Browsing by Subject "Ricci curvature"

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    On the projective Ricci curvature
    (Springer, 2020-07) Shen, Zhongmin; Sun, Liling; Mathematical Sciences, School of Science
    The notion of the Ricci curvature is defined for sprays on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. In this paper, we introduce the notion of projectively Ricci-flat sprays. We establish a global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature. Then we study and characterize projectively Ricci-flat Randers metrics.
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    Some Inequalities on Finsler Manifolds with Weighted Ricci Curvature Bounded Below
    (Springer, 2022-02-07) Cheng, Xinyue; Shen, Zhongmin; Mathematical Sciences, School of Science
    We establish some important inequalities under a lower weighted Ricci curvature bound on Finsler manifolds. Firstly, we establish a relative volume comparison of Bishop–Gromov type. As one of the applications, we obtain an upper bound for volumes of the Finsler manifolds. Further, when the S-curvature is bounded on the whole manifold, we obtain a theorem of Bonnet–Myers type on Finsler manifolds. Finally, we obtain a sharp Poincaré–Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a better lower bound for the first eigenvalue on the Finsler manifolds.