Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T05:28:07.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

Published online by Cambridge University Press:  20 November 2018

Philippe Delanoë  and
François Rouvière
Philippe Delanoë
Affiliation:
Université de Nice-Sophia Antipolis,Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice Cedex 2, e-mail: Philippe.Delanoe@unice.fr, Francois.Rouviere@unice.fr
François Rouvière
Affiliation:
Université de Nice-Sophia Antipolis,Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice Cedex 2, e-mail: Philippe.Delanoe@unice.fr, Francois.Rouviere@unice.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The squared distance curvature is a kind of two-point curvature the sign of which turned out to be crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, and an indirect one (via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

Keywords

53C3553C2153C2649N60symmetric spacesrank onepositive curvaturealmost-positive c-curvature
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 4 , 01 August 2013 , pp. 757 - 767
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Besse, A. L., Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete, 93, Springer-Verlag, Berlin, 1978.Google Scholar
[2] Brenier, Y., Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 305(1987), no. 19, 805–808.Google Scholar
[3] Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44(1991), no. 4, 375–417. http://dx.doi.org/10.1002/cpa.3160440402 Google Scholar
[4] Cartan, E., Sur une classe remarquable d’espaces de Riemann. Bull. Soc. Math. France 54(1926), 216–264.Google Scholar
[5] Brenier, Y., Sur une classe remarquable d’espaces de Riemann. II. Bull. Soc. Math. France 55(1927), 114–134.Google Scholar
[6] Chavel, I., On Riemannian symmetric spaces of rank one. Advances in Math. 4(1970), 236–263. http://dx.doi.org/10.1016/0001-8708(70)90025-3 Google Scholar
[7] Chavel, I., Riemannian symmetric spaces of rank one. Lecture Notes in Pure and Applied Mathematics, 5, Marcel Dekker Inc., New York, 1972.Google Scholar
[8] Cheeger, J. and Ebin, D. G., Comparison theorems in Riemannian geometry. North-Holland Mathematical Library, 9, North-Holland Publishing Co., Amsterdam, 1975.Google Scholar
[9] Crittenden, R. J., Minimum and conjugate points in symmetric spaces. Canad. J. Math. 14(1962), 320–328. http://dx.doi.org/10.4153/CJM-1962-024-8 Google Scholar
[10] Delanoë, Ph., Gradient rearrangement for diffeomorphisms of a compact manifold. Differential Geom. Appl. 20(2004), no. 2, 145–165. http://dx.doi.org/10.1016/j.difgeo.2003.10.003 Google Scholar
[11] Delanoë, Ph. and Ge, Y., Regularity of optimal transport on compact, locally nearly spherical, manifolds. J. Reine Angew. Math. 646(2010), 65–115. http://dx.doi.org/10.1515/crelle.2010.066 Google Scholar
[12] Delanoë, Ph., Locally nearly spherical surfaces are almost-positively c-curved. Methods Appl. Anal. 18(2011), no. 3, 269–302.Google Scholar
[13] Delanoë, Ph. and Loeper, G., Gradient estimates for potentials of invertible gradient-mappings on the sphere. Calc. Var. Partial Differential Equations 26(2006), no. 3, 297–311. http://dx.doi.org/10.1007/s00526-006-0006-4 Google Scholar
[14] Falcitelli, M., Ianus, S., and Pastore, A. M., Riemannian submersions and related topics. World Scientific Publishing Co. Inc., River Edge, NJ, 2004. http://dx.doi.org/10.1142/9789812562333 Google Scholar
[15] Figalli, A. and Rifford, L., Continuity of optimal transport maps and convexity of injectivity domains on small deformations of S2. Comm. Pure Appl. Math. 62(2009), no. 12, 1670–1706. http://dx.doi.org/10.1002/cpa.20293 Google Scholar
[16] Figalli, A., Rifford, L., and Villani, C., Nearly round spheres look convex. Amer. J. Math. 134(2012), no. 1, 109–139. http://dx.doi.org/10.1353/ajm.2012.0000 Google Scholar
[17] Gluck, H., Warner, F., and Ziller, W., The geometry of the Hopf fibrations. Enseign. Math. (2) 32(1986), no. 3–4, 173–198.Google Scholar
[18] Helgason, S., Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978.Google Scholar
[19] Karcher, H., A geometric classification of positively curved symmetric spaces and the isoparametric construction of the Cayley plane. On the geometry of differentiable manifolds (Rome 1986). Astérisque 163/164(1988), no. 6, 111–135, 282.Google Scholar
[20] Kim, Y.-H. and McCann, R. J., Continuity, curvature, and the general covariance of optimal transportation. J. Eur. Math. Soc. (JEMS) 12(2010), no. 4, 1009–1040. http://dx.doi.org/10.4171/JEMS/221 Google Scholar
[21] Kim, Y.-H., Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular). J. Reine Angew. Math., to appear.Google Scholar
[22] Klingenberg, W., Riemannian geometry. de Gruyter Studies in Mathematics, 1 ,Walter de Gruyter & Co., Berlin-New York, 1982.Google Scholar
[23] Kobayashi, S. and Nomizu, K., Foundations of differential geometry. Vol I. Interscience Publishers, a division of JohnWiley & Sons, New York-London, 1963.Google Scholar
[24] Kobayashi, S., Foundations of differential geometry. Vol. II. Interscience Tracts in Pure and Applied Mathematics, 15, Interscience Publishers JohnWiley & Sons, Inc., New York-London-Sydney, 1969.Google Scholar
[25] Loeper, G., On the regularity of solutions of optimal transportation problems. Acta Math. 202(2009), no. 2, 241–283. http://dx.doi.org/10.1007/s11511-009-0037-8 Google Scholar
[26] Loeper, G., Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna. Arch. Ration. Mech. Anal. 199(2011), no. 1, 269–289. http://dx.doi.org/10.1007/s00205-010-0330-x Google Scholar
[27] Ma, X.-N., Trudinger, N. S., and Wang, X.-J., Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177(2005), no. 2, 151–183. http://dx.doi.org/10.1007/s00205-005-0362-9 Google Scholar
[28] McCann, R. J., Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal. 11(2001), no. 3, 589–608. http://dx.doi.org/10.1007/PL00001679 Google Scholar
[29] Trudinger, N. S. and Wang, X.-J., On the second boundary value problem for Monge-Ampère type equations and optimal transportation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(2009), no. 1, 143–174.Google Scholar
[30] Villani, C., Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. http://dx.doi.org/10.1007/978-3-540-71050-9 Google Scholar