Morse-Sard type results in sub-Riemannian geometry
- 122 Downloads
Let (M,Δ,g) be a sub-Riemannian manifold and x0 ∈ M. Assuming that Chow’s condition holds and that M endowed with the sub-Riemannian distance is complete, we prove that there exists a dense subset N1 of M such that for every point x of N1, there is a unique minimizing path steering x0 to x, this trajectory admitting a normal extremal lift. If the distribution Δ is everywhere of corank one, we prove the existence of a subset N2 of M of full Lebesgue measure such that for every point x of N2, there exists a minimizing path steering x0 to x which admits a normal extremal lift, is nonsingular, and the point x is not conjugate to x0. In particular, the image of the sub-Riemannian exponential mapping is dense in M, and in the case of corank one is of full Lebesgue measure in M.
KeywordsManifold Lebesgue Measure Dense Subset Exponential Mapping Full Lebesgue Measure
Unable to display preview. Download preview PDF.
- 1.Agrachev, A.: Compactness for sub-Riemannian length-minimizers and subanalyticity. Rend. Sem. Mat. Univ. Politec. Torino 56(4), 1–12 (1998)Google Scholar
- 2.Agrachev, A., Bonnard, B., Chyba, M., Kupka, I.: Sub-Riemannian sphere in Martinet flat case. ESAIM Cont. Optim. Calc. Var. 2, 377–448 (1997)Google Scholar
- 3.Bellaïche, A.: Tangent space in sub-Riemannian geometry. Sub-Riemannian geometry, (Birkhäuser, 1996)Google Scholar
- 4.Bismut, J.-M.: Large deviations and the Malliavin calculus. Progress in Mathematics 45, (Birkhäuser, 1984)Google Scholar
- 5.Borwein, J.M., Preiss, D.: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Amer. Math. Soc. 303(2), 517–527 (1987)Google Scholar
- 6.Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics 178, (Springer-Verlag, New York 1998)Google Scholar
- 7.Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91, (American Mathematical Society, Providence, RI, 2002)Google Scholar
- 8.Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mischenko, E.: The mathematical theory of optimal processes. (Wiley Interscience, 1962)Google Scholar
- 9.Trélat, E.: Some properties of the value function and its level sets for affine control systems with quadratic cost. J. Dyn. Cont. Syst. 6(4), 511–541 (2000)Google Scholar