## A new curvature invariant and entropy of geodesic flows.(English)Zbl 0536.53048

Let M be a compact Riemannian manifold with negative sectional curvature. Denote by SM the unit tangent bundle to M, let $$\mu$$ be the normalized $$(\mu(SM)=1)$$ Liouville measure on SM, and $$h_{\mu}$$ the entropy of the geodesic flow on SM with respect to the measure $$\mu$$. For each $$p\in M$$ and each $$v\in T_ pM$$, let $$Q_ v$$ be the quadratic form on $$T_ pM$$ satisfying $$Q_ v(v)=0$$ and for each unit vector w orthogonal to v, $$Q_ v(w)$$ is the sectional curvature of the two-plane spanned by v and w. Let $$\{\lambda_ i\}$$ be the set of eigenvalues of the positive semi- definite form $$-Q_ v$$, and set $$\sigma(v)=-\sum \sqrt{\lambda_ i}$$. Then $$\sigma$$ (v) is a pointwise curvature invariant on SM whose average with respect to $$\mu$$ we denote by $$\alpha$$ (M). Theorem. $$h_{\mu}\geq - \alpha(M)$$, with equality if and only if M is locally symmetric.

### MSC:

 53C20 Global Riemannian geometry, including pinching 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53D25 Geodesic flows in symplectic geometry and contact geometry 53C22 Geodesics in global differential geometry
Full Text:

### References:

  [B-G] Brin, M. Gromov, M.: On the ergodicity of frame flows. Invent. Math.60, 1-7 (1980) · Zbl 0445.58023  [C] Chavel, I.: Riemannian symmetric spaces of rank one. Lecture Notes in Pure and Applied Mathematics, vol. 5. New York: Marcel Dekker 1972 · Zbl 0239.53032  [E] Eberlein, P.: When is a geodesic flow of Anosov type? J. Diff. Geom.8, 437-463 (1973) · Zbl 0285.58008  [F-M] Freire, A., Mañé, R.: On the entropy of the geodesic flow in manifolds without conjugate points. Invent. Math.69, 375-392 (1982) · Zbl 0488.58017  [G] Green, L.W.: A theorem of E. Hopf. Michigan Math. J.5, 31-34 (1958) · Zbl 0134.39601  [H] Heintze, E.: On homogeneous manifolds of negative curvature. Math. Ann.211, 23-34 (1974) · Zbl 0282.53036  [K] Katok, A.: Entropy and closed geodesics. Ergodic Th. and Dynam. Sys.2, 339-367 (1982)  [M] Manning, A.: Curvature bounds for the entropy of the geodesie flow on a surface. J. Lond. Math. Soc. (2)24, 351-357 (1981) · Zbl 0443.53035  [P] Pesin, Ya.B.: Characteristic Lyapunov exponents and smooth ergodic theory Russ. Math. Surveys (4)32, 55-114 (1977) · Zbl 0383.58011  [SA1] Sarnak, P.: Prime geodesic theorems. Ph. D. Thesis, Stanford 1980  [SA2] Sarnak, P.: Entropy estimates for geodesic flows. Ergod. Th. and Dynam. Systems2, 513-524 (1982) · Zbl 0525.58028  [S] Sinai, Ya.G.: The asymptotic behavior of the number of clsed geodesics on a compact manifold of negative curvature. Izv. Akad. Nauk SSSR, Ser. Math.30, 1275-1295 (1966); English translation, A.M.S. Trans.73,(2) 229-250 (1968)  [SP] Spatzier, R.: Dynamical properties of algebraic systems; a study in closed geodesics. Ph.D. Thesis, Warwick 1983
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.