On a class of projective Ricci curvature of Finsler metrics

2021 ◽  
pp. 2150084
Author(s):  
Hongmei Zhu
Keyword(s):  
General Formula ◽  
Ricci Curvature ◽  
Finsler Geometry ◽  
Finsler Metrics ◽  
Volume Form ◽  
Projective Invariant

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we investigate the projective Ricci curvature of a class of general [Formula: see text]-metrics satisfying a certain condition, which is invariant under the change of volume form. Moreover, we construct a class of new nontrivial examples on such Finsler metrics.

Projective Ricci flat spherically symmetric Finsler metrics

2018 ◽  
Vol 29 (11) ◽  
pp. 1850078 ◽  
Author(s):  
Hongmei Zhu ◽  
Haixia Zhang
Keyword(s):  
Ricci Curvature ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Spherically Symmetric ◽  
Projective Invariant ◽  
Ricci Flat ◽  
Douglas Metric

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we characterize projective Ricci flat spherically symmetric Finsler metrics. Under a certain condition, we prove that a projective Ricci flat spherically symmetric Finsler metric must be a Douglas metric. Moreover, we construct a class of new nontrivial examples on projective Ricci flat Finsler metrics.

On general (α,β)-metrics of Landsberg type

2016 ◽  
Vol 13 (06) ◽  
pp. 1650085 ◽  
Author(s):  
M. Zohrehvand ◽  
H. Maleki
Keyword(s):  
General Formula ◽  
Open Problem ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Finsler Metrics

In this paper, we study a class of Finsler metrics, which are defined by a Riemannian metric [Formula: see text] and a one-form [Formula: see text]. They are called general [Formula: see text]-metrics. We have proven that, every Landsberg general [Formula: see text]-metric is a Berwald metric, under a certain condition. This shows that the hunting for an unicorn, one of the longest standing open problem in Finsler geometry, cannot be successful in the class of general [Formula: see text]-metrics.

On a class of locally projectively flat Finsler metrics

2016 ◽  
Vol 27 (06) ◽  
pp. 1650052 ◽  
Author(s):  
Benling Li ◽  
Zhongmin Shen
Keyword(s):  
General Formula ◽  
Finsler Geometry ◽  
Nonlinear Pdes ◽  
Finsler Metrics ◽  
Regular Case ◽  
Important Group ◽  
Projectively Flat ◽  
Riemann Metric

Locally projectively flat Finsler metrics compose an important group of metrics in Finsler geometry. The characterization of these metrics is the regular case of the Hilbert’s Fourth Problem. In this paper, we study a class of Finsler metrics composed by a Riemann metric [Formula: see text] and a [Formula: see text]-form [Formula: see text] called general ([Formula: see text], [Formula: see text])-metrics. We classify those locally projectively flat when [Formula: see text] is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totally determined. Then a new group of locally projectively flat Finsler metrics is found.

On conformally related spherically symmetric Finsler metrics

2016 ◽  
Vol 13 (10) ◽  
pp. 1650118 ◽  
Author(s):  
Maryam Maleki ◽  
Nasrin Sadeghzadeh ◽  
Tahereh Rajabi
Keyword(s):  
Einstein Metrics ◽  
Finsler Geometry ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finsler Metrics ◽  
Spherically Symmetric ◽  
Conformal Transformations ◽  
Conformal Change ◽  
Projective Invariant ◽  
Necessary And Sufficient

In this paper, we study the projective invariant quantities in Finsler geometry which remain invariant under the conformal change of metrics. In particular, we obtain the necessary and sufficient conditions of a given Douglas and Weyl and generalized Douglas–Weyl (GDW) metric to be invariant under the conformal transformations. Finally, we introduce some explicit examples of these metrics. Also, some of these [Formula: see text]-conformal transformations of Einstein metrics are considered.

Homogeneous Einstein Finsler Metrics on -dimensional Spheres

2018 ◽  
Vol 62 (3) ◽  
pp. 509-523
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Constant Sectional Curvature ◽  
Einstein Metric ◽  
Finsler Metrics ◽  
Dimensional Sphere ◽  
Order Ordinary Differential Equation ◽  
Canonical Metric ◽  
Homogeneous Einstein Metrics

AbstractIn this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.

scholarly journals Fiberwise convexity of Hill’s lunar problem

2017 ◽  
Vol 09 (04) ◽  
pp. 571-630 ◽  
Author(s):  
Junyoung Lee
Keyword(s):  
Energy Level ◽  
Contact Structure ◽  
Finsler Geometry ◽  
Critical Energy ◽  
Critical Value ◽  
Finsler Metrics ◽  
Tight Contact ◽  
Systolic Inequality ◽  
Geodesic Problem ◽  
Hill's Lunar Problem

In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transformation of a geodesic problem on [Formula: see text] with a family of Finsler metrics. Therefore the compactified energy hypersurfaces below the critical energy level have the unique tight contact structure on [Formula: see text]. Also one can apply the systolic inequality of Finsler geometry to the regularized Hill’s lunar problem.

On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature

2010 ◽  
Vol 62 (1) ◽  
pp. 3-18
Author(s):  
Boudjemâa Anchouche
Keyword(s):  
Asymptotic Behavior ◽  
Line Bundle ◽  
Ricci Curvature ◽  
Projective Variety ◽  
Volume Form ◽  
Positive Ricci Curvature ◽  
Standard Type ◽  
Number Of Components ◽  
Holomorphic Sections ◽  
Volume Forms

AbstractLet (X, g) be a complete noncompact Kähler manifold, of dimension n ≥ 2, with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that X can be compactified, i.e., X is biholomorphic to a quasi-projective variety. The aim of this paper is to prove that the L2 holomorphic sections of the line bundle K−qXand the volume form of the metric g have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the Kähler metric g and of the Fubini-Study metric induced on X. In the case of dimC X = 2, we establish a relation between the number of components of the divisor D and the dimension of the.

scholarly journals A NEW QUANTITY IN RIEMANN-FINSLER GEOMETRY

2012 ◽  
Vol 54 (3) ◽  
pp. 637-645 ◽  
Author(s):  
XIAOHUAN MO ◽  
ZHONGMIN SHEN ◽  
HUAIFU LIU
Keyword(s):  
Scalar Curvature ◽  
Simple Proof ◽  
Finsler Geometry ◽  
Finsler Manifold ◽  
Flag Curvature ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Riemannian Curvature

AbstractIn this note, we study a new Finslerian quantity Ĉ defined by the Riemannian curvature. We prove that the new Finslerian quantity is a non-Riemannian quantity for a Finsler manifold with dimension n = 3. Then we study Finsler metrics of scalar curvature. We find that the Ĉ-curvature is closely related to the flag curvature and the H-curvature. We show that Ĉ-curvature gives, a measure of the failure of a Finsler metric to be of weakly isotropic flag curvature. We also give a simple proof of the Najafi-Shen-Tayebi' theorem.

scholarly journals New classes of projectively related Finsler metrics of constant flag curvature

2020 ◽  
Vol 17 (05) ◽  
pp. 2050068
Author(s):  
Georgeta Creţu
Keyword(s):  
Curvature Tensor ◽  
Flag Curvature ◽  
Finsler Metrics ◽  
Text Type ◽  
Constant Flag Curvature ◽  
Projective Invariant

We define a Weyl-type curvature tensor of [Formula: see text]-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect, we construct new families of projectively related Finsler metrics that have constant flag curvature.

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