December  2013, 3(4): 467-487. doi: 10.3934/mcrf.2013.3.467

Ricci curvatures in Carnot groups

1. 

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

Received  December 2012 Revised  February 2013 Published  September 2013

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.
Citation: Ludovic Rifford. Ricci curvatures in Carnot groups. Mathematical Control and Related Fields, 2013, 3 (4) : 467-487. doi: 10.3934/mcrf.2013.3.467
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.

show all references

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.

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