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Higher rank rigidity for Berwald spaces

Part of: Dynamical systems with hyperbolic behavior Symplectic geometry, contact geometry Global differential geometry

Published online by Cambridge University Press:  18 December 2018

WEISHENG WU
WEISHENG WU*
Affiliation:
Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, 100083, PR China email wuweisheng@cau.edu.cn
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Abstract

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded non-positive flag curvature with rank at least two whose universal cover is irreducible is a locally symmetric space or a locally Minkowski space.

Keywords

higher rank rigiditygeodesic flowBerwald spacesFinsler spacesnon-positive flag curvature

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 53D25: Geodesic flows 53C24: Rigidity results
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020 , pp. 1991 - 2016
Copyright
© Cambridge University Press, 2018

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References

Ballmann, W.. Nonpositively curved manifolds of higher rank. Ann. of Math. (2) 122(3) (1985), 597–609.Google Scholar
Ballmann, W., Brin, M. and Eberlein, P.. Structure of manifolds of nonpositive curvature. I. Ann. of Math. (2) 122(2) (1985), 171–203.Google Scholar
Ballmann, W., Brin, M. and Spatzier, R.. Structure of manifolds of nonpositive curvature. II. Ann. of Math. (2) 122(2) (1985), 205–235.Google Scholar
Ballmann, W., Gromov, M. and Schroeder, V.. Manifolds of Nonpositive Curvature (Progress in Mathematics, 61). Birkhäuser, Boston, MA, 1985.Google Scholar
Bao, D., Chern, S.-S. and Shen, Z.. An Introduction to Riemann–Finsler Geometry (Graduate Texts in Mathematics, 200). Springer, New York, 2000.Google Scholar
Burns, K. and Spatzier, R.. On topological Tits buildings and their classification. Publ. Math. Inst. Hautes Études Sci. 65(1) (1987), 5–34.Google Scholar
Burns, K. and Spatzier, R.. Manifolds of nonpositive curvature and their buildings. Publ. Math. Inst. Hautes Études Sci. 65(1) (1987), 35–59.Google Scholar
Deng, S. and Hou, Z.. Positive definite Minkowski Lie algebras and bi-invariant Finsler metrics on Lie groups. Geom. Dedicata 136(1) (2008), 191–201.Google Scholar
Eberlein, P. and O’Neill, B.. Visibility manifolds. Pacific J. Math. 46(1) (1973), 45–109.Google Scholar
Egloff, D.. Uniform Finsler Hadamard manifolds. Ann. Inst. Henri Poincaré Phys. Théor. 66(3) (1997), 323–357.Google Scholar
Foulon, P.. Curvature and global rigidity in Finsler manifolds. Houston J. Math. 28(2) (2002), 263–292.Google Scholar
Heintze, E. and Im Hof, H.-C.. Geometry of horospheres. J. Differential Geom. 12(4) (1977), 481–491.Google Scholar
Jost, J.. Nonpositive Curvature: Geometric and Analytic Aspects (Lectures in Mathematics ETH Zürich). Birkhäuser, Basel, 1997.Google Scholar
Kristály, A. and Kozma, L.. Metric characterization of Berwald spaces of non-positive flag curvature. J. Geom. Phys. 56(8) (2006), 1257–1270.Google Scholar
Mostow, G. D.. Strong Rigidity of Locally Symmetric Spaces (Annals of Mathematics Studies, 78). Princeton University Press, Princeton, NJ, 1973.Google Scholar
Shen, Z.. Differential Geometry of Spray and Finsler Spaces. Kluwer Academic, Dordrecht, 2001.Google Scholar