5, page 69]. (We recall the definition of a semianalytic set in “Appendix B”, where we describe some geometric properties of semianalytic curves. For more details on semianalytic sets, we refer the reader to [11, 26].) Furthermore, it is always possible to choose such stratification Whitney regular, i.e., that satisfies Whitney regularity conditions (a) and (b), cf. [26, Section 25] or [43]. For our purpose we need a stratification of the Martinet surface \(\Sigma \) that, in addition, is compatible with the distribution \(\Delta \) in the sense of the following lemma.

Lemma 2.4

(Stratification of \(\Sigma \)) There exists a regular semianalytic stratification of \(\Sigma \),

$$\begin{aligned} \Sigma = \Sigma ^0 \cup \Sigma ^1_{tr} \cup \Sigma ^1_{tan} \cup \Sigma ^2, \end{aligned}$$

which satisfies the following properties:

  1. (i)

    \(S = \Sigma ^0 \cup \Sigma ^1_{tr} \cup \Sigma ^1_{tan}\) (cf. Lemma 2.2).

  2. (ii)

    \(\Sigma ^0\) is a locally finite union of points.

  3. (iii)

    \(\Sigma ^1_{tan}\) is a locally finite union of 1-dimensional strata with tangent spaces everywhere contained in \(\Delta \) (that is, \(T_p\Sigma ^1_{tan} \subset \Delta _p\) for all \(p\in \Sigma ^1_{tan}\)).

  4. (iv)

    \(\Sigma ^1_{tr}\) is a locally finite union of 1-dimensional strata transverse to \(\Delta \) (that is, \(T_p\Sigma ^1_{tr} \oplus \Delta _p = T_pM\) for all \(p\in \Sigma ^1_{tr}\));

  5. (v)

    \(\Sigma ^2\) is a locally finite union of 2-dimensional strata transverse to \(\Delta \) (that is, \(T_p\Sigma ^2 + \Delta _p = T_pM\) for all \(p\in \Sigma ^2\)).

Moreover, every 1-dimensional stratum \(\Gamma \) satisfies the following local triviality property: For each point p in \(\Gamma \) there exists a neighborhood \({\mathcal {V}} \) of p in M such that \(\Sigma ^2 \cap {\mathcal {V}}\) is the disjoint union of finitely many 2-dimensional analytic submanifolds \(\Pi _1, \ldots , \Pi _r\) (\(\Sigma ^2 \cap {\mathcal {V}}\) could be empty) such that, for each i, \(\Pi _i \cup \Gamma \) is a closed \(C^1\)-submanifold of \({\mathcal {V}}\) with boundary, denoted by \({{\overline{\Pi }}}_i\), with \(\Gamma = \partial {{\overline{\Pi }}}_i\).

Proof of Lemma 2.4

By Lemma 2.2, the set \(\Sigma ^2:= \Sigma \setminus S\) is smooth and the line foliation \({\mathscr {L}}\) restricted to \(\Sigma ^2\) is non-singular. Now, we recall that S is an analytic set of dimension at most 1, so it admits a semianalytic stratification \(S_0 \cup S_1\), where \(S_0\) is a locally finite union of points and \(S_1\) is a locally finite union of analytic curves (i.e., connected non-singular semianalytic sets of dimension 1). Moreover, by [26] or [43], we may assume that \(\Sigma ^2\), \(S_1\), and \(S_0\), is a Whitney regular stratification of \(\Sigma \).

Fixed a 1-dimensional stratum \(\Gamma \) in \(S_1\), its closure \({\overline{\Gamma }}\) is a closed semianalytic set. Note that condition \(T_p\Gamma \subset \Delta _p\) is semianalytic (that is, it is locally given in terms of analytic equations and inequalities). Therefore, up to removing from \(\Gamma \) a locally finite number of points, we can assume that:

  • either \(\Delta _p\) contains \(T_p{\Gamma }\) for every \(p\in \Gamma \);

  • or \(\Delta _p\) transverse to \(T_p\Gamma \) for every \(p\in \Gamma \).

In other words, up to adding a locally finite union of points to \(S_0\), and correspondingly refining \(S_1\), we can suppose that the above dichotomy is constant along connected components of \(S_1\). Then, it suffices to define as \(\Sigma ^1_{tan}\) the union of all connected components \(\Gamma \) of \(S_1\) where \(T_p\Gamma \subset \Delta _p\) for every \(p \in \Gamma \), and as \(\Sigma ^1_{tr}\) the union of all connected components where the transversality condition \(\Delta _p \oplus T_p\Gamma = T_p M\) holds.

The last claim of Lemma follows from [43, Proposition p. 342]. \(\square \)

Remark 2.5

(Puiseux with parameter) As follows from [35, Proposition 2] or [43] (proof of Proposition p.342), we may require in Lemma 2.4 the following stronger version of local triviality of \(\Sigma \) along \(\Gamma \): Given \(p \in \Gamma \), there exist a positive integer k and a local system of analytic coordinates \(x=(x_1,x_2,x_3)\) at p such that \(\Gamma = \{x_2=x_3=0\}\) and each \(\Pi _i\) is the graph \(x_3= \varphi _i (x_1,x_2)\), defined locally on \(\{(x_1,x_2)\, \vert \, x_2\ge 0 \}\) (or \(\{(x_1,x_2)\, \vert \, x_2\le 0 \}\)), such that \(\varphi _i \) is \(C^1\) and the mapping \((t,x_1)\mapsto \varphi _i (x_1, t^k)\) is analytic.

One may remark that the latter two conditions imply that, in fact, \(\varphi _i\) is of class \(C^{1,1/k}\). Indeed, we may write for \(x_2\ge 0\)

$$\begin{aligned} \varphi _i (x_1,x_2) = \sum _{i \in {\mathbb {N}},j\in {\mathbb {N}}} a_{i,j} x_1^i x_2 ^{j/k}. \end{aligned}$$

The fact that \(\varphi _i \) is \(C^1\) implies that in this sum \(j=0\) or \(j\ge k\). Therefore the derivative \(\partial \varphi _i/ \partial x_2\) is Hölder continuous with exponent 1/k.

By the local triviality property stated in Lemma 2.4 and by Remark 2.5, the restriction of \(\Delta \) to a neighborhood of a point of \(\Sigma ^1_{tr}\) satisfies the following property (we recall that M is equipped with a metric g):

Fig. 2
figure 2

\({\mathcal {X}}^{p,\epsilon }_{\Delta ,g}\) for \(p\in \Sigma _{tr}^1\) and \(\epsilon >0\) small

Lemma 2.6

(Local triviality of \(\Delta \) along \(\Sigma _{tr}^1\)) Let \(\Gamma \) be a 1-dimensional stratum in \(\Sigma ^1_{tr}\) and let \(p \in \Gamma \) be fixed. Then the following properties hold:

  1. (i)

    There exists a neighborhood \({\mathcal {V}}\) of p and \(\delta >0\) such that, for every point \(q\in {\mathcal {V}} \cap \Sigma _{tr}^1\) and every injective singular horizontal path \(\gamma :[0,1]\rightarrow \Sigma \) such that \(\gamma (0)=q\), \(\gamma (1) \in \Sigma _{tr}^1\), and \(\gamma ((0,1)) \subset \Sigma ^2\), the length of \(\gamma \) is larger than \(\delta \).

  2. (ii)

    The image of a singular horizontal path \(\gamma : [0,1] \rightarrow M\) such that \(\gamma ([0,1)) \subset \Sigma ^{2}\) and \(\gamma (1)\in \Sigma _{tr}^1\) is semianalytic.

In particular, if \({\mathcal {V}} \) is a neighborhood of p in M such that \(\Sigma ^2 \cap {\mathcal {V}}\) is the disjoint union of the 2-dimensional analytic submanifolds \(\Pi _1, \ldots , \Pi _r\) as in Lemma 2.4, then for \(\epsilon >0\) small enough there are singular horizontal paths \(\gamma _1, \ldots , \gamma _r :[0,1] \rightarrow \Sigma \), with \(\gamma _i(0)=p\) and \(\gamma _i((0,1]) \subset \Pi _i\) for \(i=1, \ldots ,r\), such that

$$\begin{aligned} {\mathcal {X}}^{p,\epsilon }_{\Delta ,g} = \bigcup _{i=1}^r \gamma _i\left( [0,1]\right) \end{aligned}$$

(see Fig. 2), where \({\mathcal {X}}^{p,\epsilon }_{\Delta ,g}\) is defined in (1.1).

Proof of Lemma 2.6

The lemma follows readily from the following observation. Let \(x=(x_1,x_2,x_3)\) denote the system of coordinates at p introduced in Remark 2.5. Suppose that the distribution \(\Delta \) is locally defined by the 1-form \(\mu \) as in Remark 2.1. Then the pull-back of \(\mu \) on \(\Pi _i\) by the map \((x_1,t) \rightarrow (x_1,t^k, \varphi _i (x_1, t^k))\) is an analytic 1-form: \(\mu _i= a(x_1,t) dx_1 + b(x_1,t) dt\). The condition of transversality of \(\Delta \) and \(\Gamma \) at p means \(a(0,0) \ne 0\) and therefore the integral curves of \(\Delta _i\) (that is, the singular horizontal paths of \(\Delta \) in \(\Pi _i\)) are uniformly transverse to \(\Gamma \) in a neighborhood of p. \(\square \)

It remains now to introduce some definitions related to singular horizontal paths or more precisely singular trajectories (i.e., trajectories of the characteristic line foliation) converging to the set

$$\begin{aligned} {\widetilde{\Sigma }} := \Sigma ^0 \cup \Sigma ^1_{tan}. \end{aligned}$$

This is the purpose of the next section.

2.4 Characteristic and monodromic transverse-singular trajectories

We restrict our attention to a special type of trajectories of the characteristic foliation \({\mathscr {L}}\).

Definition 2.7

(Convergent transverse-singular trajectory) We call transverse-singular trajectory any absolutely continuous path \(\gamma :[0,1) \rightarrow \Sigma \) such that

$$\begin{aligned} {\dot{\gamma }}(t) \in {\mathscr {L}}_{\gamma (t)} \qquad \text{ for } \text{ a.e. } t \in [0,1), \end{aligned}$$


$$\begin{aligned} \gamma (t) \in \Sigma ^2 \cup \Sigma ^1_{tr} \qquad \forall \,t \in [0,1). \end{aligned}$$

Moreover, we say that \(\gamma \) is convergent if it admits a limit in \({\widetilde{\Sigma }}=\Sigma ^1_{tan} \cup \Sigma ^0\) as \(t\rightarrow 1^-\).

We are going to introduce a dichotomy between two types of convergent transverse-singular trajectories which is inspired by the following well-known result (see [21, Theorem 9.13] and [21, Definitions 9.4 and 9.6]):

Proposition 2.8

(Topological dichotomy for planar analytic vector-fields) Let \({\mathcal {Z}}\) be an analytic vector field defined over an open neighborhood U of the origin 0 in \({\mathbb {R}}^2\), and suppose that 0 is a singular point of \({\mathcal {Z}}\). Given a regular orbit \(\gamma (t)\) of \({\mathcal {Z}}\) converging to 0, we have:

  1. (i)

    either \(\gamma \) is a characteristic orbit, that is, the secant curve \(\psi (t) := \gamma (t) /|\gamma (t)| \in {\mathbb {S}}^{1}\) has a unique limit point;

  2. (ii)

    or \(\gamma \) is a monodromic orbit, that is, there exists an analytic section \(\Lambda \) of the vector-field \({\mathcal {Z}}\) at 0 (that is, \(\Lambda \) is a connected segment whose boundary contains 0 and the vector field \({\mathcal {Z}}\) is transverse to \(\Lambda \) everywhere outside of 0) such that \(\gamma \cap \Lambda \) is the disjoint union of an infinite number of points.

Here is our definition.

Definition 2.9

(Characteristic and monodromic convergent transverse-singular trajectories) Let \(\gamma : [0,1) \rightarrow \Sigma \) be a convergent transverse-singular trajectory such that \({\bar{y}}:=\lim _{t\rightarrow 1} \gamma (t)\) belongs to \({\widetilde{\Sigma }}\) (see (2.2)). Then we say that:

  1. (i)

    \(\gamma \) is monodromic if there exists a section \(\Lambda \subset \Sigma \) of \({\mathscr {L}}\) at \({\bar{y}}\) (that is, \(\Lambda \) is a connected 1-dimensional semianalytic manifold with boundary contained in \(\Sigma \), whose boundary contains \({\bar{y}}\) and such that \(\Lambda \setminus \{{\bar{y}}\} \subset \Sigma ^2\cup \Sigma ^1_{tr}\) is everywhere transverse to \({\mathscr {L}}\)) such that \(\gamma ([0,1)) \cap \Lambda \) is the disjoint union of infinitely many points. In addition, we say that \(\gamma \) is final if \(\gamma ([0,1)) \cap \Sigma ^1_{tr}\) is empty or infinite. In the latter case, we may choose as \(\Lambda \) a branch of \(\Sigma ^1_{tr}\).

  2. (ii)

    \(\gamma \) is characteristic if it is not monodromic.

From now on, we call monodromic (resp. characteristic) trajectory any convergent transverse-singular trajectory with a limit in \({\widetilde{\Sigma }}\) which is monodromic (resp. characteristic). The next section is devoted to the study of characteristic and monodromic trajectories, and to the proof of Theorem 1.1.

3 Proof of Theorem 1.1

The proof of Theorem 1.1 proceeds in three steps. Firstly, we describe some properties of regularity and finiteness satisfied by the characteristic trajectories. Secondly, we rule out monodromic trajectories as possible horizontal paths starting from the limit point. Finally, combining all together, we are able to describe precisely the singular horizontal curves and the sets of the form \({\mathcal {X}}^{x,L}_{\Delta ,g}\) (see (1.1)).

3.1 Description of characteristic trajectories

The following result is a consequence of the results on resolution of singularities stated in Theorem 4.7 and the fact that the characteristic trajectories correspond, in the resolution space, to characteristics of an analytic vector field with singularities of saddle type.

Proposition 3.1

Let \(\Sigma ^0\) and \({{\widetilde{\Sigma }}}\) be as in Lemma 2.4 and (2.2). There exist a locally finite set of points \({\widetilde{\Sigma }}^0\), with \(\Sigma ^0 \subset {\widetilde{\Sigma }}^0\subset {\widetilde{\Sigma }}\), such that the following properties hold:

  1. (i)

    If \(\gamma : [0,1) \rightarrow \Sigma \) is a convergent transverse-singular trajectory such that \({\bar{y}}:=\lim _{t\rightarrow 1} \gamma (t)\) belongs to \({\widetilde{\Sigma }}\) then \({\bar{y}}\) belongs to \( {\widetilde{\Sigma }}^0\). Moreover, if \(\gamma \) is characteristic then \(\gamma ([0,1))\) is semianalytic and there is \({\bar{t}} \in [0,1)\) such that \(\gamma ([{\bar{t}},1)) \subset \Sigma ^2\).

  2. (ii)

    For every \({\bar{y}} \in {\widetilde{\Sigma }}^{0}\) there exist only finitely many (possibly zero) characteristic trajectories converging to \({\bar{y}}\) and all of them are semianalytic curves.

The proof of Proposition 3.1 is given in Sect. 4.3, as a consequence of Theorem 4.7.

Remark 3.2

(On Proposition 3.1and its proof)

  1. (i)

    There exist elementary proofs of Proposition 2.8. However, for proving Proposition 3.1(ii) we cannot use the topological simplicity of the plane. In this context it is natural to use resolution of singularities (cf. [21, Proof of Theorem 9.13]).

  2. (ii)

    Proposition 3.1(ii) is specific to characteristic line foliations, and does not hold for arbitrary line foliations over a surface. In our situation we can show that there exists a (locally defined) vector field which generates the characteristic foliation \({\mathscr {L}}\) and whose divergence is controlled by its coefficients (see Sect. 4.1, cf. [9, Lemmas 2.3 and 3.2]). This guarantees that, after resolution of singularities, all singular points of the pull back of \({\mathscr {L}}\) are saddles (see Theorem 4.7(II), cf. Lemma 4.3).

3.2 Monodromic trajectories have infinite length

The main objective of this subsection is to prove the following crucial result:

Proposition 3.3

(Length of monodromic trajectories) The length of any monodromic trajectory is infinite.

Remark 3.4

If we assume that the distribution \(\Delta \) is generic (with respect to the \({\mathcal {C}}^{\infty }\)-Whitney topology), then the Martinet surface is smooth and the above result corresponds to [47, Lemma 2.1].

The proof of Proposition 3.3 is done by contradiction. The first step consists in showing that if \(\gamma \) has finite length, then every monodromic trajectory which is “topologically equivalent” to \(\gamma \) (see Definition 3.5 below) also has finite length (see Proposition 3.7 below). Hence, as discussed in the introduction, the assumption of finiteness on the length of \(\gamma \) implies that \({\mathcal {X}}^{{\bar{y}}}_{\Delta }\) has positive 2-dimensional Hausdorff measure (cf. Lemma 3.6). Then, the second step consists in using an analytic argument based on Stokes’ Theorem to obtain a contradiction.

Let us consider a monodromic trajectory \(\gamma :[0,1) \rightarrow \Sigma \) with limit \({\bar{y}} \in {\widetilde{\Sigma }}\) and assume that \(\gamma \) is injective and final (cf. Definition 2.9(i)), and that its image is contained in a neighborhood \({\mathcal {V}}\) of \({\bar{y}}\) where the line foliation \({\mathscr {L}}\) is generated by a vector-field \({\mathcal {Z}}\) (see Remark 2.3). Denote by \(\varphi ^{{\mathcal {Z}}}_s(x)\) the flow associated to \({\mathcal {Z}}\) with time s and initial condition \(x\in {\mathcal {V}} \cap \Sigma \), by \(\Lambda \) a fixed section as in Definition 2.9, and by \(d^{\Lambda }: \Lambda \rightarrow {\mathbb {R}}\) the function which associates to each point \(p \in \Lambda \) the length of the half-arc contained in \(\Lambda \) which joins p to \({\bar{y}}\) (we may also assume that \(\Lambda \cap {\mathcal {V}}\) is a curve connecting \({\bar{y}}\) to a point of the boundary of \({\mathcal {V}}\)). Moreover assume that \(\gamma (0)\) belongs to \(\Lambda \). By monodromy, there exists an infinite increasing sequence \(\{t_k^{\gamma }\}_{k\in {\mathbb {N}}}\) in [0, 1) with \(t_0^{\gamma }=0\) such that

$$\begin{aligned} \gamma (t) \in \Lambda \quad \text{ if } \text{ and } \text{ only } \text{ if } \quad t=t_k^{\gamma } \quad \text{ for } \text{ some } k \in {\mathbb {N}}\end{aligned}$$


$$\begin{aligned} \lim _{k \rightarrow \infty } t_k^{\gamma }= 1. \end{aligned}$$

We shall introduce a sequence of Poincaré mappings adapted to \(\gamma \), to do so we need to distinguish two cases, depending on whether the set \(\gamma ([0,1)) \cap \Sigma _{tr}^1\) is finite or not. Note that, if \(\gamma ([0,1)) \cap \Sigma _{tr}^1\) is a finite set, then up to restricting \(\gamma \) to an interval of the form \([t_0,1)\) for some \(t_0\in [0,1)\), we can assume that \(\gamma ([0,1)) \cap \Sigma _{tr}^1=\emptyset \). Hence the two cases to analyze are the case where \(\gamma ([0,1)) \cap \Sigma _{tr}^1\) is empty and the case where \(\gamma ([0,1)) \cap \Sigma _{tr}^1\) is infinite (Figs. 3, 4).

Fig. 3
figure 3

The case \(\gamma ([0,1)) \cap \Sigma _{tr}^1=\emptyset \)

First case: \(\gamma ([0,1)) \cap \Sigma ^1_{tr} = \emptyset \).

This is the classical case where we can consider the Poincaré first return map from \(\Lambda \) to \(\Lambda \) (see e.g. [21, Definition 9.8]). By a Poincaré-Bendixon type argument, up to shrinking \({\mathcal {V}}\) and changing the orientation of \({\mathcal {Z}}\) we may assume that the mapping

$$\begin{aligned} T^{\Lambda } \, : \, \Lambda \cap {\mathcal {V}} \, \longrightarrow \, \Lambda \cap {\mathcal {V}} \end{aligned}$$

which assigns to each \(p\in \Lambda \cap {\mathcal {V}}\) the first point \(\varphi ^{{\mathcal {Z}}}_{t}(p) \in \Lambda \) with \(t>0\) and \(d^{\Lambda }(\varphi ^{{\mathcal {Z}}}_{t}(p))<d^{\Lambda }(p)\) is well-defined, continuous, and satisfies

$$\begin{aligned} T^{\Lambda }(\gamma (t_k^{\gamma })) = \gamma (t_{k+1}^{\gamma }) \qquad \forall \,k \in {\mathbb {N}}\end{aligned}$$


$$\begin{aligned}&d^{\Lambda }(p_1)< d^{\Lambda }(q)< d^{\Lambda }(p_2) \nonumber \\&\quad \implies \quad d^{\Lambda }(T^{\Lambda }(p_1))< d^{\Lambda }(T^{\Lambda }(q)) < d^{\Lambda }(T^{\Lambda }(p_2)) \end{aligned}$$

for every \(p_1, p_2, q\) in \(\Lambda \cap {\mathcal {V}}\).

Fig. 4
figure 4

The case \(\gamma ([0,1))\cap \Sigma _{tr}^1\) infinite

Second case: \(\gamma ([0,1)) \cap \Sigma ^1_{tr}\) is infinite.

In this case, up to shrinking \({\mathcal {V}}\), by semianalyticity of \(\Sigma ^{1}_{tr}\) and Lemma 2.4 we can assume that \(\Sigma ^1_{tr}\cap {\mathcal {V}}\) is the union of r connected components, say \(\Gamma ^1, \ldots , \Gamma ^r\), whose boundaries are given by \({\bar{y}}\) and a point in the boundary of \({\mathcal {V}}\) (this point is distinct for each \(i=1,\ldots ,r\)). In addition, for each \(i=1, \ldots , r\) there exists a neighborhood \({\mathcal {V}}^i\) of \(\Gamma ^i\) such that \(\left( \Sigma \setminus \Gamma ^i\right) \cap {\mathcal {V}}^i\) is the union of \(s_i\) connected smooth subsets of \(\Sigma ^2\), say \(S_{j}^i\) for \(j = 1, \ldots , s_i\). Furthermore, as in the first case and up to shrinking \({\mathcal {V}}\) again, by a Poincaré-Bendixon type argument (see [17]) we may assume that for every \(i=1, \ldots , r\), if a piece of \(\gamma ([0,1))\) joins \(\Gamma ^i\) to some \(\Gamma ^{i'}\) through some \(S^i_j\) then the corresponding Poincaré mapping from \(\Gamma ^{i}\) to \(\Gamma ^{i'}\) is well-defined. To be more precise, for each \(i=1, \ldots ,r\) we consider the maximal subset of the \(S^i_j\)’s, relabeled \(S^i_1, \ldots , S^i_{{\hat{s}}_i}\), with the jump correspondence

$$\begin{aligned} J(i,\cdot ) \, : \, j \in \left\{ 1, \ldots , {\hat{s}}_i\right\} \, \longrightarrow \, \{1, \ldots , r\}, \end{aligned}$$

such that the transition maps

$$\begin{aligned} T^{i}_j \, : \, \Gamma ^i \longrightarrow \Gamma ^{J(i,j)} \ \forall \,i=1, \ldots , r, \quad \forall \,j=1, \ldots , {\hat{s}}_i, \end{aligned}$$

that assign to each \(p\in \Gamma ^i\) the point \(q\in \Gamma ^{J(i,j)}\) such that there is an absolutely continuous path \(\alpha :[0,1] \rightarrow \Sigma \) tangent to \({\mathscr {L}}\) over (0, 1) satisfying \(\alpha (0)=p\), \(\alpha (1)=q\), \(\alpha ((0,1)) \subset \Sigma ^2\) and \(\alpha ((0,\epsilon )) \subset S^i_j\) for some \(\epsilon >0\), are well-defined and continuous. Similarly as before, if we denote by \(d^i:\Gamma ^i \rightarrow {\mathbb {R}}\) the function which associates to each point p the length of the half-arc contained in \(\Gamma ^i\) which joins p to \({\bar{y}}\), then we may also assume that for every \(i=1, \ldots ,r \) and every \(p,q \in \Gamma ^i\),

$$\begin{aligned} d^{i}(p)< d^{i}(q) \implies d^{J(i,j)}(T^{i}_j(p)) < d^{J(i,j)}(T^{i}_j(q)) \quad \forall \,j=1, \ldots , {\hat{s}}_i.\nonumber \\ \end{aligned}$$

By construction, for each integer k, there are \(i_k \in \{1, \ldots ,r\}\) and \(j_k \in \{1, \ldots , {\hat{s}}_{i_k}\}\) such that \(\gamma (t_k^{\gamma }) \in \Gamma ^{i_k}\) and \(\gamma (t_{k+1}^{\gamma }) \in \Gamma ^{J(i_k,j_k)}=\Gamma ^{i_{k+1}}\). We call sequence of jumps of \(\gamma \) the sequence \(\{(i_k,j_k)\}_{k\in {\mathbb {N}}}\) associated with \(\{t_k^{\gamma }\}_{k\in {\mathbb {N}}}\).

We can now introduce the equivalence class on the set of monodromic trajectories.

Definition 3.5

(Equivalence of monodromic paths) Let \(\gamma _1, \gamma _2: [0,1) \rightarrow \Sigma \cap {\mathcal {V}}\) be two final and injective monodromic trajectories with the same limit point \({\bar{y}}\) and which share the same section \(\Lambda \), where \(\gamma _i(0) \in \Lambda \) for \(i=1,2\). We say that \(\gamma _1\) and \(\gamma _2\) are jump-equivalent if:

  • either \(\gamma _1([0,1)) \cap \Sigma ^{1}_{tr} = \gamma _2 ([0,1)) \cap \Sigma _{tr}^1 = \emptyset \);

  • or they have the same sequence of jumps.

By Eq. (3.2) in the first case, and by a concatenation of orbits of \({\mathscr {L}}\) connecting \(\Gamma ^i\) to \(\Gamma ^j\) and Eq. (3.3) in the second case, the following holds:

Lemma 3.6

(One parameter families of equivalent monodromic paths) Let \(\gamma : [0,1) \rightarrow \Sigma \cap {\mathcal {V}}\) be a final and injective monodromic trajectory with limit point \({\bar{y}}\), and let \(\Lambda \) be a section such that \(\gamma (0) \in \Lambda \). Then, for every point \(p \in \Lambda \) with \(d^{\Lambda }(p)<d^{\Lambda }(\gamma (0))\), there exists a final and injective monodromic trajectory \(\lambda :[0,1) \rightarrow \Sigma \cap {\mathcal {V}}\), with \(\lambda (0)=p\), which is jump-equivalent to \(\gamma \). Moreover, such a trajectory is unique as a curve (that is, up to reparametrization).

Lemma 3.6 plays a key role in the proof of Proposition 3.3. Indeed, from the existence of one monodromic trajectory, it allows us to infer the existence of a parametrized set of monodromic trajectories filling a 2-dimensional surface. The next result will also be crucial to control the length of the monodromic trajectories in such a set (we denote by \(\text{ length}^g\) the length of a curve with respect to the metric g).

Proposition 3.7

(Comparison of equivalent monodromic paths) Let \(\gamma \) be a monodromic trajectory with limit point \({\bar{y}}\) and section \(\Lambda \) such that \(\gamma (0) \in \Lambda \). Suppose that the length of \(\gamma \) is finite. Then there exists a constant \(K>0\) such that, for every monodromic trajectory \(\lambda \) jump-equivalent to \(\gamma \) satisfying \(d^{\Lambda }(\lambda (0)) < d^{\Lambda }(\gamma (0))\), we have

$$\begin{aligned} \mathrm{length}^g(\lambda ) \le K\, \mathrm{length}^g(\gamma ) < \infty . \end{aligned}$$

The proof of Proposition 3.7 is given in Sect. 4.4 as a consequence of Theorem 4.7. We give here just an idea of the proof.

Remark 3.8

(Idea of the proof of Proposition 3.7)

  1. (i)

    If \(\gamma ([0,1)) \cap \Sigma ^1_{tr} = \emptyset \), then Proposition 3.7 can be proved by a much more elementary argument based on the following observation. By properties (3.1)–(3.2) we note that, for all \(k\in {\mathbb {N}}\) and all \(p\in \Lambda \),

    $$\begin{aligned}&d^{\Lambda }\left( \gamma (t_k^{\gamma })\right)>d^{\Lambda }(p)>d^{\Lambda } \left( \gamma (t_{k+1}^{\gamma })\right) \quad \\&\quad \implies \quad d^{\Lambda }\left( \gamma (t_{k+1}^{\gamma })\right)>d^{\Lambda }\left( T^{\Lambda }(p) \right) >d^{\Lambda } \left( \gamma (t_{k+2}^{\gamma })\right) . \end{aligned}$$

    So, if we denote by \(\lambda _k\) the half-leaf of \({\mathscr {L}}\) connecting p and \(T^{\Lambda }(p)\), it follows by elementary (although non-trivial) geometrical arguments that there exist \(K>0\) and \(\epsilon _k\ge 0\) such that

    $$\begin{aligned} \mathrm{length}^g(\lambda _k) \le K\, \mathrm{length}^g \left( \gamma ([t_k^{\gamma },t_{k+2}^{\gamma }])\right) + \epsilon _k \qquad \forall \,k \in {\mathbb {N}}, \end{aligned}$$

    where \(\sum _{k\in {\mathbb {N}}}\epsilon _k< \infty \) (since we will not use this fact, we do not prove it). This bound essentially allows one to prove 3.7, up to an extra additive constant in the bound \(\mathrm{length}^g(\lambda ) \le K\, \mathrm{length}^g(\gamma )\) that anyhow is inessential for our purposes; note that this argument depends essentially on the fact that \(\gamma (t_k^{\gamma })\) belongs to the same section \(\Lambda \) for every k.

  2. (ii)

    In the case where \(\gamma ([0,1)) \cap \Sigma ^1_{tr} \ne \emptyset \) is infinite, the situation is much more delicate. One needs to work with the countable composition of transition maps \(T^{i_k}_{j_k}\) (in order to replace the Poincaré return), and the sequence of maps that one needs to consider is arbitrary. In particular, paths \(\gamma \) whose jump sequences are non-periodic are specially challenging because we can not adapt the argument of the first part of the remark to this case. This justifies our use of more delicate singularity techniques (e.g. the regularity of transition maps [38] and the bi-Lipschitz class of the pulled-back metric [8]). This leads to the more technical statement in Theorem 4.7(IV) (see also Lemma 4.14).

We are now ready to prove Proposition  3.3.

Proof of Proposition 3.3

Consider a monodromic trajectory \(\gamma :[0,1) \rightarrow \Sigma \) with limit \({\bar{y}} \in {\widetilde{\Sigma }}\) as above, and assume that it has finite length. As before, we may assume that \(\gamma \) is final, injective, and that \({\bar{z}}:=\gamma (0)\in \Lambda \). By Lemma 3.6, for every \(z\in \Lambda \) such that \(d^{\Lambda }(z)<d^{\Lambda }({\bar{z}}) \) there exists a unique final monodromic singular trajectory \(\gamma ^z :[0,1] \rightarrow \Sigma \), with \(\gamma ^z(0)=z\), which is jump-equivalent to \(\gamma \). Moreover, by Proposition 3.7 there exists \(K>0\) such that

$$\begin{aligned} \mathrm{length}^g(\gamma ^z) \le K\, \mathrm{length}^g(\gamma ). \end{aligned}$$

Let \(\{(i_k,j_k)\}_k\) be the sequence of jumps associated with \(\gamma \). For every \(z \in \Lambda \) with \(d^{\Lambda }(z)<d^{\Lambda }({\bar{z}}) \) the path \(\gamma ^z :[0,1] \rightarrow \Sigma \) is a singular horizontal path starting at z, so it admits an abnormal lift \(\psi ^z=(\gamma ^z,p^z):[0,1] \rightarrow T^*M\) such that \(\psi ^z(0)=(z,p)\) with \(p \perp \Delta _z\) and \(|p|^*=1\) (see Proposition A.1). Moreover, by (3.4) and Proposition A.3, there exists \({\tilde{K}}>0\) such that

$$\begin{aligned} |\psi ^z(t)|^* \le {\tilde{K}} \qquad \forall \, t \in [0,1], \quad \forall \,z \in \Lambda \text{ with } d^{\Lambda }(z)<d^{\Lambda }({\bar{z}}). \end{aligned}$$

Fix \(z \in \Lambda \cap {\mathcal {V}}\) such that \(d^{\Lambda }(z)<d^{\Lambda }({\bar{z}})\). Then there is an injective smooth path \(\xi =(\alpha ,\beta ): [0,1] \rightarrow T^*M\) which satisfies the following properties:

$$\begin{aligned}&\alpha (0)={\bar{z}},\quad \alpha (1) = z, \quad \alpha (s) \in \Lambda , \quad \text{ and }\nonumber \\&\quad d^{\Lambda }(\alpha (s)) \le d^{\Lambda }(\alpha (s')) \qquad \forall \,0\le s'\le s \le 1, \end{aligned}$$
$$\begin{aligned}&\beta (s) \perp \Delta _{\alpha (s)} \quad \text{ and } \quad |\beta (s)|^*=1 \qquad \forall \, s \in [0,1], \end{aligned}$$


$$\begin{aligned} A:= \int _{0}^1 \beta (s) \cdot {\dot{\alpha }}(s) \, ds >0. \end{aligned}$$

Note that (3.8) can be satisfied because \(\Delta \) is transverse to \(\Lambda \). For every \(s\in [0,1]\), set \(\gamma ^s:=\gamma ^{\alpha (s)}\) and note that \(\gamma ^0=\gamma \). By construction, each path \(\gamma ^s\) has the same sequence of jumps \(\{(i_k,j_k)\}_{k\in {\mathbb {N}}}\) which is associated to sequences of times \(\{t^s_k:=t_k^{\gamma ^s}\}_{k\in {\mathbb {N}}}\). For every \(s\in [0,1]\), denote by \(\psi ^s=(\gamma ^s,p^s)\) the abnormal lift associated to \(\gamma ^s\) starting at \((\alpha (s),\beta (s))=\xi (s)\). We may assume without loss of generality that \(p^s=p^{\alpha (s)}\) for all \(s\in [0,1]\), so that \(\psi ^s=\psi ^{\alpha (s)}\).

From \(t_0=0\) to \(t_1^s\), the set of lifts \(\psi ^s=(\gamma ^s,p^s):=\psi ^{\alpha (s)}\) of the paths \(\gamma ^s\) starting at \((\alpha ^s,\beta ^s)\) span a surface \({\mathcal {S}}^0\) homeomorphic to a 2-dimensional disc whose boundary is composed by \(\xi \), the lift \(\psi ^0|_{[0,t_1^0]}\), the lift \(\psi ^1|_{[0,t_1^1]}\), and a path \(\xi ^1=(\alpha ^1,\beta ^1):[0,1] \rightarrow T^*M\) whose projection is contained in \(\Gamma ^{J(i_0,j_0)}\) and which connects \(\psi ^0(t_1^0)\) to \(\psi ^1(t_1^1)\) (see Fig. 5). Thus, by Stokes’ Theorem we have, using the canonical symplectic form \(\omega \) on \(T^*M\) (see “Appendix A”),

$$\begin{aligned} \int _{{\mathcal {S}}^0} \omega= & {} \int _{0}^1 \beta (s) \cdot {\dot{\alpha }}(s) \, ds + \int _{0}^{t_1^1} p^1(t) \cdot {\dot{\gamma }}^1(t) \, dt \\&- \int _{0}^1 \beta ^1(s) \cdot {\dot{\alpha }}^1(s) \, ds - \int _{0}^{t_1^0} p^0(t) \cdot {\dot{\gamma }}(t) \, dt. \end{aligned}$$

Since \(\gamma \) and \(\gamma ^1\) are both singular horizontal paths we have \(p^0(t) \cdot {\dot{\gamma }}(t) = p^1(t) \cdot {\dot{\gamma }}^1(t)=0\) for all \(t\in [0,1]\). Moreover, since the derivative of the lifts \(\psi ^s\) is always contained in the kernel of \(\omega _{\vert \Delta ^{\perp }}\) (see “Appendix A”), we have \(\int _{{\mathcal {S}}^0} \omega =0\). As a consequence, we infer that

$$\begin{aligned} \int _{0}^1 \beta ^1(s) \cdot {\dot{\alpha }}^1(s) \, ds = \int _{0}^1 \beta (s) \cdot {\dot{\alpha }}(s) \, ds =A. \end{aligned}$$

Repeating this argument and recalling (3.5), we get a sequence of arcs \(\xi ^k=(\alpha ^k,\beta ^k):[0,1] \rightarrow T^*M\) such that

$$\begin{aligned} \int _{0}^1 \beta ^k(s) \cdot {\dot{\alpha }}^k(s) \, ds =A>0 \qquad \forall \, k \in {\mathbb {N}}, \end{aligned}$$


$$\begin{aligned} \mathrm{length}^g(\alpha ^k) \rightarrow 0 \quad \text{ and } \quad |\beta ^k(s)|^* \le {\tilde{K}} \qquad \forall \, k \in {\mathbb {N}}. \end{aligned}$$

This provides the desired contradiction, which proves the result. \(\square \)

Fig. 5
figure 5

The projection of \({\mathcal {S}}^0\) in M

The proof of Theorem 1.1 is given hereafter as a consequence of both Proposition 3.1 and Proposition 3.3.

3.3 Proof of Theorem 1.1

Before starting the proof let us summarize the different types of points \(y\in \Sigma \) that can be crossed by a singular horizontal path. We distinguish four cases.

First case: \(y\in \Sigma ^2\).

The line foliation is regular in a neighborhood of y, so there is an analytic curve such that any singular path containing y is locally contained in this curve.

Second case: \(y\in {\widetilde{\Sigma }}^1_{tan}:=\Sigma ^1_{tan}\setminus {\widetilde{\Sigma }}^0\).

By Proposition 3.1 and the fact that \({\widetilde{\Sigma }}^0\) is locally finite, any singular path passing through y is contained in \({\widetilde{\Sigma }}^1_{tan}\), that is locally analytic.

Third case: \(y\in {\widetilde{\Sigma }}^0\).

The singular paths that contain y are either the branches of \({\Sigma }^1_{tan}\) or the characteristic singular paths. In the first case, these branches are actually contained inside \({{\widetilde{\Sigma }}}^1_{tan}\) with the exception of y. In the second case, there are only finitely many characteristic singular paths by Proposition 3.1, and they are semianalytic by Proposition 4.12.

Fourth case: \(y\in \Sigma ^1_{tr}\).

By Lemma 2.6, there are finitely many semianalytic singular horizontal curves that can cross y.

In conclusion, if we travel along a given singular path \(\gamma :[0,1]\rightarrow M\) then bifurcation points may happen only when \(\gamma \) crosses the set \({\widetilde{\Sigma }}^0 \cup \Sigma ^1_{tr}\). Since \({\mathcal {X}}^{x,L}_{\Delta ,g}\) is compact, there are only finitely many points of \({\widetilde{\Sigma }}^0\) to consider. Moreover, by Lemma 2.6, from every bifurcation point in \(\Sigma ^1_{tr}\) there are only finitely many curves exiting from it. By Proposition 3.1 any singular horizontal path interesect \(\Sigma ^1_{tr}\) finitely many times, but we need to show that the intersection of \({\mathcal {X}}^{x,L}_{\Delta ,g}\) with \(\Sigma ^1_{tr}\) is finite. This follows from the fact that \({\mathcal {X}}^{x,L}_{\Delta ,g}\) can be constructed from finitely many singular path emanating from x, by successive finite branching at the points of \({\widetilde{\Sigma }}^0 \cup \Sigma ^1_{tr}\) met by the paths. Let us present this argument precisely.

We associate to \({\mathcal {X}}^{x,L}_{\Delta ,g}\) a tree T constructed recursively as follows. Let the initial vertex \(v_0\) of the tree represent the point x and let the edges from \(v_0\) be in one-to-one correspondence with different singular horizontal paths of length \(\le L\) starting from x. If such path arrives to a branching point, that is a point of \({{\widetilde{\Sigma }}}_0\cup \Sigma ^1_{tr}\), we represent this point as another vertex of the tree (even if this point is again x). If a singular path does not arrive at \({{\widetilde{\Sigma }}}_0\cup \Sigma ^1_{tr}\) we just add formally a (final) vertex. In this way we construct a connected (a priori infinite) locally finite tree. We note that any injective singular horizontal path starting at x, of length bounded by L, is represented in T by a finite simple path of the tree (a path with no repeated vertices).

Suppose, by contradiction, that T is infinite. By König’s Lemma (see, e.g. [45]), the tree T contains a simple path \(\omega _\infty \) that starts at \(v_0\) and continues from it through infinitely many vertices. Such path corresponds to a singular horizontal trajectory \(\gamma _\infty \) that passes infinitely many times through \({{\widetilde{\Sigma }}}_0\cup \Sigma ^1_{tr}\). Since any finite subpath of \(\omega _\infty \) corresponds to a singular horizontal path of length bounded by L, \(\gamma _\infty \) itself has length bounded by L and crosses infinitely many times \(\Sigma ^1_{tr}\) (a finite length path cannot pass infinitely many times through \({{\widetilde{\Sigma }}}_0\cap {\mathcal {X}}^{x,L}_{\Delta ,g}\) that is finite). Hence:

  • either \(\gamma _\infty \) is monodromic of finite length, and this contradicts Proposition 3.1;

  • or the limit point of \(\gamma _\infty \) belongs to \(\Sigma ^1_{tr}\), which contradicts Lemma 2.6(i).

Therefore, the tree T is finite, and \({\mathcal {X}}^{x,L}_{\Delta ,g}\) consist of finitely many singular horizontal curves by  2.6(ii).

The last part of Theorem 1.1 follows from the fact that any smooth manifold can be equipped with a complete Riemannian metric (see [34]).

4 Singularities of the characteristic line-foliation

4.1 Divergence property

In this subsection we introduce some basic results about the divergence of vector fields. The subsection follows a slightly more general setting than the previous section, but which relates to the study of the Sard Conjecture via the local model given in Remark 2.1(i).

We start by considering a nonsingular analytic surface \({{\mathscr {S}}}\) with a volume form \(\omega _{{{\mathscr {S}}}}\). Denote by \({\mathcal {O}}_{{\mathscr {S}}}\) the sheaf of analytic functions over \({{\mathscr {S}}}\). We note that there exists a one-to-one correspondence between 1-differential forms \(\eta \in \Omega ^1({{\mathscr {S}}})\) and vector fields \({\mathcal {Z}} \in Der_{{\mathscr {S}}}\) given by

$$\begin{aligned} {\mathcal {Z}} \longleftrightarrow \eta \qquad \text{ if } \eta = i_{{{\mathcal {Z}}}}\omega _{{{\mathscr {S}}}}. \end{aligned}$$

This correspondence gives the following formula on the divergence:

$$\begin{aligned} \text{ div}_{\omega _{{{\mathscr {S}}}}}({{\mathcal {Z}}}) \omega _{{{\mathscr {S}}}}= d\eta . \end{aligned}$$

Remark 4.1

(Basic properties)

  1. (i)

    Suppose that uv are local coordinates on \({{\mathscr {S}}}\) such that \(\omega _{{{\mathscr {S}}}}= du\wedge dv\). Then the form \(\eta = \alpha du + \beta dv\) corresponds to \({\mathcal {Z}} = \alpha \partial _y - \beta \partial _x\).

  2. (ii)

    Given an analytic function \(f: {\mathscr {S}} \rightarrow {\mathbb {R}}\), we have

    $$\begin{aligned} df\wedge \eta = df\wedge i_{{{\mathcal {Z}}}} \omega _{{{\mathscr {S}}}}= i_{{{\mathcal {Z}}}} df \wedge \omega _{{{\mathscr {S}}}}= {\mathcal {Z}}(f) \omega _{{{\mathscr {S}}}}\end{aligned}$$

    where \(i_{{{\mathcal {Z}}}}\omega \) denotes the contraction of a form \(\omega \) by \({{\mathcal {Z}}}\) (see, e.g. [2, page 198]).

  3. (iii)

    The above results can easily be generalized to d-dimensional analytic manifolds, where the one-to-one correspondence is between \(d-1\) forms and vector fields (that is, between \(\Omega ^{d-1}(M)\) and \(Der_M\)).

We denote by \({{\mathcal {Z}}}({\mathcal {O}}_{{\mathscr {S}}})\) the ideal sheaf generated by the derivation \({{\mathcal {Z}}}\) applied to the analytic functions in \({\mathcal {O}}_{{\mathscr {S}}}\), that is, the ideal sheaf locally generated by the coefficients of \({{\mathcal {Z}}}\). In what follows, we study closely the property \(\text{ div}_{\omega _{{{\mathscr {S}}}}}({\mathcal {Z}}) \in {\mathcal {Z}}({\mathcal {O}}_{{\mathscr {S}}})\), following [9, Lemma 2.3 and 3.2]. The next result shows that the property is independent of the volume form.

Lemma 4.2

(Intrinsicality) Let \(\omega _{{{\mathscr {S}}}}\) and \(\omega _{{{\mathscr {S}}}}'\) be two volume forms over \({\mathscr {S}}\). Then \(\text{ div}_{\omega _{{{\mathscr {S}}}}}({\mathcal {Z}}) \in {\mathcal {Z}}({\mathcal {O}}_{{\mathscr {S}}})\) if and only if \(\text{ div}_{\omega _{{{\mathscr {S}}}}'}({\mathcal {Z}}) \in {\mathcal {Z}}({\mathcal {O}}_{{\mathscr {S}}})\).


Given a point \(p \in {\mathscr {S}}\), there exist an open neighborhood U of p and a smooth function \(F: U \rightarrow {\mathbb {R}}\) which is everywhere non-zero and such that \(\omega _{{{\mathscr {S}}}}' = F\cdot \omega _{{{\mathscr {S}}}}\) in U. Therefore,

$$\begin{aligned} \begin{aligned} \text{ div}_{\omega _{{{\mathscr {S}}}}'}({\mathcal {Z}}) \cdot \omega _{{{\mathscr {S}}}}'&= d\left( i_{{\mathcal {Z}}}\omega _{{{\mathscr {S}}}}' \right) = d\left( F \cdot i_{{\mathcal {Z}}}\omega _{{{\mathscr {S}}}}\right) \\&= \left[ {\mathcal {Z}}(F)/F + \text{ div}_{\omega _{{{\mathscr {S}}}}}({\mathcal {Z}}) \right] \omega _{{{\mathscr {S}}}}', \end{aligned} \end{aligned}$$

and we conclude easily. \(\square \)

Lemma 4.3 below illustrates the importance of this property; in its statement we use the notion of elementary singularities (see, e.g. [21, Definition 4.27])), that we recall in “Appendix D.4” (Definition D.8).

Lemma 4.3

(Final singularities) Let \({\mathcal {Z}}\) be a real analytic vector-field defined in an open neighborhood \(U \subset {\mathbb {R}}^2\) of the origin and \(\omega _U\) to be a volume form over U. Let (xy) be a coordinate system defined over U and suppose that:

  1. (i)

    \(\text{ div}_{\omega _U}({\mathcal {Z}}) \in {\mathcal {Z}}({\mathcal {O}}_{U})\);

  2. (ii)

    \({\mathcal {Z}} = x^{\alpha }y^{\beta }\, \widetilde{{\mathcal {Z}}}\), for some \(\alpha \) and \(\beta \in {\mathbb {N}}\) and a real analytic vector field \(\widetilde{{\mathcal {Z}}}\) all of whose singularities are isolated and elementary.

Then the vector field \(\widetilde{{\mathcal {Z}}}\) is tangent to the set \(\{x^{\alpha }y^{\beta }=0\}\) and all of its singularities are saddles.

Proof of Lemma 4.3

By Lemma 4.2, up to shrinking U we can suppose that \(\omega _U = dx\wedge dy\). We set by \(A ={\mathcal {Z}}(x)\) and \(B={\mathcal {Z}}(y)\) so that \({\mathcal {Z}}= A(x,y) \partial _x+B(x,y) \partial _y\). By assumption (ii), these functions are divisible by \(x^{\alpha }y^{\beta }\), namely \(A=x^{\alpha }y^{\beta } {\widetilde{A}}\) and \(B=x^{\alpha }y^{\beta }{\widetilde{B}}\). By assumption (i), there exist smooth functions f and g such that

$$\begin{aligned} \partial _xA + \partial _yB= & {} f\cdot A + g\cdot B, \quad \text { so that } \nonumber \\&\quad \alpha \cdot {\widetilde{A}}/x + \partial _x{\widetilde{A}} + \beta \cdot {\widetilde{B}}/y + \partial _y{\widetilde{B}} = f\cdot {\widetilde{A}} + g\cdot {\widetilde{B}} \end{aligned}$$

In particular \(\alpha {\widetilde{A}}/x + \beta {\widetilde{B}}/y \) does not have poles, which implies that \({\widetilde{A}}\) is divisible by x if \(\alpha \ne 0\), and \({\widetilde{B}}\) is divisible by y if \(\beta \ne 0\). In other words, \(\widetilde{{\mathcal {Z}}} = {\widetilde{A}}(x,y) \partial _x+{\widetilde{B}}(x,y) \partial _y\) is tangent to \(\{x^{\alpha }y^{\beta }=0\}\).

Without loss of generality, we can suppose that the origin is the only singularity of \(\widetilde{{\mathcal {Z}}}\). We consider the determinant and the trace of the Jacobian of \(\widetilde{{\mathcal {Z}}}\) at the origin:

$$\begin{aligned} \begin{aligned} \det \bigl (\text{ Jac }(\widetilde{{\mathcal {Z}}})(0)\bigr )&= \partial _x{\widetilde{A}}(0)\cdot \partial _y{\widetilde{B}}(0) - \partial _y{\widetilde{A}}(0)\cdot \partial _x{\widetilde{B}}(0),\\ \text{ tr }\bigl (\text{ Jac }(\widetilde{{\mathcal {Z}}})(0)\bigr )&= \partial _x{\widetilde{A}}(0)+ \partial _y{\widetilde{B}}(0). \end{aligned} \end{aligned}$$

In order to conclude, thanks to Remark D.9(i) it is enough to prove that \(\det \bigl (\text{ Jac }(\widetilde{{\mathcal {Z}}})(0)\bigr )<0\). We distinguish two cases, depending on the value of \(\alpha \) and \(\beta \).

First, suppose that \(\alpha =\beta =0\) (in particular \(A = {\widetilde{A}}\) and \(B={\widetilde{B}}\)). Then, thanks to (4.1),

$$\begin{aligned} \text{ tr }\bigl (\text{ Jac }(\widetilde{{\mathcal {Z}}})(0)\bigr ) = \partial _xA(0) + \partial _yB(0) = 0. \end{aligned}$$

Since the origin is an elementary singularity of \(\widetilde{{\mathcal {Z}}}\), using Remark D.9(ii) we conclude that the determinant is negative. Thus, the singularity is a saddle point.

Second, without loss of generality we suppose that \(\alpha \ne 0\). In this case x divides \({\widetilde{A}}\), which implies that \(\partial _y{\widetilde{A}}(0)=0\) and \(\partial _x{\widetilde{A}}(0)= \bigl ({\widetilde{A}}/x\bigr )(0)\). In particular, this yields

$$\begin{aligned} \det \bigl (\text{ Jac }(\widetilde{{\mathcal {Z}}})(0)\bigr ) = \partial _x{\widetilde{A}}(0)\cdot \partial _y{\widetilde{B}}(0). \end{aligned}$$

Also, since \(\partial _x{\widetilde{A}}(0)= \bigl ({\widetilde{A}}/x\bigr )(0)\), and either \(\beta =0\) or \(\partial _y{\widetilde{B}}(0)= \bigl ({\widetilde{B}}/y\bigr )(0)\), using (4.1) we get

$$\begin{aligned} (\alpha +1)\partial _x{\widetilde{A}}(0) + (\beta +1)\partial _y{\widetilde{B}}(0)=0. \end{aligned}$$

It follows that \(\partial _x{\widetilde{A}}(0)\) and \(\partial _y{\widetilde{B}}(0)\) have opposite signs (if they are both zero then the determinant and the trace are zero, contradicting the definition of elementary singularity), and therefore the determinant is negative (see (4.2)). Once again, since the origin is an elementary singularity of \(\widetilde{{\mathcal {Z}}}\), using Remark D.9(ii) we conclude that the singularity is a saddle point. \(\square \)

Next, suppose that M is a 3-dimensional analytic manifold and denote by \(\omega _M\) its volume form. We now return to the study of the Martinet surface \(\Sigma \), cf. Remark 2.1(i).

Let \(\mu \in \Omega ^1(M)\) be an everywhere non-singular analytic 1-form and denote by h the analytic function defined as in Eq. (2.1). Denote by \(*: \Omega ^1 (M) \rightarrow \Omega ^2(M)\) the Hodge star operator, cf. [44, Ch. V]. We start by a known characterization of \(d\mu \) in terms of \(\mu \) and \(*\mu \):

Lemma 4.4

There exists an analytic form \(a\in \Omega ^1 (M)\) such that:

$$\begin{aligned} d\mu = a\wedge \mu + h \langle \mu , *\mu \rangle ^{-1} (*\mu ). \end{aligned}$$

Proof of Lemma 4.4

Since \(\mu \) is nonsingular, the induced scalar product \(\langle \mu , *\mu \rangle \) is a nowhere vanishing function and we have

$$\begin{aligned} \mu \wedge *\mu = \langle \mu , *\mu \rangle \omega _M . \end{aligned}$$

The form \(d\mu \) can be decomposed as

$$\begin{aligned} d\mu = a\wedge \mu + b (*\mu ), \end{aligned}$$

where a is an analytic 1-form and b is an analytic function. Combining (2.1), (4.4), and (4.5), we deduce that \(h= b \langle \mu , *\mu \rangle \), which proves (4.3). \(\square \)

Now, we consider an analytic map \(\pi : {\mathscr {S}}\rightarrow \Sigma \subset M\) from an analytic surface \({\mathscr {S}}\) to the Martinet surface \(\Sigma \), and we set \(\eta := \pi ^* (\mu )\). It follows from Lemma 4.4 that

$$\begin{aligned} d\eta = {{\tilde{a}}} \wedge \eta \end{aligned}$$

with \({{\tilde{a}}} = \pi ^* a\), because \( \pi ^*h \equiv 0\). Let \({\mathcal {Z}}\) be the vector field associated to \(\eta \), and denote by \({\mathcal {Z}}(\pi )\) the ideal subsheaf of \({\mathcal {Z}}({\mathcal {O}}_{{\mathscr {S}}})\) generated by the derivation \({\mathcal {Z}}\) applied to the pullback by \(\pi \) of analytic functions on M.

Remark 4.5

  1. (i)

    For our applications, the map \(\pi \) is either going to be an inclusion of the regular part of \(\Sigma \) into M, or a resolution of singularities of (the analytic space) \(\Sigma \) (cf. Theorem 4.7).

  2. (ii)

    If we write (locally) \(\pi =(\pi _1, \pi _2, \pi _3)\), then \({\mathcal {Z}}(\pi )\) is locally generated by \({\mathcal {Z}} (\pi _1), {\mathcal {Z}} (\pi _2), {\mathcal {Z}} (\pi _3)\).

The next proposition shows that, in the local setting (following Remarks 2.1(i) and 4.5 (i)), the property “\(\text{ div}_{\omega _{{{\mathscr {S}}}}}({{\mathcal {Z}}}) \in {{\mathcal {Z}}}({\mathcal {O}}_{{\mathscr {S}}})\)” is always satisfied. This can be seen as a reformulation of [9, Lemmas 2.3, 3.1, and 4.3]

Proposition 4.6

(Divergence bound) Let \(\eta \in \Omega ^1({\mathscr {S}})\), and let \({\mathcal {Z}}\) be the vector field associated to \(\eta \).

  1. (i)

    If \(\eta \) satisfies (4.6), then \(\text{ div}_{\omega _{{{\mathscr {S}}}}}({{\mathcal {Z}}}) \in {{\mathcal {Z}}}({\mathcal {O}}_{{\mathscr {S}}})\).

  2. (ii)

    If in addition \(\eta = \pi ^* (\mu )\), then \(\text{ div}_{\omega _{{{\mathscr {S}}}}}({{\mathcal {Z}}}) \in {\mathcal {Z}}(\pi )\). In particular, for every compact subset \({\mathcal {K}}\subset {\mathscr {S}}\) there is a constant \(K>0\) such that

    $$\begin{aligned} \left| \text{ div}_{\omega _{{{\mathscr {S}}}}}({{\mathcal {Z}}}) \right| \le K\left| \pi _* ({\mathcal {Z}})\right| \qquad \text { on }{\mathcal {K}}. \end{aligned}$$

Proof of Proposition 4.6

Let a be as in (4.5), and write it in local coordinates on M as \(a = \sum g_i dx_i\). Then \(\pi ^* (a) = \sum (g_i\circ \pi )\, d\pi _i\), which implies that, in local coordinates on \({\mathscr {S}}\), we have

$$\begin{aligned} \text{ div}_{\omega _{{{\mathscr {S}}}}}({{\mathcal {Z}}}) \omega _{{{\mathscr {S}}}}= \sum (g_i\circ \pi ) d\pi _i\wedge \eta = \sum (g_i\circ \pi ) {\mathcal {Z}} (\pi _i) \omega _{{{\mathscr {S}}}}. \end{aligned}$$

The bound follows from the fact that \(\pi _* ({\mathcal {Z}}) = ( {\mathcal {Z}} (\pi _1), {\mathcal {Z}} (\pi _2), {\mathcal {Z}} (\pi _3))\). \(\square \)

4.2 Resolution of singularities

Here we follow the notation and framework introduced in “Appendix D” and in Sects. 2.1 and 2.2 . All definitions and concepts concerning resolution of singularities (e.g. blowings-up, simple normal crossing divisors, strict transforms, etc) are recalled in “Appendix D”.

Fig. 6
figure 6

A saddle point as in the first case of Theorem 4.7(IV.i.b)

Theorem 4.7

There exist an analytic surface \({\mathscr {S}}\), and a simple normal crossing divisor E in \({\mathscr {S}}\) (see Sect. D.1), and a sequence of admissible blowings-up \(\pi : {\mathscr {S}} \rightarrow \Sigma \) with exceptional divisor E (see Definition D.2) such that:

  1. (I)

    The restriction of \(\pi \) to \({\mathscr {S}}\setminus E\) is a diffeomorphism onto its image \(\Sigma \setminus S\) (c.f. Lemma 2.2).

  2. (II)

    Denote by \(\widetilde{{\mathscr {L}}}\) the strict transform of the foliation \({\mathscr {L}}\) (cf. Sect. D.4). Then all singularities of \(\widetilde{{\mathscr {L}}}\) are saddle points.

  3. (III)

    The exceptional divisor E is given by the union of two locally finite union of divisors, \(E_{tan}\) and \(E_{tr}\), such that \(E_{tan} \cap E_{tr}\) is a locally finite set of points, \(\widetilde{{\mathscr {L}}}\) is tangent to \(E_{tan}\), and \(\widetilde{{\mathscr {L}}}\) is everywhere transverse to \(E_{tr}\). Furthermore, the log-rank of \(\pi \) at \(E_{tr}\setminus E_{tan}\) is constant equal to 1 (we recall the definition of log-rank in “Appendix D.3”).

  4. (IV)

    At each point \({\bar{z}}\in E_{tan}\), there exists an open neighborhood \(U_{{\bar{z}}}\) of \({\bar{z}}\) such that:

    1. (i)

      Suppose that there exists only one irreducible component of \(E_{tan}\) passing through \({\bar{z}}\). Then there exists a coordinate system (uv) centered at \({\bar{z}}\) and defined in \(U_{{\bar{z}}}\), such that:

      1. (a)

        The exceptional divisor \(E_{tan}\) restricted to \(U_{{\bar{z}}}\) coincides with \(\{u=0\}\).

      2. (b)

        Either \({\bar{z}}\) is a saddle point of \(\widetilde{{\mathscr {L}}}\) (see Fig. 6); or at each half-plane (bounded by \(E_{tan}\)) there exist two smooth analytic semi-segments \(\Lambda _{{\bar{z}}}^1\) and \(\Lambda _{{\bar{z}}}^2\) which are transverse to \(\widetilde{{\mathscr {L}}}\) and \(E_{tan}\), such that the flow (of a local generator \({\widetilde{{\mathcal {Z}}}}\)) associated to \(\widetilde{{\mathscr {L}}}\) gives rise to a bi-analytic transition map

        $$\begin{aligned} \phi _{{\bar{z}}} : \Lambda _{{\bar{z}}}^1 \rightarrow \Lambda _{{\bar{z}}}^2, \end{aligned}$$

        and there exist a rectangle \(V_{{\bar{z}}}\) bounded by \(E_{tan}\), \(\Lambda _{{\bar{z}}}^1\), \(\Lambda _{{\bar{z}}}^2\) and a regular leaf \({\mathcal {L}} \not \subset E_{tan}\) of \(\widetilde{{\mathscr {L}}}\) such that \({\bar{z}} \in \partial V_{{\bar{z}}} \setminus (\Lambda _{{\bar{z}}}^1 \cup \Lambda _{{\bar{z}}}^2 \cup {\mathcal {L}})\) (see Fig. 7).

      3. (c)

        If \({\bar{z}} \in E_{tr}\), then \({\bar{z}}\) is a regular point of \(\widetilde{{\mathscr {L}}}\) and \(E_{tr} \cap U_{{\bar{z}}} = \{v=0\}\) does not intersect \(\Lambda _{{\bar{z}}}^1\) nor \(\Lambda _{{\bar{z}}}^2\). Furthermore, the map \(\phi _{{\bar{z}}}\) is the composition of two analytic maps (see Fig. 8):

        $$\begin{aligned} \phi _{{\bar{z}}}^1: \Lambda _{{\bar{z}}}^1 \rightarrow E_{tr}, \qquad \phi _{{\bar{z}}}^2: E_{tr} \rightarrow \Lambda _{{\bar{z}}}^2. \end{aligned}$$
    2. (ii)

      Suppose that there exist two irreducible components of \(E_{tan}\) passing through \({\bar{z}}\). Then there exists a coordinate system \(\pmb {u}=(u_1,u_2)\) centered at \({\bar{z}}\) and defined in \(U_{{\bar{z}}}\), such that:

      1. (a)

        The exceptional divisor \(E_{tan}\) restricted to \(U_{{\bar{z}}}\) coincides with \(\{u_1\cdot u_2=0\}\).

      2. (b)

        At each quadrant (bounded by \(E_{tan}\)) there exist two smooth analytic semi-segments \(\Lambda _{{\bar{z}}}^1\) and \(\Lambda _{{\bar{z}}}^2\) which are transverse to \(\widetilde{{\mathscr {L}}}\) and to \(E_{tan}\), such that the flow (of a local generator \(\widetilde{{\mathcal {Z}}}\)) associated to \(\widetilde{{\mathscr {L}}}\) gives rise to a bijective (but not necessarily analytic) transition map

        $$\begin{aligned} \phi _{{\bar{z}}} : \Lambda _{{\bar{z}}}^1 \rightarrow \Lambda _{{\bar{z}}}^2 \end{aligned}$$

        and there exist a region \(V_{{\bar{z}}}\) bounded by \(E_{tan}\), \(\Lambda _{{\bar{z}}}^1\), \(\Lambda _{{\bar{z}}}^2\) and a regular leaf \({\mathcal {L}} \not \subset E\) of \(\widetilde{{\mathscr {L}}}\) such that \({\bar{z}} \in \partial V_{{\bar{z}}} \setminus (\Lambda _{{\bar{z}}}^1 \cup \Lambda _{{\bar{z}}}^2 \cup {\mathcal {L}})\) (see Fig. 9).

      3. (c)

        There exist \(\alpha ,\,\beta \in {\mathbb {N}}^2\) such that the pulled-back metric \(\pi ^{*}(g) = g^{*}\) is locally bi-Lipschitz equivalent to:

        $$\begin{aligned} \begin{aligned} h_{{\bar{z}}}&= (d \pmb {u}^{\alpha })^2 + (d \pmb {u}^{\beta })^2,\quad \text { where } \quad \pmb {u}^{\alpha } = u_1^{\alpha _{1}}u_2^{\alpha _{2}} \text { and } \, \pmb {u}^{\beta }=u_1^{\beta _{1}}u_2^{\beta _{2}}. \end{aligned} \end{aligned}$$

        Furthermore, there exists a vector field \(\widetilde{{\mathcal {Z}}}\), which locally generates \(\widetilde{{\mathscr {L}}}\), such that:

        • either \(|\widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha })| \ge |\widetilde{{\mathcal {Z}}}(\pmb {u}^{\beta })|\) everywhere in \(V_{{\bar{z}}}\), and \(\widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha }) \ne 0\) everywhere in \(V_{{\bar{z}}}\setminus E\);

        • or \(|\widetilde{{\mathcal {Z}}}(\pmb {u}^{\beta })| \ge |\widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha })|\) everywhere in \(V_{{\bar{z}}}\), and \(\widetilde{{\mathcal {Z}}}(\pmb {u}^{\beta }) \ne 0\) everywhere in \(V_{{\bar{z}}} \setminus E\).

Fig. 7
figure 7

Second case of Theorem 4.7(IV.i.b)

Fig. 8
figure 8

Case (IV.i.c) in Theorem 4.7

Fig. 9
figure 9

Case (IV.ii) in Theorem 4.7

Proof of Theorem 4.7

Denote by \({\mathscr {M}}\) the reduced analytic space associated with \(\Sigma \). By [8, Theorem 1.3] (we recall the details in Theorem D.6 below), there exists a resolution of singularities \(\pi : {\mathscr {S}} \rightarrow \Sigma \subset M\) via admissible blowings-up which satisfies the Hsiang–Pati property (see “Appendix D.3”). All blowings-up project into the singular set \(Sing({\mathscr {M}}) \subset S\); we can further suppose that the pre-image of S is contained in the exceptional divisor (which is useful for combinatorial reasons), which guarantees (I). These properties are preserved by further real blowings-up described in Theorem D.6(i)-(ii).

Next, by [21, Theorem 8.14] (we recall the details in Theorem D.10 below) we can further compose \(\pi \) with a locally finite number of blowings-up of points in the exceptional divisor so that the strict transform of \({\mathscr {L}}\), which we denote by \(\widetilde{{\mathscr {L}}}\), has only elementary singularities and is either tangent or transverse to connected components of the exceptional divisor E. Denote by \(E_{tan}\) the union of connected exceptional divisors tangent to \({\mathscr {L}}\), and by \(E_{tr}\) the remaining ones.

Now, fix a point \({\bar{z}} \in {\mathscr {S}}\) and let \({\mathcal {W}}\) be a sufficiently small neighborhood of \({\bar{z}}\) so that \({\mathcal {W}}\) is orientable; in particular, fix a volume form \(\omega _{{\mathcal {W}}}\) defined over \({\mathcal {W}}\). Next, up to shrinking \({\mathcal {W}}\), there exists a relatively compact open set \({\mathcal {V}} \subset M\), with \(\pi ({\mathcal {W}}) \subset {\mathcal {V}}\), such that \(\Delta \) is generated on \({\mathcal {V}}\) by a 1-form \(\mu \in \Omega ^1({\mathcal {V}})\) (cf. Remark 2.1(i)).

Consider the vector field \({\mathcal {Z}}\) (which is defined over \({\mathcal {W}}\)) given by:

$$\begin{aligned} i_{{\mathcal {Z}}} \omega _{{\mathcal {W}}} = \pi ^{*} \mu . \end{aligned}$$

By Proposition 4.6 we have that \(\text{ div}_{\omega _{{\mathcal {S}}}}({\mathcal {Z}}) \in {\mathcal {Z}}(\pi )\). Now, denote by \(\widetilde{{\mathcal {Z}}}\) a local generator of \(\widetilde{{\mathscr {L}}}\) defined over \({\mathcal {W}}\); we note that \(\widetilde{{\mathcal {Z}}}\) is given by the division of \({\mathcal {Z}}\) by as many powers as possible of the exceptional divisor (c.f. Lemma 4.3(ii)). It follows from Lemma 4.3 that all singularities of \(\widetilde{{\mathscr {L}}} \cap {\mathcal {W}}\) are saddles, and that the foliation is tangent to connected components of the exceptional divisors where the log-rank of \(\pi \) is zero (because, by Eq. (4.7), the vector field \({\mathcal {Z}}\) is divisible by powers of these exceptional divisors). In particular, the log-rank over \(E_{tr}\setminus E_{tan}\cap {\mathcal {W}}\) must be equal to 1. Since \({\mathcal {V}}\) was arbitrary, we conclude Properties (II) and (III).

Next, we provide an argument over 2-points in order to prove (IV)(ii). Let \({\bar{z}}\) be a point in the intersection of two connected components of \(E_{tan}\). Since \(\widetilde{{\mathscr {L}}}\) is tangent to \(E_{tan}\), we deduce that \({\bar{z}}\) is a saddle point of \(\widetilde{{\mathscr {L}}}\). Now, by Lemma D.7, the pulled back metric \(g^{*}\) is locally (at \({\bar{z}}\)) bi-Lipschitz equivalent to the metric

$$\begin{aligned} h_{{\bar{z}}} = (d \pmb {u}^{\alpha })^2 + (d \pmb {u}^{\beta })^2. \end{aligned}$$

Recalling that \(\widetilde{{\mathcal {Z}}}\) is a local generator of \(\widetilde{{\mathscr {L}}}\), we consider the locally defined analytic set \(T = \{\pmb {u}\,|\,|\widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha })| = |\widetilde{{\mathcal {Z}}}(\pmb {u}^{\beta })|\}\).

If T is a 2-dimensional set then \(|\widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha })| = |\widetilde{{\mathcal {Z}}}(\pmb {u}^{\beta })|\) everywhere on a neighborhood of \({\bar{z}}\), and by the existence of transition maps close to saddle points (see, e.g. [3, Section 2.4]), we conclude easily Properties (IV.ii.a), (IV.ii.b), and (IV.ii.c).

Therefore, we can suppose that T is an analytic curve. We claim that, up to performing combinatorial blowings-up (that is, blowings-up whose centers are the intersection of exceptional divisors), we can suppose that \(T \subset E_{tan}\) (we recall that the argument is only for 2-points). As a result, without loss of generality, we locally have: either \(|\widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha })|> | \widetilde{{\mathcal {Z}}}(\pmb {u}^{\beta })|\), or \(|\widetilde{{\mathcal {Z}}}(\pmb {u}^{\beta })|> |\widetilde{ {\mathcal {Z}}}(\pmb {u}^{\alpha })|\) everywhere outside the exceptional divisor \(E_{tan}\). Hence, again by the existence of transition maps close to saddle points, we conclude the proof of Properties (IV.ii.a), (IV.ii.b), and (IV.ii.c).

In order to prove the claim, consider a sequence of combinatorial blowings-up so that the strict transform \(T^{st}\) of T does not intersect 2-points. Indeed, by direct computation over local charts, the pull-back of the metric \(h_{{\bar{z}}}\) again satisfies the Hsiang–Pati coordinate system over 2-points in the pre-image of \({\bar{z}}\) (more explicitly, this follows by combining Eqs. (4.8) with (D.2)). Now, denote by \({\bar{T}}\) the analogue of the set T, but computed after the sequence of combinatorial blowings-up; since \(\widetilde{{\mathscr {L}}}\) is a line foliation (therefore, generated by one vector field), we conclude that \(T^{st}\) and \({\bar{T}}\) coincide everywhere outside of the exceptional divisor, which proves the claim.

Finally, let \({\bar{z}}\) be a point contained in only one connected component of \(E_{tan}\) and assume that \({\bar{z}}\) is not a singularity of \(\widetilde{{\mathscr {L}}}\). Then, up to taking a sufficiently small neighborhood of \({\bar{z}}\), the flow-box Theorem (see e.g. [3, Theorem 1.12] or [21, Theorem 1.14]) implies properties (IV.i.a), (IV.i.b), and (IV.i.c). This concludes the proof. \(\square \)

Remark 4.8

As follows from Lemma 4.3, \(\widetilde{{\mathscr {L}}}\) is tangent to a component F of the exceptional divisor if and only if so is the pull-back foliation \(\pi ^* {\mathscr {L}}\). Indeed, in the language of the local vector fields generating these foliations, Lemma 4.3 yields “\({\mathcal {Z}}\) is tangent to F if and only if \(\widetilde{{\mathcal {Z}}}\) is”.

Lemma 4.9

(Compatibility of stratifications) Recall the notation of Lemma 2.4 and of Theorem 4.7. Up to adding a locally finite number of points to \(\Sigma ^0\), we have:

$$\begin{aligned} \pi (E_{tr}\setminus E_{tan}) \subset \Sigma _{tr}^1 , \text { and } \pi (E_{tan}) \subset \Sigma ^0 \cup \Sigma ^1_{tan}. \end{aligned}$$

Proof of Lemma 4.9

We start by making two remarks:

(1) up to adding a locally finite number of points to \(\Sigma ^0\), without loss of generality we can assume that \(\Sigma ^0\) contains all points \({\bar{w}}\in \Sigma \) where \(\pi \) has \(\text{ log } \text{ rk }\) equal to 0 over the fiber of \({\bar{w}}\).

(2) Let F be an irreducible exceptional divisor of E where the log-rank is constant equal to 1. Then \(\pi (F)\) is an analytic curve over \(\Sigma \); furthermore, by expression (D.3), it follows that \(d\pi |_{T F}: TF \rightarrow T\pi (F)\) is an isomorphism. In particular, \(\Delta \) is tangent to \(\pi (F)\) at \({\bar{w}} = \pi ({\bar{z}})\) if and only if \(\pi ^* {\mathscr {L}}\) is tangent to F at \({\bar{z}}\). By Remark 4.8, this latter property is equivalent to the tangency of \(\widetilde{{\mathscr {L}}}\) to F at \({\bar{z}}\).

Now, by Theorem 4.7(I), we know that \(\pi ( E_{tan} \cup E_{tr}) \subset \Sigma ^0 \cup \Sigma ^1_{tan}\cup \Sigma ^1_{tr}\). Therefore, by the second remark, it is clear that \(\pi (E_{tr}) \subset \Sigma ^1_{tr}\). Next, let \({\bar{w}} \in \Sigma _{tr}^1\) and note that the \(\text{ log } \text{ rk }\) can be assumed to be constant equal to 1 along the fiber \(\pi ^{-1}({\bar{w}})\), thanks to the first remark. Moreover, if we assume by contradiction that there exists \({\bar{z}} \in \pi ^{-1}({\bar{w}})\) which belongs to \(E_{tan}\), we get a contradiction with the second remark. We conclude easily. \(\square \)

Remark 4.10

Unlike for the complex analytic spaces, a resolution map of a real analytic space is not necessarily surjective and its image equals the closure of the regular part. For instance for the Whitney umbrella \(\{(x,y,z)\in {\mathbb {R}}^3; y^2= z x^2 \}\), the singular part is the vertical line \(\{x=y=0\}\), and the image of any resolution map equals \(\{(x,y,z)\in {\mathbb {R}}^3; y^2= z x^2, z\ge 0 \}\) and does not contain “the handle” \(\{x=y=0, z<0\}\).

4.3 Proof of Proposition 3.1

We follow the notation of Theorem 4.7. Without loss of generality, we may suppose that the pre-image of \(\Sigma ^0\) contain all points over which \(\pi \) has log-rank equal to 0. Next, we note that the singular set of \(\widetilde{{\mathscr {L}}}\) is a locally finite set of discrete points contained in \(E_{tan}\). By Lemma 4.9 and the fact that \(\pi \) is proper, we conclude that there exists a locally finite set of points \({\widetilde{\Sigma }}^0 \subset \Sigma ^0 \cup \Sigma ^1_{tan} = {\widetilde{\Sigma }}\) whose pre-image contain all singular points of \(\widetilde{{\mathscr {L}}}\) and all points where log-rank of \(\pi \) is zero. Apart from adding a locally finite number of points to \({\widetilde{\Sigma }}^0\), we can suppose that \(\Sigma ^0 \subset {\widetilde{\Sigma }}^0\).

Now, let \(\gamma : [0,1) \rightarrow \Sigma \) be a convergent transverse-singular trajectory such that \({\bar{y}}:=\lim _{t\rightarrow 1} \gamma (t) \in {\widetilde{\Sigma }}\). Denote by \({\widetilde{\gamma }}\) the strict transform of \(\gamma ([0,1))\) under \(\pi \), that is

$$\begin{aligned} {\widetilde{\gamma }} := \overline{\pi ^{-1}(\gamma ([0,1))\setminus \Sigma ^{1}_{tr})}. \end{aligned}$$

By hypothesis, we know that the topological limit of \({\widetilde{\gamma }}\), which is defined by

$$\begin{aligned} \omega ({\widetilde{\gamma }}) := \bigcap _{t\in [0,1)} \overline{\pi ^{-1}(\gamma ([t,1))\setminus \Sigma ^{1}_{tr})}, \end{aligned}$$

is contained in the pre-image of \({\bar{y}}\), say \(F =\pi ^{-1}({\bar{y}}) \subset E_{tan}\).

Now, suppose for a contradiction that \({\bar{y}} \notin {\widetilde{\Sigma }}^0\). In this case, \(\widetilde{{\mathscr {L}}}\) is an everywhere regular foliation over F, and \(\pi \) has log-rank equals to 1 over F. By Eq. (D.3) and Theorem 4.7(IV.i.b), we conclude that the topological limit of \({\widetilde{\gamma }}\) must contain an open neighborhood of F in \(E_{tan}\), which projects into a 1-dimensional analytic set over \(\Sigma \). This is a contradiction with the definition of convergent transverse-singular trajectory, which implies that \({\bar{y}} \in {\widetilde{\Sigma }}^0\).

We now need the following:

Proposition 4.11

A convergent transverse-singular trajectory \(\gamma :[0,1) \rightarrow \Sigma \) is characteristic, if and only if, the topological limit \(\omega ({\tilde{\gamma }})\) is a singular point \({\bar{z}}\) of \(\widetilde{{\mathscr {L}}}\) and in this case \({{\widetilde{\gamma }}}\) is a characteristic orbit of an analytic vector field that generates \(\widetilde{{\mathscr {L}}}\) in a neighborhood of \({\bar{z}}\).

Proof of Proposition 4.11

Let \(\gamma \) be a convergent transverse-singular trajectory. Since \(\omega (\gamma ) \in {\widetilde{\Sigma }}\), we conclude that \(\omega ({\widetilde{\gamma }}) \subset E_{tan}\). Now, if \(\omega ({\widetilde{\gamma }})\) contains a regular point \({\bar{z}}\) of \(\widetilde{{\mathscr {L}}}\), then it follows by the Flow-Box Theorem that there exists a neighborhood U of \({\bar{z}}\) where \({\widetilde{\gamma }} \cap U\) intersects an infinite number of times a transverse section of \(\widetilde{{\mathscr {L}}}\) passing by \({\bar{z}}\). Therefore, since all singular points of \(\widetilde{{\mathscr {L}}}\) are isolated, we may assume that \(\omega ({\widetilde{\gamma }})= \{{\bar{z}}\}\), where \({\bar{z}}\) is singular point of \(\widetilde{{\mathscr {L}}}\). Since all singular points are saddles, given a local generator \(\widetilde{{\mathcal {Z}}}\) of \(\widetilde{{\mathscr {L}}}\) defined in a sufficiently small open neighborhood of \({\bar{z}}\), there are only a finite number of characteristic orbits of \(\widetilde{{\mathcal {Z}}}\) converging to \({\bar{z}}\). We conclude easily. \(\square \)

As a consequence of the last proposition, we can now prove the following result, which concludes the proof of Proposition 3.1.

Proposition 4.12

Let \(\gamma :[0,1) \rightarrow \Sigma \) be a (convergent transverse-singular) characteristic trajectory, then \(\gamma ([0,1))\) is a semianalytic curve.

Proof of Proposition 4.12

The strict transform \({{\widetilde{\gamma }}}\) of \(\gamma \) is a characteristic orbit of a saddle singularity, and therefore, it is semianalytic by the stable manifold Theorem of Briot and Bouquet [10] (see, e.g. [3, Theorem 2.7]). To conclude, we note that the image of a semianalytic curve by a proper analytic map is semianalytic, see Remark B.2. \(\square \)

4.4 Proof of Proposition 3.7

We follow the notation of Theorem 4.7, Proposition 3.7, and Remark 3.8. Without loss of generality, we may assume that there exists an open neighborhood \({\mathcal {W}}\) of \({\bar{y}}\) such that \(\gamma ([0,1)) \subset {\mathcal {W}}\), \(\lambda ([0,1)) \subset {\mathcal {W}}\), and either \(\gamma ([0,1))\cap \Sigma _{tr}^1 = \lambda ([0,1))\cap \Sigma _{tr}^1 = \emptyset \) with sequences of times \(\{t_k^{\gamma }\}_{k\in {\mathbb {N}}}, \{t_k^{\lambda }\}_{k\in {\mathbb {N}}}\), or \(\gamma ([0,1))\cap \Sigma _{tr}^1 \ne \emptyset \), \(\lambda ([0,1))\cap \Sigma _{tr}^1 \ne \emptyset \) are infinite with a common sequence of jumps \(\{(i_k,j_k)\}_{k\in {\mathbb {N}}}\) associated respectively with \(\{t_k^{\gamma }\}_{k\in {\mathbb {N}}}\) and \(\{t_k^{\lambda }\}_{k\in {\mathbb {N}}}\).

The Riemmanian metric g is bi-lipschitz equivalent to an analytic metric over \({\mathcal {W}}\). Since Proposition 3.7 is invariant by local bi-lipschitz equivalence of metrics, we suppose without loss of generality that g is analytic. We denote by \({\mathcal {V}}\) the pre-image of \({\mathcal {W}}\) under \(\pi \), and by \(g^{*}\) the pull-back of g via \(\pi \) (which is analytic).

We recall that \(\gamma (0)\) and \(\lambda (0)\) are assumed to belong to the same section \(\Lambda \) and that \(d^{\Lambda }(\lambda (0))<d^{\Lambda }(\gamma (0))\). We denote by \({\widetilde{\gamma }}\) and \({\widetilde{\lambda }}\) the strict transform of \(\gamma \) and \(\lambda \) (defined as in the proof of Proposition 3.1).

Since the transition maps \(T^{\Lambda }\) and \(T^{j_k}_{i_k}\) satisfy property (3.2) and (3.3) respectively, we note that

$$\begin{aligned} d^{\Lambda }\left( \gamma (t^{\gamma }_k)\right) < d^{\Lambda } \left( \lambda (t^{\lambda }_k)\right) \qquad \forall \,k\in {\mathbb {N}}\end{aligned}$$

in the case \(\gamma ([0,1))\cap \Sigma _{tr}^1 = \lambda ([0,1))\cap \Sigma _{tr}^1 = \emptyset \), and

$$\begin{aligned} d^{i_k}\left( \gamma (t_k^{\gamma })\right) < d^{i_k}\left( \lambda (t_k^{\lambda })\right) \qquad \forall k\in {\mathbb {N}}\end{aligned}$$

in the case where \(\gamma ([0,1))\cap \Sigma _{tr}^1 \ne \emptyset \) and \(\lambda ([0,1))\cap \Sigma _{tr}^1 \ne \emptyset \) are infinite.

Finally, since \(\pi \) is a proper morphism, in order to prove Proposition 3.7 it is enough to show a similar result, locally, at every point on the resolution space which belongs to the topological limit \(\omega ({\widetilde{\gamma }}) = \omega ({\widetilde{\lambda }})\).

Since \(\gamma \) is monodromic, if a point \({\bar{z}} \in \omega ({\widetilde{\gamma }})\) is a saddle of \(\widetilde{{\mathscr {L}}}\), then there are two connected components of \(E_{tan}\) which contain \({\bar{z}}\) (in other words, it satisfies the normal form given in Theorem 4.7(IV.ii)). Therefore, for each \({\bar{z}} \in \omega ({\widetilde{\gamma }})\), either the normal form (IV.i) or (IV.ii) of Theorem 4.7 is verified. We study these two possibilities separately (we do not distinguish (IV.i.b) and (IV.i.c) in this part of the proof). In both cases, given local sections \(\Lambda _{{\bar{z}}}^{j}\) for \(j=1,2\) we consider the distance functions

$$\begin{aligned} d_{{\bar{z}}}^j: \Lambda _{{\bar{z}}}^{j} \rightarrow {\mathbb {R}}, \end{aligned}$$

where \(d_{{\bar{z}}}^j(p)\) is the length (via \(g^{*}\)) of the half arc contained in \(\Lambda _{{\bar{z}}}^j\) that joins p to \(E_{tan}\), and \(\text{ length}^{g^{*}}\) denotes the length with respect to \(g^{*}\).

The next lemma handles the first case.

Lemma 4.13

Recalling the notation of Theorem 4.7(IV.i), assume that there exists only one connected component of \(E_{tan}\) which contains \({{\bar{z}}}\). For each point \(p \in \Lambda ^1_{{\bar{z}}}\), denote by L(p) the half-leaf of \(\widetilde{{\mathscr {L}}}\) whose boundary is given by p and \(\phi _{{\bar{z}}}(p)\). Then there exists \(\epsilon >0\) (which depends only on the neighborhood of \({\bar{z}}\)) such that

$$\begin{aligned} d^1_{{\bar{z}}}(p)< d^1_{{\bar{z}}}(q)< \epsilon \quad \implies \quad \mathrm{length}^{g^{*}}(L( p)) \le \mathrm{length}^{g^{*}}(L(q)). \end{aligned}$$

Proof of Lemma 4.13

Note that \(\widetilde{{\mathscr {L}}}\) is non-singular, so there exists a non-singular locally defined vector field \(\widetilde{{\mathcal {Z}}}\) which generates \(\widetilde{{\mathscr {L}}}\). Denote by \(\varphi ^{\widetilde{{\mathcal {Z}}}}_p(t)\) the flow of \(\widetilde{{\mathcal {Z}}}\) with time t and initial condition p. Since \(\widetilde{{\mathcal {Z}}}\) is non-zero, for each \(p\in \Lambda ^1_{{\bar{z}}}\) the minimal time \(t_p\) so that \(\varphi _p^{\widetilde{{\mathcal {Z}}}}(t_p) \in \Lambda ^2_{{\bar{z}}}\) is an analytic function in p. It follows that the function

$$\begin{aligned} \mathrm{length}^{g^{*}}(L(p)) = \int _{0}^{t_p} \left| {\widetilde{{\mathcal {Z}}}}(\varphi ^{\widetilde{{\mathcal {Z}}}}_p(s))\right| ^{g^{*}}ds \end{aligned}$$

is analytic over \(\Lambda ^1_{{\bar{z}}}\), since all objects are analytic. Furthermore, \(\mathrm{length}^{g^{*}}(L(p)) \ge 0\), and it is equal to zero if and only if \(p\in E\). This implies the desired monotonicity property. \(\square \)

We now handle the second case, which is the only point of the paper where we use the regularity result from [38].

Lemma 4.14

Recalling the notation of Theorem 4.7(IV.ii), assume that there exist two connected component of \(E_{tan}\) which contains \({{\bar{z}}}\). For each point \(p \in \Lambda ^1_{{\bar{z}}}\), denote by L(p) the half-leaf of \(\widetilde{{\mathscr {L}}}\) whose boundary is p and \(\phi _{{\bar{z}}}(p)\). Then there exist \(K>0\) and \(\epsilon >0\) (which depends only on the neighborhood of \({\bar{z}}\)) such that

$$\begin{aligned} d^1_{{\bar{z}}}(p)< d^1_{{\bar{z}}}(q)<\epsilon \quad \implies \quad \mathrm{length}^{g^{*}}(L( p)) \le K\, \mathrm{length}^{g^{*}}(L(q)). \end{aligned}$$

Proof of Lemma 4.14

Up to changing \(K>0\), it suffices to prove the result for a metric that is bi-lipschitz equivalent to \(g^{*}\). Therefore, without loss of generality we assume that \(g^{*} = h_{{\bar{z}}} = (d \pmb {u}^{\alpha })^2+(d \pmb {u}^{\beta })^2\) (see Theorem 4.7(IV.ii.b)). Although the definition of \(\alpha \) and \(\beta \) are not symmetric (see (D.5)), this does not interfere in this part of the proof; so we assume that \(\widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha }) \ne 0\) everywhere outside of E, and that \(|\widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha })|\ge |\widetilde{{\mathcal {Z}}}(\pmb {u}^{\beta })|\) everywhere in the neighborhood of \({\bar{z}}\), where \(\widetilde{{\mathcal {Z}}}\) is a local generator of \(\widetilde{{\mathscr {L}}}\) (the other case is analogous).

Denote by \(\varphi ^{\widetilde{{\mathcal {Z}}}}_p(t)\) the flow of \(\widetilde{{\mathcal {Z}}}\) with time t and initial condition p. For each \(p\in \Lambda ^1_{{\bar{z}}}\setminus E\) the minimal time \(t_p\) so that \(\varphi _p^{\widetilde{{\mathcal {Z}}}}(t_p) \in \Lambda ^2_{{\bar{z}}}\) is an analytic function over \(\Lambda ^1_{{\bar{z}}}\setminus E\), but it does not admit an analytic extension to \(\Lambda ^1_{{\bar{z}}}\cap E\). In particular, the function

$$\begin{aligned} \mathrm{length}^{g^{*}}(L(p)) = \int _{0}^{t_p} \left| \widetilde{{\mathcal {Z}}}(\varphi _p^{\widetilde{{\mathcal {Z}}}}(s))\right| ^{g^{*}}ds \end{aligned}$$

does not admit an analytic extension to \(\Lambda ^1_{{\bar{z}}}\cap E\). Nevertheless, we note that:

$$\begin{aligned} \begin{aligned} \mathrm{length}^{g^{*}}(L(p))&= \int _{0}^{t_p} \sqrt{\left[ \widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha })\circ \varphi _p^{\widetilde{{\mathcal {Z}}}}(s)\right] ^2 + \left[ \widetilde{{\mathcal {Z}}}(\pmb {u}^{\beta })\circ \varphi _p^{\widetilde{{\mathcal {Z}}}}(s)\right] ^2}\,ds\\&=\int _{0}^{t_p} \sqrt{ d(\pmb {u}^{\alpha } \circ \varphi _p^{\widetilde{{\mathcal {Z}}}}(s) )^2 + d(\pmb {u}^{\beta } \circ \varphi _p^{\widetilde{{\mathcal {Z}}}}(s) )^2}\, ds. \end{aligned} \end{aligned}$$

Now, since \(\widetilde{{\mathcal {Z}}}(\pmb {u}^{\alpha })\) is analytic and vanishes only on E, we conclude that \(d (\pmb {u}^{\alpha } \circ \varphi _p^{\widetilde{{\mathcal {Z}}}}(s)) \) is of constant sign outside of E. On the one hand, this implies that

$$\begin{aligned} \begin{aligned} \mathrm{length}^{g^{*}}(L(p))&\ge \int _{0}^{t_p} |d(\pmb {u}^{\alpha } \circ \varphi _p^{\widetilde{{\mathcal {Z}}}}(s) |ds\\&= \left| \pmb {u}^{\alpha }(p) - \pmb {u}^{\alpha }\circ \phi _{{\bar{z}}}(p)\right| . \end{aligned} \end{aligned}$$

On the other hand, from the fact that \(|d(\pmb {u}^{\alpha } \circ \varphi _p^{\widetilde{{\mathcal {Z}}}}(s)) | \ge |d(\pmb {u}^{\beta } \circ \varphi _p^{\widetilde{{\mathcal {Z}}}}(s)) |\), we conclude that

$$\begin{aligned} \begin{aligned} \mathrm{length}^{g^{*}}(L(p))&\le \sqrt{2} \int _{0}^{t_p} |d(\pmb {u}^{\alpha } \circ \varphi _p^{\widetilde{{\mathcal {Z}}}}(s) |ds\\&= \sqrt{2} \left| \pmb {u}^{\alpha }(p) - \pmb {u}^{\alpha }\circ \phi _{{\bar{z}}}(p)\right| . \end{aligned} \end{aligned}$$

Although the function \(\phi _{{\bar{z}}}(p)\) is analytic outside of E, it does not admit an analytic extension to E and the treatment of this case differs from the one in Lemma 4.13.

In order to be precise, denote by \(\lambda _i:[0,1] \rightarrow \Lambda ^i_{{\bar{z}}}\) an analytic parametrizations of the sections \(\Lambda ^i_{{\bar{z}}}\) such that \(\lambda _i(0) \in E\), for \(i=1,2\). They can be always chosen so that \(\pmb {u}^{\alpha } \circ \lambda _i(t) = t^{a_i}\) for some \(a_i \in {\mathbb {N}}\). Now, by [38, Theorem 1.4 and Remark 4.13], the composition \(\lambda _2^{-1} \circ \phi _{{\bar{z}}} \circ \lambda _1 \circ (\exp (-x))\) belongs to a Hardy field \({\mathcal {F}}\) of germs of function at infinity which also contains the exponential function, see “Appendix C” for some more details. Thus, since \({\mathcal {F}}\) is a field, it follows that also the function

$$\begin{aligned} \begin{aligned}&\pmb {u}^{\alpha } \circ \lambda _1 \circ (\exp (-x))- \pmb {u}^{\alpha } \circ \phi _{{\bar{z}}}\circ \lambda _1 \circ (\exp (-x))\\&\quad = \left( \exp (-x)\right) ^{a_1} - \left( \lambda _2^{-1} \circ \phi _{{\bar{z}}}\circ \lambda _1 \circ (\exp (-x))\right) ^{a_2} \end{aligned} \end{aligned}$$

belongs to it. Since \({\mathcal {F}}\) is a Hardy field, Lemma C.1 implies that the function

$$\begin{aligned} \pmb {u}^{\alpha }\circ \lambda _1(t) - \pmb {u}^{\alpha }\circ \phi _{{\bar{z}}} \circ \lambda _1(t) \end{aligned}$$

is monotone for t sufficiently close to 0 (that is, for \(p = \lambda _1(t)\) sufficiently close to E). We conclude easily from this observation and the inequalities (4.11) and (4.12). \(\square \)

As observed before, Proposition 3.7 now follows from the two lemmas above and the previous considerations made in all this section.