# American Institute of Mathematical Sciences

September  2013, 33(9): 3903-3913. doi: 10.3934/dcds.2013.33.3903

## Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane

 1 School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University

Received  September 2012 Revised  January 2013 Published  March 2013

This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. I prove that the only convex billiard without conjugate points on the hyperbolic plane or on the hemisphere is a circular billiard.
Citation: Misha Bialy. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3903-3913. doi: 10.3934/dcds.2013.33.3903
##### References:
 [1] M.-C. Arnaud, A particular minimization property implies $C^0$ -integrability, J. Differential Equations, 250 (2011), 2389-2401. doi: 10.1016/j.jde.2010.12.002. [2] M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154. doi: 10.1007/BF02572397. [3] M. Bialy, Maximizing orbits for higher dimensional convex billiards, Journal of Modern Dynamics, 3 (2009), 51-59. doi: 10.3934/jmd.2009.3.51. [4] V. Blumen, K. Y. Kim, J. Nance and V. Zharnitsky, Three-period orbits in billiards on the surfaces of constant curvature Int. Math. Res. Notices, 2012 (2012), 5014-5024. [5] Yu. Burago and V. Zalgaller, "Geometric Inequalities," Translated from the Russian by A. B. Sosinskiĭ, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 285, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1988. [6] D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal., 4 (1994), 259-269. doi: 10.1007/BF01896241. [7] B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature, Comm. Math. Phys., 208 (1999), 65-90. doi: 10.1007/s002200050748. [8] E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries, J. Geom. Phys., 40 (2002), 277-301. doi: 10.1016/S0393-0440(01)00039-0. [9] E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Regul. Chaotic Dyn., 8 (2003), 1-13. doi: 10.1070/RD2003v008n01ABEH000222. [10] S. Elaydi, "An Introduction to Difference Equations," Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005. [11] E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U. S. A., 34 (1948), 47-51. [12] J. Heber, On the geodesic flow of tori without conjugate points, Math. Z., 216 (1994), 209-216. doi: 10.1007/BF02572318. [13] N. Innami, Integral formulas for polyhedral and spherical billiards, J. Math. Soc. Japan, 50 (1998), 339-357. doi: 10.2969/jmsj/05020339. [14] V. Kaloshin and A. Sorrentino, On conjugacy of convex billiards, preprint, arXiv:1203.1274. [15] A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3 (1990), 961-973. [16] R. MacKay, J. Meiss and J. Strark, Converse KAM theory for symplectic twist maps, Nonlinearity, 2 (1989), 555-570. [17] A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107. doi: 10.1016/0393-0440(90)90021-T. [18] M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164. [19] S. Tabachnikov, Billiards, Panor. Synth., 1 (1995), vi+142 pp.

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##### References:
 [1] M.-C. Arnaud, A particular minimization property implies $C^0$ -integrability, J. Differential Equations, 250 (2011), 2389-2401. doi: 10.1016/j.jde.2010.12.002. [2] M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154. doi: 10.1007/BF02572397. [3] M. Bialy, Maximizing orbits for higher dimensional convex billiards, Journal of Modern Dynamics, 3 (2009), 51-59. doi: 10.3934/jmd.2009.3.51. [4] V. Blumen, K. Y. Kim, J. Nance and V. Zharnitsky, Three-period orbits in billiards on the surfaces of constant curvature Int. Math. Res. Notices, 2012 (2012), 5014-5024. [5] Yu. Burago and V. Zalgaller, "Geometric Inequalities," Translated from the Russian by A. B. Sosinskiĭ, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 285, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1988. [6] D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal., 4 (1994), 259-269. doi: 10.1007/BF01896241. [7] B. Gutkin, U. Smilansky and E. Gutkin, Hyperbolic billiards on surfaces of constant curvature, Comm. Math. Phys., 208 (1999), 65-90. doi: 10.1007/s002200050748. [8] E. Gutkin and S. Tabachnikov, Billiards in Finsler and Minkowski geometries, J. Geom. Phys., 40 (2002), 277-301. doi: 10.1016/S0393-0440(01)00039-0. [9] E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Regul. Chaotic Dyn., 8 (2003), 1-13. doi: 10.1070/RD2003v008n01ABEH000222. [10] S. Elaydi, "An Introduction to Difference Equations," Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005. [11] E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U. S. A., 34 (1948), 47-51. [12] J. Heber, On the geodesic flow of tori without conjugate points, Math. Z., 216 (1994), 209-216. doi: 10.1007/BF02572318. [13] N. Innami, Integral formulas for polyhedral and spherical billiards, J. Math. Soc. Japan, 50 (1998), 339-357. doi: 10.2969/jmsj/05020339. [14] V. Kaloshin and A. Sorrentino, On conjugacy of convex billiards, preprint, arXiv:1203.1274. [15] A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3 (1990), 961-973. [16] R. MacKay, J. Meiss and J. Strark, Converse KAM theory for symplectic twist maps, Nonlinearity, 2 (1989), 555-570. [17] A. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107. doi: 10.1016/0393-0440(90)90021-T. [18] M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164. [19] S. Tabachnikov, Billiards, Panor. Synth., 1 (1995), vi+142 pp.
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