# 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 & 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.  Google Scholar [2] M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154. doi: 10.1007/BF02572397.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [6] D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal., 4 (1994), 259-269. doi: 10.1007/BF01896241.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Regul. Chaotic Dyn., 8 (2003), 1-13. doi: 10.1070/RD2003v008n01ABEH000222.  Google Scholar [10] S. Elaydi, "An Introduction to Difference Equations," Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005.  Google Scholar [11] E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U. S. A., 34 (1948), 47-51.  Google Scholar [12] J. Heber, On the geodesic flow of tori without conjugate points, Math. Z., 216 (1994), 209-216. doi: 10.1007/BF02572318.  Google Scholar [13] N. Innami, Integral formulas for polyhedral and spherical billiards, J. Math. Soc. Japan, 50 (1998), 339-357. doi: 10.2969/jmsj/05020339.  Google Scholar [14] V. Kaloshin and A. Sorrentino, On conjugacy of convex billiards,, preprint, ().   Google Scholar [15] A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3 (1990), 961-973.  Google Scholar [16] R. MacKay, J. Meiss and J. Strark, Converse KAM theory for symplectic twist maps, Nonlinearity, 2 (1989), 555-570.  Google Scholar [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.  Google Scholar [18] M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.  Google Scholar [19] S. Tabachnikov, Billiards, Panor. Synth., 1 (1995), vi+142 pp.  Google Scholar

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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.  Google Scholar [2] M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154. doi: 10.1007/BF02572397.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [6] D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal., 4 (1994), 259-269. doi: 10.1007/BF01896241.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] E. Gutkin, Billiard dynamics: A survey with the emphasis on open problems, Regul. Chaotic Dyn., 8 (2003), 1-13. doi: 10.1070/RD2003v008n01ABEH000222.  Google Scholar [10] S. Elaydi, "An Introduction to Difference Equations," Third edition, Undergraduate Texts in Mathematics, Springer, New York, 2005.  Google Scholar [11] E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U. S. A., 34 (1948), 47-51.  Google Scholar [12] J. Heber, On the geodesic flow of tori without conjugate points, Math. Z., 216 (1994), 209-216. doi: 10.1007/BF02572318.  Google Scholar [13] N. Innami, Integral formulas for polyhedral and spherical billiards, J. Math. Soc. Japan, 50 (1998), 339-357. doi: 10.2969/jmsj/05020339.  Google Scholar [14] V. Kaloshin and A. Sorrentino, On conjugacy of convex billiards,, preprint, ().   Google Scholar [15] A. Knauf, Closed orbits and converse KAM theory, Nonlinearity, 3 (1990), 961-973.  Google Scholar [16] R. MacKay, J. Meiss and J. Strark, Converse KAM theory for symplectic twist maps, Nonlinearity, 2 (1989), 555-570.  Google Scholar [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.  Google Scholar [18] M. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.  Google Scholar [19] S. Tabachnikov, Billiards, Panor. Synth., 1 (1995), vi+142 pp.  Google Scholar
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