Browsing by Author "Shen, Zhongmin"

Now showing 1 - 10 of 12
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    A conclusive theorem on Finsler metrics of sectional flag curvature
    (arXiv, 2018-12-22) Huang, Libing; Shen, Zhongmin; Mathematical Sciences, School of Science
    If the flag curvature of a Finsler manifold reduces to sectional curvature, then locally either the Finsler metric is Riemannian, or the flag curvature is isotropic.
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    Conformal Vector Fields on Some Finsler Manifolds
    (Springer, 2016-01) Shen, Zhongmin; Yuan, Mingao; Department of Mathematical Sciences, School of Science
    We study conformal vector fields on a Finsler manifold whose metric is defined by a Riemannian metric, a 1-form and its norm. We find PDEs characterizing conformal vector fields. Then we obtain the explicit expressions of conformal vector fields for certain spherically symmetric metrics on R n .
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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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    Nonlinear spectrums of Finsler manifolds
    (Springer, 2022-01) Kristály, Alexandru; Shen, Zhongmin; Yuan, Lixia; Zhao, Wei; Mathematical Sciences, School of Science
    In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler–Laplacian operator, we introduce faithful dimension pairs by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigenvalues are provided in the spirit of Cheng, Buser and Gromov, which extend in several aspects the results of Hassannezhad, Kokarev and Polterovich. Moreover, we construct several faithful dimension pairs based on Lusternik–Schnirelmann category, Krasnoselskii genus and essential dimension, respectively; however, we also show that the Lebesgue covering dimension pair is not faithful. As an application, we show that the Bakry–Émery spectrum of a closed weighted Riemannian manifold can be characterized by the faithful Lusternik–Schnirelmann dimension pair.
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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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    On a class of weakly weighted Einstein metrics
    (World Scientific, 2022-09) Shen, Zhongmin; Zhao, Runzhong; Mathematical Sciences, School of Science
    The notion of general weighted Ricci curvatures appears naturally in many problems. The N-Ricci curvature and the projective Ricci curvature are just two special ones with totally different geometric meanings. In this paper, we study general weighted Ricci curvatures. We find that Randers metrics of certain isotropic weighted Ricci curvature must have isotropic S-curvature. Then we classify them via their navigation expressions. We also find equations that characterize Randers metrics of almost isotropic weighted Ricci curvature.
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    On Concircular Transformations in Finsler Geometry
    (Springer, 2019) Shen, Zhongmin; Yang, Guojun; Mathematical Sciences, School of Science
    A geodesic circle in Finsler geometry is a natural extension of that in a Euclidean space. In this paper, we study geodesic circles and (infinitesimal) concircular transformations on a Finsler manifold. We characterize a concircular vector field with some PDEs on the tangent bundle, and then we obtain respectively necessary and sufficient conditions for a concircular vector field to be conformal and a conformal vector field to be concircular. We also show conditions for two conformally related Finsler metrics to be concircular, and obtain some invariant curvature properties under conformal and concircular transformations.
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    On sprays with vanishing χ-curvature
    (World Scientific, 2021) Shen, Zhongmin; Mathematical Sciences, School of Science
    Every Riemannian metric or Finsler metric on a manifold induces a spray via its geodesics. In this paper, we discuss several expressions for the χ-curvature of a spray. We show that the sprays obtained by a projective deformation using the S-curvature always have vanishing χ-curvature. Then we establish the Beltrami Theorem for sprays with χ=0.
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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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    Ricci Curvature of Finsler Metrics by Warped Product
    (2020-05) Marcal, Patricia; Shen, Zhongmin; Buse, Olguta; Ramras, Daniel; Roeder, Roland
    In the present work, we consider a class of Finsler metrics using the warped product notion introduced by B. Chen, Z. Shen and L. Zhao (2018), with another “warping”, one that is consistent with the form of metrics modeling static spacetimes and simplified by spherical symmetry over spatial coordinates, which emerged from the Schwarzschild metric in isotropic coordinates. We will give the PDE characterization for the proposed metrics to be Ricci-flat and construct explicit examples. Whenever possible, we describe both positive-definite solutions and solutions with Lorentz signature. For the latter, the 4-dimensional metrics may also be studied as Finsler spacetimes.