Ricci curvatures in Carnot groups

Ludovic Rifford

doi:10.3934/mcrf.2013.3.467

Article Contents

2013, Volume 3, Issue 4: 467-487. Doi: 10.3934/mcrf.2013.3.467

This issue Previous Article Sparse stabilization and optimal control of the Cucker-Smale model Next Article

Ricci curvatures in Carnot groups

Ludovic Rifford^1,

1.
Université de Nice-Sophia Antipolis, Labo. J.-A. Dieudonné, UMR CNRS 6621, Parc Valrose, 06108 Nice Cedex 02

Received: November 30, 2012

Revised: January 31, 2013

Published: November 2013

Abstract Related Papers Cited by

Abstract

We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy such properties and give a lower bound of possible curvature exponents in terms of the datas.

Keywords:

Mathematics Subject Classification: 53C23, 58C15, 22E25.

Citation:

Related Papers

Cited by

References

[1]	A. Agrachev, Any sub-Riemannian metric has points of smoothness, Dokl. Akad. Nauk, 424 (2009), 295-298; Translation in Dokl. Math., 79 (2009), 45-47.doi: 10.1134/S106456240901013X.
[2]	A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, to appear.
[3]	A. Agrachev and P. Lee, Optimal transportation under nonholonomic constraints, Trans. Amer. Math. Soc., 361 (2009), 6019-6047.doi: 10.1090/S0002-9947-09-04813-2.
[4]	A. Agrachev and P. Lee, Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds, preprint, (2009).
[5]	V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361.doi: 10.5802/aif.233.
[6]	A. Bellaïche, The tangent space in sub-Riemannian geometry, in "Sub-Riemannian Geometry," Progr. Math., 144, Birkhäuser, Basel, (1996), 1-78.doi: 10.1007/978-3-0348-9210-0_1.
[7]	P. Cannarsa and L. Rifford, Semiconcavity results for optimal control problems admitting no singular minimizing controls, Ann. Inst. H. Poincaré Non Linéaire, 25 (2008), 773-802.doi: 10.1016/j.anihpc.2007.07.005.
[8]	P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.
[9]	M. Castelpietra and L. Rifford, Regularity properties of the distance function to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry, ESAIM Control Optim. Calc. Var., 16 (2010), 695-718.doi: 10.1051/cocv/2009020.
[10]	A. Figalli and L. Rifford, Mass transportation on sub-Riemannian manifolds, Geom. Funct. Anal., 20 (2010), 124-159.doi: 10.1007/s00039-010-0053-z.
[11]	S. Gallot, D. Hulin and J. Lafontaine, "Riemannian Geometry," Third edition, Universitext, Springer-Verlag, Berlin, 2004.doi: 10.1007/978-3-642-18855-8.
[12]	C. Golé and R. Karidi, A note on Carnot geodesics in nilpotent Lie groups, J. Dynam. Control Systems, 1 (1995), 535-549.doi: 10.1007/BF02255895.
[13]	M. Gromov, "Metric Structures for Riemannian and Non-Riemannian Spaces," Progress in Mathematics, Vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999.
[14]	J. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus, Trans. Amer. Math. Soc., 353 (2001), 21-40.doi: 10.1090/S0002-9947-00-02564-2.
[15]	N. Juillet, Geometric inequalities and generalized Ricci bounds in the Heisenberg group, Int. Math. Res. Not. IMRN, (2009), 2347-2373.doi: 10.1093/imrn/rnp019.
[16]	N. Juillet, On a method to disprove generalized Brunn-Minkowski inequalities, in "Probabilistic Approach to Geometry," Adv. Stud. Pure. Math., 57, Math. Soc. Japan, Tokyo, (2010), 189-198.
[17]	E. Le Donne, Lecture notes on sub-Riemannian geometry, preprint, (2010).
[18]	Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math., 58 (2005), 85-146.doi: 10.1002/cpa.20051.
[19]	J. Lott and C. Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal., 245 (2007), 311-333.doi: 10.1016/j.jfa.2006.10.018.
[20]	J. Milnor, Curvatures of left-invariant metrics on Lie groups, Advances in Math., 21 (1976), 293-329.doi: 10.1016/S0001-8708(76)80002-3.
[21]	J. Mitchell, On Carnot-Carathéodory spaces, J. Differential Geom., 21 (1985), 35-45.
[22]	R. Montgomery, A tour of sub-Riemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, Vol. 91, American Mathematical Society, Providence, RI, 2002.
[23]	S. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv., 82 (2007), 805-828.doi: 10.4171/CMH/110.
[24]	L. Rifford, Sub-Riemannian geometry and optimal transport, preprint, (2012).
[25]	K. -T. Sturm, On the geometry of metric measure spaces. I, Acta Math., 196 (2006), 65-131.doi: 10.1007/s11511-006-0002-8.
[26]	K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math., 196 (2006), 133-177.doi: 10.1007/s11511-006-0003-7.
[27]	C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.doi: 10.1007/978-3-540-71050-9.