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Weighted Ricci Curvature

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Comparison Finsler Geometry

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Abstract

In Part I, we saw that the natural notions of Finsler curvatures (the flag and Ricci curvatures) can be introduced through the behavior of geodesics, and then several comparison theorems follow smoothly by similar arguments to the Riemannian case, or through the characterizations of these curvatures from the Riemannian geometric point of view.

In order to proceed further in this direction, we would like to equip our Finsler manifold with a measure on it. At this point, however, we face a difficulty in choosing a measure, because a Finsler manifold does not necessarily have a unique canonical measure like the volume measure in the Riemannian case.

Then our standpoint is that, instead of choosing some constructive measure, we begin with an arbitrary measure and modify the Ricci curvature into the weighted Ricci curvature according to the choice of a measure. This is motivated by the theory deeply investigated in the Riemannian case by Lichnerowicz, Bakry and others. It will turn out that this strategy fits the Finsler setting very well.

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Authors and Affiliations

  1. Department of Mathematics, Osaka University, Osaka, Japan

    Shin-ichi Ohta

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  1. Shin-ichi Ohta
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Ohta, Si. (2021). Weighted Ricci Curvature. In: Comparison Finsler Geometry. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-80650-7_9

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