A first integral for 𝐶^{∞}, k-basic Finsler surfaces and applications to rigidity

Foulon, P.; Ruggiero, R.

doi:10.1090/proc/13079

A first integral for $C^{\infty }$, k-basic Finsler surfaces and applications to rigidity
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Proc. Amer. Math. Soc. 144 (2016), 3847-3858 Request permission

Abstract:

We show that a compact $C^{\infty }$, k-basic Finsler surface without conjugate points and genus greater than one is Riemannian. This result is a $C^{\infty }$ version of the fact, proved by G. Paternain, that analytic, compact, k-basic Finsler surfaces with genus greater than one are Riemannian. The proof in the $C^{\infty }$ case relies mainly on two facts: first of all the existence of a first integral for the geodesic flow of any k-basic Finsler surface, one of the main contributions of this note; and secondly the triviality of every first integral assuming the absence of conjugate points.

References

H. Akbar-Zadeh, Sur les espaces de Finsler à courbures sectionnelles constantes, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), no. 10, 281–322 (French, with English summary). MR 1052466
D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000. MR 1747675, DOI 10.1007/978-1-4612-1268-3
David Bao and Colleen Robles, Ricci and flag curvatures in Finsler geometry, A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, Cambridge, 2004, pp. 197–259. MR 2132660
Alessandro G. Chimenton, José Barbosa Gomes, and Rafael O. Ruggiero, Transitivity of Finsler geodesic flows of compact surfaces without conjugate points and higher genus, and applications to Finsler rigidity problems, Houston J. Math. 41 (2015), no. 2, 523–551. MR 3403744
Gonzalo Contreras and Renato Iturriaga, Convex Hamiltonians without conjugate points, Ergodic Theory Dynam. Systems 19 (1999), no. 4, 901–952. MR 1709426, DOI 10.1017/S014338579913387X
G. Contreras, R. Iturriaga, G. P. Paternain, and M. Paternain, Lagrangian graphs, minimizing measures and Mañé’s critical values, Geom. Funct. Anal. 8 (1998), no. 5, 788–809. MR 1650090, DOI 10.1007/s000390050074
José Barbosa Gomes and Rafael O. Ruggiero, Smooth k-basic Finsler compact surfaces with expansive geodesic flows are Riemannian, Houston J. Math. 37 (2011), no. 3, 793–806. MR 2844450
José Barbosa Gomes and Rafael O. Ruggiero, On Finsler surfaces without conjugate points, Ergodic Theory Dynam. Systems 33 (2013), no. 2, 455–474. MR 3035293, DOI 10.1017/S0143385711001027
P. Foulon, Oral communication: On the flag curvature of Katok-Ziller metrics.
José Barbosa Gomes and Rafael O. Ruggiero, Weak integrability of Hamiltonians on the two torus and rigidity, Nonlinearity 26 (2013), no. 7, 2109–2129. MR 3078109, DOI 10.1088/0951-7715/26/7/2109
A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822
Harold Marston Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924), no. 1, 25–60. MR 1501263, DOI 10.1090/S0002-9947-1924-1501263-9
Gabriel P. Paternain, Finsler structures on surfaces with negative Euler characteristic, Houston J. Math. 23 (1997), no. 3, 421–426. MR 1690049
Zhongmin Shen, Lectures on Finsler geometry, World Scientific Publishing Co., Singapore, 2001. MR 1845637, DOI 10.1142/9789812811622