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The Pressure of Ricci Curvature

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Abstract

Let (M n,g) be a closed Riemannian manifold and let κ0 be any positive upper bound for the sectional curvature. We prove that

$$P\left( {\frac{{r_g }}{{2\sqrt {\kappa _0 } }}} \right) \leqslant \frac{{n - 1}}{2}\sqrt {\kappa _0 } ,$$

where P(f) stands for the topological pressure of a function f on the unit sphere bundle SM and r g(v) is the Ricci curvature in the direction of v ∈ SM. This result gives rise to several estimates for the various entropies of the geodesic flow which in turn have several consequences. One of them is entropy rigidity for those metrics in a hyperbolic manifold whose normalized total scalar curvature is bigger than that of the hyperbolic metric.

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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, England

    Gabriel P. Paternain

  2. CIMAT, A.P. 402, 36000, Guanajuato. Gto., México

    Jimmy Petean

Authors
  1. Gabriel P. Paternain
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  2. Jimmy Petean
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Paternain, G.P., Petean, J. The Pressure of Ricci Curvature. Geometriae Dedicata 100, 93–102 (2003). https://doi.org/10.1023/A:1025842932050

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