Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity (original) (raw)
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Journal of Mathematical Sciences, 2006
The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way first we obtain a new elementary proof of classical Levi-Civita's Theorem about the classification of all Riemannian geodesically equivalent metrics in a neighborhood of so-called regular (stable) point w.r.t. these metrics. Secondly we prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally we make the classification of all pairs of geodesically equivalent Riemannian metrics on a surface, which proportional in an isolated point. This is the simplest case, which was not covered by Levi-Civita's Theorem. * S.I.S.S.A.,
SOME RESULTS ON SUB-RIEMANNIAN GEOMETRY
Sub-Riemannian structures naturally occur in different branches of Mathematics in the study of constrained systems in classical mechanics, in optimal control, geometric measure theory and differential geometry. In this paper, we show that Sub-Riemannian structures on three manifolds locally depend on two functions 1 and K of three variables and we investigate how these differential invariants influence the geometry.
On projective and affine equivalence of sub-Riemannian metrics
Geometriae Dedicata
Consider a smooth connected manifold M equipped with a bracket generating distribution D. Two sub-Riemannian metrics on (M, D) are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric g is called rigid (resp. conformally rigid) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to g is constantly proportional to g (resp. conformal to g). In the Riemannian case the local classification of projectively (resp. affinely) equivalent metrics was done in the classical work (Levi-Civita in Ann Mat Ser 2a 24:255-300, 1896; resp. Eisenhart in Trans Am Math Soc 25 : [297][298][299][300][301][302][303][304][305][306] 1923). In particular, a Riemannian metric which is not rigid with respect to one of the above equivalences satisfies the following two special properties: its geodesic flow possesses a collection of nontrivial integrals of special type and the metric induces certain canonical product structure on the ambient manifold. The only proper sub-Riemannian cases to which these classification results were extended so far are sub-Riemannian metrics on contact and quasi-contact distributions (Zelenko in J Math Sci (NY) 135(4): 2006). The general goal is to extend these results to arbitrary sub-Riemannian manifolds. In this article we establish two types of results toward this goal: if a sub-Riemannian metric is not conformally rigid with respect to the projective equivalence, then, first, its flow of normal extremals has at least one nontrivial integral quadratic on the fibers of the cotangent bundle and, second, the nilpotent approximation of the underlying distribution at any point admits a product structure. As a consequence we obtain two types of genericity results for rigidity: first, we show that a generic sub-Riemannian metric on a fixed pair (M, D) is conformally rigid with respect to projective equivalence. Second, we prove that, except for special pairs (m, n), for a generic distribution D of rank m on an ndimensional manifold, every sub-Riemannian metric on D is conformally rigid with respect
The Schouten Curvature for a Nonholonomic Distribution in Sub-Riemannian Geometry and Jacobi Fields
2018
The paper shows that if the distribution is defined on a manifold with the special smooth structure and does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature tensor. The Schouten curvature tensor is used to write the Jacobi equation for the distribution. This leads to studies on second-order optimality conditions for the horizontal geodesics in subRiemannian geometry. Conjugate points are defined by the solutions of the Jacobi equation. If a geodesic passed a point conjugated with its beginning then this geodesic ceases to be optimal.
End-Point Equations and Regularity of Sub-Riemannian Geodesics
Geometric and Functional Analysis, 2008
For a large class of equiregular sub-Riemannian manifolds, we show that length-minimizing curves have no corner-like singularities. Our first result is the reduction of the problem to the homogeneous, rank-2 case, by means of a nilpotent approximation. We also identify a suitable condition on the tangent Lie algebra implying existence of a horizontal basis of vector fields whose coefficients depend only on the first two coordinates x 1 , x 2 . Then, we cut the corner and lift the new curve to a horizontal one, obtaining a decrease of length as well as a perturbation of the end-point. In order to restore the end-point at a lower cost of length, we introduce a new iterative construction, which represents the main contribution of the paper. We also apply our results to some examples.
On Weyl's type theorems and genericity of projective rigidity in sub-Riemannian Geometry
2020
H. Weyl in 1921 demonstrated that for a connected manifold of dimension greater than 1, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper, we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl projective rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl projectively rigid and genericity of Weyl projectively rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this al...
Morse theory for normal geodesics in sub-Riemannian manifolds with codimension one distributions
2003
We consider a Riemannian manifold (M, g) and a codimension one distribution ∆ ⊂ T M on M which is the orthogonal of a unit vector field Y on M. We do not make any nonintegrability assumption on ∆. The aim of the paper is to develop a Morse Theory for the sub-Riemannian action functional E on the space of horizontal curves, i.e. everywhere tangent to the distribution ∆. We consider the case of horizontal curves joining a smooth submanifold P of M and a fixed point q ∈ M. Under the assumption that P is transversal to ∆, it is known (see ) that the set of such curves has the structure of an infinite dimensional Hilbert manifold and that the critical points of E are the so called normal extremals (see ). We compute the second variation of E at its critical points, we define the notions of P-Jacobi field, of P-focal point and of exponential map and we prove a Morse Index Theorem. Finally, we prove the Morse relations for the critical points of E under the assumption of completeness for (M, g).