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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
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References:

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