The Sard conjecture on Martinet surfaces
Given a totally nonholonomic distribution of rank two on a three-dimensional manifold we investigate the size of the set of points that can be reached by singular horizontal paths starting from a same point. In this setting, the Sard conjecture states that that set should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero. We prove that the conjecture holds in the case where the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities.
Let be a smooth connected manifold of dimension and let be a totally nonholonomic distribution of rank on , that is a smooth subbundle of of dimension such that for every there is an open neighborhood of where is locally parametrized by linearly independent smooth vector fields satisfying the so-called Hörmander or bracket generating condition
An absolutely continuous curve is called horizontal with respect to if it satisfies
By Chow-Rashevsky’s Theorem, total nonholonomicity plus connectedness implies horizontal path-connectedness. In other words, for any two points there is an horizontal path such that and . Given , the set of horizontal paths starting from whose derivative is square integrable (with respect to a given metric on ) can be shown to enjoy the structure of a Hilbert manifold. However, in general given the set of paths in which join to fails to be a submanifold of globally, it may have singularities. It happens to be a submanifold only in neighborhoods of horizontal paths which are not singular. The Sard conjecture for totally nonholonomic distributions is concerned with the size of the set of points that can be reached by those singular paths in . It is related to some of the major open problems in sub-Riemannian geometry, see [1, 14, 17, 18, 19].
The aim of the present paper is to solve partially this conjecture in the case of rank-two distributions in dimension three. Before stating precisely our result, we wish to define rigorously the notion of singular horizontal path. For further details on the material presented here, we refer the reader to Bellaïche’s monograph , or to the books by Montgomery , by Agrachev, Baralilari and Boscain , or by the second author . For specific discussions about the Sard conjecture and sub-Riemannian geometry, we suggest , [14, Chapter 10] and [17, 18].
To introduce the notion of singular horizontal path, it is convenient to identify the horizontal paths with the trajectories of a control system. It can be shown that there is a finite family (with ) of smooth vector fields on such that
For every , there is a non-empty maximal open set such that for every control , the solution to the Cauchy problem
is well-defined. By construction, for every and every control the trajectory is an horizontal path in . Moreover the converse is true, any can be written as the solution of (1.1) for some . Of course, since in general the vector fields are not linearly independent globally on , the control such that is not necessarily unique. For every point , the End-Point Mapping from (associated with in time ) is defined as
It shares the same regularity as the vector fields , it is of class . Given , a control is said to be singular (with respect to ) if the linear mapping
is not onto, that is if is not a submersion at . Then, we call an horizontal path singular if and only if for some which is singular (with respect to ). It is worth noting that actually the property of singularity of an horizontal path does depend only on , it is independent of the choice of and of the control which is chosen to parametrize the path. For every , we denote by the set of controls which are singular with respect to . As we said, the Sard conjecture is concerned with the size of the set of end-points of singular horizontal paths in given by
The set is defined as the set of critical values of the smooth mapping , so according to Sard’s theorem we may expect it to have Lebesgue measure zero in which is exactly the statement of the Sard conjecture. Unfortunately, Sard’s theorem is known to fail in infinite dimension (see ), so we cannot prove by ”abstract nonsense” that has measure zero. In fact, singular horizontal paths can be characterized as the projections of the so-called abnormal extremals, which allows, in some cases, to describe the set of singular horizontal paths as the set of orbits of some vector field in . The first case of interest is the case of rank two totally nonholonomic distributions in dimension three whose study is the purpose of the present paper.
If has dimension three and rank two, it can be shown that the singular horizontal paths are those horizontal paths which are contained in the so-called Martinet surface (see Proposition A.2)
where is the (possibly singular) distribution defined by
Moreover, by total nonholonomicity of the distribution, the set can be covered by a countable union of smooth submanifolds of codimension at least one. Consequently for every , the set is always contained in which has zero Lebesgue measure zero, so that the Sard conjecture as stated above holds true for rank two (totally nonholonomic) distributions in dimension three. In fact, since for this specific case singular horizontal paths are valued in a two-dimensional subset of the Sard conjecture for rank-two distributions in dimension three is stronger and asserts that all the sets have vanishing -dimensional Hausdorff measure. The validity of this conjecture is supported by a major contribution in the nineties made by Zelenko and Zhitomirskii who proved that for generic rank-two distributions (with respect to the Whitney topology) all the sets have indeed Hausdorff dimension at most one, see . Since then, no notable progress has been made. The purpose of the present paper is to attack the non-generic case. Our first result is concerned with distributions for which the Martinet surface is smooth.
Let be a smooth manifold of dimension and a rank-two totally nonholonomic distribution on whose Martinet surface is smooth. Then for every the set has -dimensional Hausdorff measure zero.
The key idea of the proof of Theorem 1.1 is to observe that the divergence of the vector field which generates the trace of the distribution on is controlled by its norm (see Lemma 2.3). To our knowledge, such an observation has never been made nor used before. This idea plays also a major role for our second result which is concerned with the real-analytic case.
Let us now assume that both and are real-analytic with of rank two and of dimension three. By total nonholonomicity, the set is a closed analytic set in of dimension and for every there is an open neighborbood of and a non-zero analytic function such that . In general, this analytic set admits singularities, that is some points in a neighborhood of which is not diffeomorphic to a smooth surface. To prove our second result, we will show that techniques from resolution of singularities allow to recover what is needed to apply the ideas that govern the proof of Theorem 1.1. Since resolution of singularity is an algebraic process which applies to spaces that include a functional structure and not to sets, we need, before stating Theorem 1.2, to introduce a few notions from analytic geometry, we refer the reader to [10, 15, 20] for more details. We will consider analytic spaces where is an analytic set in and is a principal reduced and coherent ideal sheaf with support , which means that admits a locally finite covering by open sets and that there is a family of analytic functions such that the following properties are satisfied:
For any , there exists an analytic function such that and for all .
For every , and the set is a set of codimension at least two in .
The set of singularities or singular set of an analytic space , denoted by , is defined as the union of the sets for . By the properties (i)-(ii) above, is an analytic subset of of codimension at least two, so it can be stratified by strata respectively of dimension zero and one, where is a locally finite union of points and is a locally finite union of analytic submanifolds of of dimension one. Then, for every we define the tangent space to at as if belongs to and if belongs to . The proof of our second result is based on the resolution of singularities in the setting of coherent ideals and analytic spaces described above which was obtained by Hironaka [9, 10] (in fact, we will follow the modern proof of Hironaka’s result given by Bierstone and Milman which includes a functorial property [7, 8] - see also [11, 23] and references therein). In particular, it requires the Martinet surface to have the structure of a coherent analytic space. This fact is proven in Appendix C, according to it we use from now the notation to refer to the analytic space defined by the Martinet surface. We are now ready to state our second result.
Let be an analytic manifold of dimension and a rank-two totally nonholonomic analytic distribution on , assume that
Then for every the set has -dimensional Hausdorff measure zero.
The assumption (1.2) means that at each singularity of the distribution generates the tangent space to . It is trivially satisfied in the case where has only isolated singularities.
Let be an analytic manifold of dimension and a rank-two totally nonholonomic analytic distribution on , assume that has only isolated singularities, that is . Then for every the set has -dimensional Hausdorff measure zero.
If the assumption (1.2) in Theorem 1.2 is not satisfied, that is if there are points in where is transverse to , then our approach leads to the study of possible concatenations of homoclinic orbits of a smooth vector field defined on which vanishes on . Let us illustrate what may happen by treating an example. In let us consider the totally nonholonomic analytic distribution spanned by the two vector fields
We check easily that , so that the Martinet surface is given by
The Martinet surface is an analytic set whose singular set is the vertical axis , see Figure 1. We observe that the distribution is transverse to the -dimensional subset of , so the assumption of Theorem 1.2 is not satisfied. Let be the loop part of , that is the set of points in of the form with , for each we can construct an horizontal path such that and belongs to with . Let us show how to proceed. First, we notice that the trace of on outside its singular set is generated by the smooth vector field (we set )
which is collinear (on ) to the vector field
If we forget about the -coordinate, the projection of onto the plane (whose phase portrait is drawn in Figure 2) has a unique equilibrium at the origin, has the vertical axis as invariant set where it coincides with , and it is equal to on the horizontal axis. Moreover, enjoyes a property of symmetry with respect to the horizontal axis, there holds where . Consequently, the quadrant is invariant with respect to and any trajectory of starting from a point of the form with satisfies
Moreover, we can check that the curvature of its projection has the sign of which is equal to so it has a constant sign. In conclusion, the curve converges to the origin as tends to , it is symmetric with respect to the horizontal axis, and it surrounds a convex surface which is contained in the rectangle (because which is on ). Then is homoclinic and its length satisfies
Moreover, we check easily that (see Appendix D)
In conclusion, if we fix , then there is such that the orbit starting from at time satisfies , for some , has length and the inequalities (1.3)-(1.6) are satisfied with . If we repeat this construction from , then we get a decreasing sequence of positive real numbers together with a sequence of lengths such that for every ,
where stands for the maximum of and . Moreover, the sequences are associated with a sequence of singular horizontal paths of length which joins to for every . Therefore, concatenating the paths we get a singular horizontal path which depends upon the starting point and which tends to as tends to , see Figure 3. The union of the all singular paths obtained in this way with will fill the loop-part of , so we may think that the set of horizontal paths starting from the origin reach a set of positive -dimensional Hausdorff measure.
Hopefully, we can prove that all the paths constructed above have infinite length and so are not admissible because they do not allow to construct horizontal paths starting from the origin. To see this, assume that there are sequences satisfying (1.7) such that is finite. By the second inequality in (1.7) the sum should be finite as well. But since
for large enough, we have for for every integers
Which means that holds when so that
and contradicts the finiteness of . As we can see, the estimates (1.7) prevent the existence of horizontal paths obtained as concatenations of infinitely many homoclinic orbits. The availability of such estimates and more generally possible extensions of Theorem 1.2 in absence of assumption (1.2) will be the subject of a subsequent paper.
The paper is organized as follows: The proof of Theorem 1.1 is given in Section 2. The Section 3 is devoted to the proof of Theorem 1.2 which is divided in two parts. The first part (Section 3.1) consists in extending to the singular analytic case an intermediate result already used in the proof of Theorem 1.1 and the second part (Section 3.3) presents the result from resolution of singularities (Proposition 3.3) which allows to apply the arguments of divergence which are used in the proof of Theorem 1.1. The Section 3.2 in between those two sections, which should be seen as a warm-up to Section 3.3, is dedicated to the study of the specific case of a conical singularity. There, we explain how a simple blowup yields Proposition 3.3 for this case. The proof of Proposition 3.3 is given in Section 4 and all the material and results from resolution of singularities in manifolds with corner which are necessary for its proof are provided in Section 5. Finally, the appendix gathers the proofs of several results which are referred to in the body of the paper.
Acknowledgments. The first author would like to thank Edward Bierstone for several useful discussions. The second author is grateful to Adam Parusinski for enlightening discussions. We would also like to thank the hospitality of the Fields Institute and the Universidad de Chile.
Throughout this section we assume that is a rank-two totally nonholonomic distribution on (of dimension three) whose Martinet surface is a smooth submanifold. First of all, we notice that, since for every the set is contained in , the result of Theorem 1.1 holds if has dimension one. So we may assume from now on that has dimension two. Let us define two subsets of by
By construction, is an open subset of and is closed. The following holds.
The set is countably smoothly -rectifiable, that is it can be covered by countably many submanifolds of dimension one.
Proof of Lemma 2.1.
We need to show that for every there is an open neighborhood of in such that is contained in a smooth submanifold of dimension one. Let be fixed. Taking a sufficiently small open neighborhood of in and doing a change of coordinates if necessary we may assume that there is a set of coordinates such that and and there are two smooth vector fields on of the form
where are smooth functions such that and
Then we have on which yields ( denotes the partial derivative of a smooth function with respect to the variable)
and moreover a point belongs to if and only if . Let be the set of for which for some pair in . The set is open in and by the Implicit Function Theorem the set can be covered by a finite union of smooth submanifolds of of dimension one. Therefore, it remains to show that the set
can be covered by countably many smooth submanifolds of dimension one. By total nonholonomicity, for every there are a least integer and a -tuple such that the vector field defined by
satisfies and all the vector fields for defined by
satisfy . Then for every we have and
Consequently, taking smaller if necessary (in order to use that are well-defined in a neighborhood of ) and using compactness, there is a finite number of smooth functions such that for every there are and such that and . We conclude easily by the Implicit Function Theorem. ∎
Define the singular distribution, that is a distribution with non-constant rank, by
We observe that by Proposition A.2, for every non-constant singular horizontal path there is a set which is open in such that for any and for any , and moreover, for almost every , . By smoothness of on , the open set can be foliated by the orbits of . Let us fix a Riemannian metric on whose geodesic distance is denoted . Through each point of there is a maximal orbit , that is a one-dimensional smooth submanifold of which is either compact and diffeomorphic to a circle, or open and parametrized by a smooth curve parametrized by arc-length (with respect to the metric ) of the form
where belong to . Note that every open orbit admits two parametrizations as above. For every whose orbit is open, we call half-orbit of , denoted by , any of the two subsets of given by or . This pair of sets does not depend on the parametrization of . We observe that if some half-orbit of the form is part of a singular horizontal path between two points, then it must have finite length, or equivalently finite measure, and in consequence is finite and necessarily has a limit as tends to . In this case, we call end of , denoted by , the limit which by construction belongs to . The following result will allow to work in a neighborhood of a point of . In the statement, and stand respectively for the and -dimensional Hausdorff measures associated with .
Assume that there is such that has positive -dimensional Hausdorff measure. Then there is such that for every neighborhood of in , there are two closed sets in satisfying the following properties:
for every , there is a half-orbit which is contained in such that and .
Proof of Lemma 2.2.
Let be fixed such that . Since by Lemma 2.1 the set has Hausdorff dimension at most one, the set is a subset of the smooth surface with positive area (for the volume associated with the restriction of to or equivalently for any Lebesgue measure on ). Let us denote by the set of Lebesgue density points of on and define the singular distance from , by
for every , where if there is no singular horizontal path joining to . By construction, is lower semicontinuous on and finite on . We claim that the set of such that any half-orbit of satisfies has measure zero in . As a matter of fact, consider some such that for any half-orbit . Since belongs to , there is a (singular) horizontal path with , and (the existence of such a minimizing arc follows easily by compactness arguments, see ). If is not contained in then it must contains an half-orbit of which is assumed to have infinite length, which contradicts . Therefore, , belongs to , and thas to be on the orbit of , which proves that has measure zero in . In conclusion, the set has positive measure in and for every there is an half-orbit with finite length. So there is and a closed subset of with positive measure such that for every there is an half-orbit such that . In other words, any can be joined to some point of by a singular horizontal path of length . Define the projection onto the set (here denotes the set of subsets of )
for every . By classical compactness results in sub-Riemannian geometry, the domain of is closed in and its graph is closed in . We have just proved that for every , the set is nonempty. Now if we consider a countable and locally finite covering of the closed set by geodesic open balls of radius , there is a closed subset of of positive measure such that for every . Considering a finite covering of by open sets of diameters less than , we get a closed set such that for some , for all . By a recursive argument, we construct a decreasing sequence of open sets of (geodesic) diameter along with a sequence of closed sets of positive measure such that for every , for any . Let us prove that such that yields the result.
Let be a neighborhood of in . By construction, there is such that and for any . Set
By construction, is closed, nonempty, and satisfies (ii). Moreover, for every there is an half-orbit with and . Furthermore, we note that if , and is an half-orbit such that , then for every in , is an end of the half-orbit of which is contained in . This means that we can replace by some set such that (i) and (iii) are satisfied. ∎
From now on, we assume that there is some such that has positive -dimensional Hausdorff measure and we fix some given by Lemma 2.2. Since is a smooth surface, there are a relatively compact open neighborhood of in and a set of coordinates in such that and , and there are two smooth vector fields on of the form
with such that for every . In addition, without loss of generality we may assume that the above property holds on a neighborhood of . Define the smooth vector field on by