Browsing by Subject "Finsler metric"

Now showing 1 - 3 of 3
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    Einstein Finsler Metrics and Killing Vector Fields on Riemannian Manifolds
    (Springer, 2017-01) Cheng, Xinyue; Shen, Zhongmin; Department of Mathematical Sciences, School of Science
    We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S3 with Ric = 2F2, Ric = 0 and Ric = -2F2, respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.
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    On a class of projectively flat Finsler metrics
    (2016) Li, Benling; Shen, Zhongmin; Department of Mathematical Sciences, School of Science
    In this paper, we study a class of Finsler metrics composed by a Riemann metric $\alpha=\sqrt{a_{ij}(x)y^i y^j}$ and a $1$-form $\beta=b_i(x)y^i$ called general ($\alpha$, $\beta$)-metrics. We classify those projectively flat when $\alpha$ is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totally determined. Then a new group of projectively flat Finsler metrics is found.
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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.