Inverse problem of sprays with scalar curvature

2019 ◽  
Vol 30 (09) ◽  
pp. 1950041
Author(s):  
Ying Li ◽  
Xiaohuan Mo ◽  
Yaoyong Yu
Keyword(s):  
Inverse Problem ◽  
Scalar Curvature ◽  
Finsler Geometry ◽  
Positive Definite ◽  
Finsler Metric ◽  
Open Domain ◽  
Isotropic Curvature ◽  
Differential Manifold

Every Finsler metric on a differential manifold induces a spray. The converse is not true. Therefore, it is one of the most fundamental problems in spray geometry to determine whether a spray is induced by a Finsler metric which is regular, but not necessary positive definite. This problem is called inverse problem. This paper discuss inverse problem of sprays with scalar curvature. In particular, we show that if such a spray [Formula: see text] on a manifold is of vanishing [Formula: see text]-curvature, but [Formula: see text] has not isotropic curvature, then [Formula: see text] is not induced by any (not necessary positive definite) Finsler metric. We also find infinitely many sprays on an open domain [Formula: see text] with scalar curvature and vanishing [Formula: see text]-curvature, but these sprays have no isotropic curvature. This contrasts sharply with the situation in Finsler geometry.

Sprays of isotropic curvature

2018 ◽  
Vol 29 (01) ◽  
pp. 1850003 ◽  
Author(s):  
Benling Li ◽  
Zhongmin Shen
Keyword(s):  
Inverse Problem ◽  
Scalar Curvature ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Geometric Quantity ◽  
Zero Curvature ◽  
Isotropic Curvature

In this paper, a new notion of isotropic curvature for sprays is introduced. We show that for a spray of scalar curvature, it is of isotropic curvature if and only if the non-Riemannian quantity [Formula: see text] vanishes. In fact, it is the first geometric quantity to show the spray of isotropic curvature even in the Finslerian case. How to determine a spray is induced by a Finsler metric or not is an interesting inverse problem. We study this problem when the spray is of isotropic curvature and show that a spray of zero curvature can be induced by a group of Finsler metrics. Further, an efficient way is given to construct a family of sprays of isotropic curvature which cannot be induced by any Finsler metric.

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.

On generalized 4-th root metrics of isotropic scalar curvature

2018 ◽  
Vol 68 (4) ◽  
pp. 907-928 ◽  
Author(s):  
Akbar Tayebi
Keyword(s):  
Scalar Curvature ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Sufficient Condition ◽  
Riemannian Curvature ◽  
Necessary And Sufficient Condition ◽  
Conformal Change ◽  
Riemannian Curvature Tensor ◽  
Necessary And Sufficient

AbstractBy an interesting physical perspective and a suitable contraction of the Riemannian curvature tensor in Finsler geometry, Akbar-Zadeh introduced the notion of scalar curvature for the Finsler metrics. A Finsler metric is called of isotropic scalar curvature if the scalar curvature depends on the position only. In this paper, we study the class of generalized 4-th root metrics. These metrics generalize 4-th root metrics which are used in Biology as ecological metrics. We find the necessary and sufficient condition under which a generalized 4-th root metric is of isotropic scalar curvature. Then, we find the necessary and sufficient condition under which the conformal change of a generalized 4-th root metric is of isotropic scalar curvature. Finally, we characterize the Bryant metrics of isotropic scalar curvature.

Inverse problem of left invariant sprays on Lie groups

2021 ◽  
pp. 2150076
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Inverse Problem ◽  
Lie Groups ◽  
Lie Group ◽  
Positive Definite ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Riemann Curvature

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray [Formula: see text] on a Lie group [Formula: see text] satisfies that [Formula: see text] is commutative or [Formula: see text] is projective, then [Formula: see text] is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.

On the conformal scalar curvature equations on Finsler manifolds

2016 ◽  
Vol 14 (01) ◽  
pp. 1750008
Author(s):  
Neda Shojaee ◽  
Morteza MirMohammad Rezaii
Keyword(s):  
Scalar Curvature ◽  
Finsler Geometry ◽  
Yamabe Problem ◽  
Finsler Manifolds ◽  
Yamabe Flow ◽  
Conformal Deformations

In this paper, we study conformal deformations and [Formula: see text]-conformal deformations of Ricci-directional and second type scalar curvatures on Finsler manifolds. Then we introduce the best equation to study the Yamabe problem on Finsler manifolds. Finally, we restrict conformal deformations of metrics to [Formula: see text]-conformal deformations and derive the Yamabe functional and the Yamabe flow in Finsler geometry.

scholarly journals The Β-Change by Finsler Metric of C-Reducible Finsler Spaces in Finsler Geometry

2019 ◽  
Vol 33 (1) ◽  
pp. 1-10
Author(s):  
Khageswar Mandal
Keyword(s):  
Finsler Space ◽  
Homogeneous Function ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Spaces

 This paper considered about the β-Change of Finsler metric L given by L*= f(L, β), where f is any positively homogeneous function of degree one in L and β and obtained the β-Change by Finsler metric of C-reducible Finsler spaces. Also further obtained the condition that a C-reducible Finsler space is transformed to a C-reducible Finsler space by a β-change of Finsler metric.

scholarly journals METRIC COMPATIBLE OR NON-COMPATIBLE FINSLER–RICCI FLOWS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250041 ◽  
Author(s):  
SERGIU I. VACARU
Keyword(s):  
Ricci Flow ◽  
Evolution Equations ◽  
Finsler Geometry ◽  
Flow Theory ◽  
Finsler Metric ◽  
Geometric Evolution ◽  
Ricci Flows ◽  
Canonical Metric ◽  
Self Consistent ◽  
Math Phys

There were elaborated different models of Finsler geometry using the Cartan (metric compatible), or Berwald and Chern (metric non-compatible) connections, the Ricci flag curvature, etc. In a series of works, we studied (non)-commutative metric compatible Finsler and non-holonomic generalizations of the Ricci flow theory [see S. Vacaru, J. Math. Phys. 49 (2008) 043504; 50 (2009) 073503 and references therein]. The aim of this work is to prove that there are some models of Finsler gravity and geometric evolution theories with generalized Perelman's functionals, and correspondingly derived non-holonomic Hamilton evolution equations, when metric non-compatible Finsler connections are involved. Following such an approach, we have to consider distortion tensors, uniquely defined by the Finsler metric, from the Cartan and/or the canonical metric compatible connections. We conclude that, in general, it is not possible to elaborate self-consistent models of geometric evolution with arbitrary Finsler metric non-compatible connections.

scholarly journals ON THE CURVATURE OF A CERTAIN RIEMANNIAN SPACE OF MATRICES

2000 ◽  
Vol 03 (02) ◽  
pp. 199-212 ◽  
Author(s):  
PETER W. MICHOR ◽  
DÉNES PETZ ◽  
ATTILA ANDAI
Keyword(s):  
Scalar Curvature ◽  
State Space ◽  
Quantum System ◽  
Curvature Tensor ◽  
Riemannian Space ◽  
Riemannian Metric ◽  
Positive Definite ◽  
The State ◽  
Inner Product ◽  
Positive Definite Matrices

Positive definite matrices of trace 1 describe the state space of a finite quantum system. This manifold can be endowed by the physically relevant Bogoliubov–Kubo–Mori inner product as a Riemannian metric. In this paper the curvature tensor and the scalar curvature are computed.

On a Yamabe Type Problem in Finsler Geometry

2017 ◽  
Vol 60 (2) ◽  
pp. 253-268
Author(s):  
Bin Chen ◽  
Lili Zhao
Keyword(s):  
Scalar Curvature ◽  
Finsler Geometry ◽  
Constant Scalar Curvature ◽  
Conformal Invariants ◽  
Type Problem ◽  
Conformal Class ◽  
Standard Sphere ◽  
Yamabe Constant ◽  
Cartan Torsion

AbstractIn this paper, a newnotion of scalar curvature for a Finslermetric F is introduced, and two conformal invariants Y(M, F) and C(M, F) are deûned. We prove that there exists a Finslermetric with constant scalar curvature in the conformal class of F if the Cartan torsion of F is suõciently small and Y(M, F)C(M, F) < Y(Sn) where Y(Sn) is the Yamabe constant of the standard sphere.

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