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Mathematische Zeitschrift

, Volume 282, Issue 1–2, pp 99–130 | Cite as

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part I

  • Erlend GrongEmail author
  • Anton Thalmaier
Article

Abstract

We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in the inequality. Using this inequality, we obtain a lower bound for the eigenvalues of the sub-Laplacian. This inequality also lays the foundation for proving several powerful results in Part II.

Keywords

Curvature-dimension inequality Sub-Riemannian geometry Hypoelliptic operator Spectral gap Riemannian foliations 

Mathematics Subject Classification

58J35 53C17 58J99 

Notes

Acknowledgments

This work has been supported by the Fonds National de la Recherche Luxembourg (AFR 4736116 and OPEN Project GEOMREV).

References

  1. 1.
    Agrachev, A., Barilari, D., Rizzi, L.: Curvature: a variational approach. To appear in: Memoirs AMS. Arxiv e-prints: arXiv:1306.5318 (2013)
  2. 2.
    Agrachev, A., Boscain, U., Gauthier, J.P., Rossi, F.: The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256(8), 2621–2655 (2009). doi: 10.1016/j.jfa.2009.01.006 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, pp. 177-206. Springer, Berlin (1985). doi: 10.1007/BFb0075847
  4. 4.
    Bakry, D., Ledoux, M.: Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123(2), 259–281 (1996). doi: 10.1007/s002220050026 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barilari, D., Rizzi, L.: Comparison theorems for conjugate points in sub-Riemannian geometry. To appear in: ESAIM Control Optim. Calc. Var. doi: 10.1051/cocv/2015013
  6. 6.
    Baudoin, F., Bonnefont, M.: Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality. J. Funct. Anal. 262(6), 2646–2676 (2012). doi: 10.1016/j.jfa.2011.12.020 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baudoin, F., Bonnefont, M., Garofalo, N.: A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality. Math. Ann. 358(3–4), 833–860 (2014). doi: 10.1007/s00208-013-0961-y MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baudoin, F., Garofalo, N.: Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. To appear in: J. Eur. Math. Soc. ArXiv e-prints: arXiv:1101.3590 (2011)
  9. 9.
    Baudoin, F., Kim, B., Wang, J.: Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves. ArXiv e-prints: arXiv:1408.0548 (2014)
  10. 10.
    Baudoin, F., Wang, J.: Curvature dimension inequalities and subelliptic heat kernel gradient bounds on contact manifolds. Potential Anal. 40(2), 163–193 (2014). doi: 10.1007/s11118-013-9345-x MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bott, R.: On a topological obstruction to integrability. In: Global Analysis (Proceedings of the Symposium Pure Math., vol. XVI, Berkeley, Calif., 1968), pp. 127-131. American Math. Soc., Providence, RI (1970)Google Scholar
  12. 12.
    Chow, W.L.: Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117, 98–105 (1939)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Émery, M.: Stochastic calculus in manifolds. Universitext. Springer, Berlin (1989). doi: 10.1007/978-3-642-75051-9. With an appendix by P.-A. Meyer
  14. 14.
    Hladky, R.K.: Bounds for the first eigenvalue of the horizontal Laplacian in positively curved sub-Riemannian manifolds. Geom. Dedicata 164, 155–177 (2013). doi: 10.1007/s10711-012-9766-5 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hsu, E.P.: Stochastic analysis on manifolds. In: Graduate Studies in Mathematics, vol. 38. American Mathematical Society, Providence, RI (2002). doi: 10.1090/gsm/038
  17. 17.
    Kolář, I., Michor, P.W., Slovák, J.: Natural operations in differential geometry. Springer, Berlin (1993). doi: 10.1007/978-3-662-02950-3
  18. 18.
    Li, C., Zelenko, I.: Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries. J. Geom. Phys. 61(4), 781–807 (2011). doi: 10.1016/j.geomphys.2010.12.009 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Meyer, P.A.: Géométrie stochastique sans larmes. In: Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), Lecture Notes in Math., vol. 850, pp. 44-102. Springer, Berlin (1981)Google Scholar
  20. 20.
    Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications. In: Mathematical Surveys and Monographs, vol. 91. American Mathematical Society, Providence, RI (2002)Google Scholar
  21. 21.
    Rashevskii, P.K.: On joining any two points of a completely nonholonomic space by an admissible line. Math. Ann. 3, 83–94 (1938)Google Scholar
  22. 22.
    Reinhart, B.L.: Foliated manifolds with bundle-like metrics. Ann. Math. 2(69), 119–132 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shigekawa, I.: On stochastic horizontal lifts. Z. Wahrsch. Verw. Gebiete 59(2), 211–221 (1982). doi: 10.1007/BF00531745 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Strichartz, R.S.: Sub-Riemannian geometry. J. Differ. Geom. 24(2), 221–263 (1986). http://projecteuclid.org/euclid.jdg/1214440436
  25. 25.
    Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and \(L^p\)-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994). doi: 10.1515/crll.1994.456.173 MathSciNetzbMATHGoogle Scholar
  26. 26.
    Wang, F.Y.: Equivalence of dimension-free Harnack inequality and curvature condition. Integral Equ. Oper. Theory 48(4), 547–552 (2004). doi: 10.1007/s00020-002-1264-y CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Zelenko, I., Li, C.: Parametrized curves in Lagrange Grassmannians. C. R. Math. Acad. Sci. Paris 345(11), 647–652 (2007). doi: 10.1016/j.crma.2007.10.034 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Mathematics Research Unit, FSTCUniversity of LuxembourgLuxembourgGrand Duchy of Luxembourg

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