Finsler conformal transformations and the curvature invariances

Abstract

This article studies the global conformal transformations f on a Finsler space (M, F ), which satisfy f * F = e c(x) F , where F := F (x, y) is a Finsler metric on M and x ∈ M , y ∈ T x M \ {0}. We obtain the relations between some important geometric quantities of F and their correspondences respectively, including Riemann curvatures, Ricci curvatures, Landsberg curvatures, mean Landsberg curvatures and S-curvatures. Then, we discuss the properties of those conformal transformations on (M, F ) which preserve Ricci curvature, Landsberg curvature, mean Landsberg curvature and S-curvature respectively.

Conformal Transformations on General (α,β)-Spaces

“…Remark 1. Note that φ = 1 + s or φ = 1 + s 2 does not satisfy (2). Thus, by the definition of general (α, β)-metrics and Theorem 1, conformal transformations that preserve the mean Landsberg curvature of Randers metrics F = α + β or square metrics F = (α+β) 2 α are homothetic.…”

“…A natural problem is knowing how to determine, given a Finsler metric with some properties on a manifold M, all conformally related Finsler metrics with the given properties. B ács ó-Cheng [2] characterized conformal transformations that preserve the Riemann curvature, the Ricci curvature, the (mean) Landsberg curvature, or the S-curvature, respectively. Chen-Cheng-Zou [3] proved that if both conformally related (α, β)-metrics are of the Douglas type or of isotropic S-curvature, then the conformal transformations between them are homothetic.…”

Kropina Metrics with Isotropic Scalar Curvature via Navigation Data

“…And we call the smooth function σ(x) the conformal factor. For conformally related Finsler metrics, B ács ó and Cheng [21] gave some transformation conclusions. Here are some related results.…”

Locally Conformally Flat Doubly Twisted Product Complex Finsler Manifolds

“…In [26], Aldea proved that the scale function of a conformal transformation between two complex Finsler metrics depends only on the position of the base manifold. Recently, many author studied conformal transformation in complex Finsler geometry [27][28][29][30]. In 2021, Li studied the locally conformal pseudo-Kähler Finsler manifolds [31].…”