Comparison theorems for conjugate points in sub-Riemannian geometry

Barilari, D.; Rizzi, L.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 439-472.

We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems, that play the role of the constant curvature spaces. As an application, we prove a version of sub-Riemannian Bonnet−Myers theorem and we obtain some new results on conjugate points for three dimensional left-invariant sub-Riemannian structures.

Classification : 53C17, 53C21, 53C22, 49N10
Mots clés : Sub-Riemannian geometry, curvature, comparison theorems, conjugate points

H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations: In Control and Systems Theory. Systems and Control: Foundations and Applications. Springer Verlag, NY (2003). | MR | Zbl

A. Agrachev, Exponential mappings for contact sub-Riemannian structures. J. Dyn. Control Syst. 2 (1996) 321–358. | DOI | MR | Zbl

A. Agrachev, Any sub-Riemannian metric has points of smoothness. Dokl. Akad. Nauk 424 (2009) 295–298. | MR | Zbl

A. Agrachev and D. Barilari, Sub-Riemannian structures on 3D Lie groups. J. Dyn. Control Syst. 18 (2012) 21–44. | DOI | MR | Zbl

A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry (Lecture Notes). Preprint SISSA 09/2012/M. Available at http://hdl.handle.net/1963/5877 (2012). | MR

A. Agrachev, D. Barilari and L. Rizzi, The curvature: a variational approach. Preprint (2013). | arXiv | MR

A. Agrachev and R.V. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry. I. Regular extremals. J. Dyn. Control Syst. 3 (1997) 343–389. | DOI | MR | Zbl

A. Agrachev and P.W.Y. Lee, Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds. Math. Ann. 360 (2014) 209–253. | DOI | MR | Zbl

A. Agrachev, L. Rizzi and P. Silveira, On conjugate times of LQ optimal control problems. J. Dyn. Control Syst. (2014) 1–17. | MR

A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint. Control Theory and Optimization, II. Vol. 87 of Encycl. Math. Sci. Springer-Verlag, Berlin (2004). | MR | Zbl

A. Agrachev and I. Zelenko, Geometry of Jacobi curves. I. J. Dyn. Control Syst. 8 (2002) 93–140. | DOI | MR | Zbl

Y. Awais Butt, Y.L. Sachkov and A.I. Bhatti, Extremal Trajectories and Maxwell Strata in Sub-Riemannian Problem on Group of Motions of Pseudo-Euclidean Plane. J. Dyn. Control Syst. 20 (2014) 341–364. | DOI | MR | Zbl

D. Barilari, U. Boscain, G. Charlot and R.W. Neel. On the heat diffusion for generic Riemannian and sub-Riemannian structures. Preprint (2013). | arXiv | MR

D. Barilari, U. Boscain and R.W. Neel, Small-time heat kernel asymptotics at the sub-Riemannian cut locus. J. Differ. Geom. 92 (2012) 373–416. | DOI | MR | Zbl

F. Baudoin, M. Bonnefont and N. Garofalo, A sub-riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality. Math. Ann. (2013) 1–28. | MR | Zbl

F. Baudoin, M. Bonnefont, N. Garofalo and I.H. Munive, Volume and distance comparison theorems for sub-Riemannian manifolds. Preprint (2014), To appear in J. Funct. Anal. (2015). | arXiv | MR | Zbl

F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. To appear in J. Eur. Math. Soc. (2014). | MR

F. Baudoin and J. Wang, Curvature-dimension inequalities and subelliptic heat kernel gradient bounds on contact manifolds. Potential Anal. 40 (2014) 163–193. | DOI | MR | Zbl

U. Boscain and F. Rossi, Invariant Carnot-Caratheodory metrics on $S^3,SO\left(3\right),SL\left(2\right)$ and lens spaces. SIAM J. Control Optim. 47 (2008) 1851–1878. | DOI | MR | Zbl

V. Gershkovich and A. Vershik, Nonholonomic manifolds and nilpotent analysis. J. Geom. Phys. 5 (1988) 407–452. | DOI | MR | Zbl

V. Jurdjevic, Geometric Control Theory. Vol. 52 of Cambridge Stud. Adv. Math. Cambridge University Press, Cambridge (1997). | MR | Zbl

P.W.Y. Lee, C. Li and I. Zelenko, Measure contraction properties of Sasakian manifolds. Preprint (2014); To appear in Discrete Contin. Dyn. Syst. Ser. A (2016). | arXiv

C. Li and I. Zelenko, Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries. J. Geom. Phys. 61 (2011) 781–807. | DOI | MR | Zbl

I. Moiseev and Y.L. Sachkov, Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 380–399. | Numdam | MR | Zbl

S.B. Myers, Riemannian manifolds with positive mean curvature. Duke Math. J. 8 (1941) 401–404. | DOI | MR | Zbl

S.-I. Ohta, On the curvature and heat flow on hamiltonian systems. Anal. Geom. Metr. Spaces 2 (2014) 81–114. | MR | Zbl

S.-I. Ohta and K.-T. Sturm, Bochner–Weitzenböck formula and Li–Yau estimates on Finsler manifolds. Adv. Math. 252 (2014) 429–448. | DOI | MR | Zbl

L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes. Translated from the Russian by K.N. Trirogoff. Edited by L.W. Neustadt. Interscience Publishers John Wiley & Sons, Inc.  New York, London (1962). | MR

L. Rifford, Ricci curvatures in Carnot groups. Math. Control Relat. Fields 3 (2013) 467–487. | DOI | MR | Zbl

Y.L. Sachkov, Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 1018–1039. | Numdam | MR | Zbl

C. Villani, Optimal Transport. In Vol. 338 of Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences]. | MR | Zbl

B.Y. Wu and Y.L. Xin, Comparison theorems in Finsler geometry and their applications. Math. Ann. 337 (2007) 177–196. | DOI | MR | Zbl

I. Zelenko and C. Li, Differential geometry of curves in Lagrange Grassmannians with given Young diagram. Differ. Geom. Appl. 27 (2009) 723–742. | DOI | MR | Zbl