In this chapter we study homogeneous Finsler spaces. In Sect. 4.1, we define the notions of Minkowski Lie pairs and Minkowski Lie algebras to give an algebraic description of invariant Finsler metrics on homogeneous manifolds and bi-invariant Finsler metrics on Lie groups. Then in Sect. 4.2, we present a sufficient and necessary condition for a coset space to have invariant non-Riemannian Finsler metrics. In Sect. 4.3, we study homogeneous Finsler spaces of negative curvature and prove that every homogeneous Finsler space with nonpositive flag curvature and negative Ricci scalar must be simply connected. In Sect. 4.4, we apply our result to study the degree of symmetry of closed manifolds. In particular, we prove that if a closed manifold is not diffeomorphic to a rank-one Riemannian symmetric space, then its degree of symmetry can be realized by a non-Riemannian Finsler metric. Finally, in Sect. 4.5, we study fourth-root homogeneous Finsler metrics. As an explicit example, we give a classification of all invariant fourth-root Finsler metrics on Grassmannian manifolds.

Homogeneous Finsler Spaces

/pdf/homogeneous-riemannian-spaces-of-four-dimensions-4370jqy8xb.pdf

Homogeneous Riemannian spaces of four dimensions.

For every Finsler metric F we associate a Riemannian metric gF (called the Binet‐ Legendre metric). The Riemannian metric gF behaves nicely under conformal deformation of the Finsler metric F , which makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a question of M Matsumoto about local conformal mapping between two Minkowski spaces, we describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds, we solve a conjecture of S Deng and Z Hou on the Berwaldian character of locally symmetric Finsler spaces, and extend a classic result by H C Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions. Most proofs in this paper go along the following scheme: using the correspondence F 7! gF we reduce the Finslerian problem to a similar problem for the Binet‐ Legendre metric, which is easier and is already solved in most cases we consider. The solution of the Riemannian problem provides us with the additional information that helps to solve the initial Finslerian problem. Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced by a weaker partial smoothness, a notion we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.

/pdf/the-binet-legendre-metric-in-finsler-geometry-4y3cdbf5l7.pdf

The Binet-Legendre Metric in Finsler Geometry

/pdf/holomorphically-projective-changes-and-their-groups-in-an-1xftunp7zn.pdf

Holomorphically projective changes and their groups in an almost complex manifold

For every Finsler metric $F$ we associate a Riemannian metric $g_F$ (called the Binet-Legendre metric). The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or bilipshitz deformation of the Finsler metric $F$. These properties makes it a powerful tool in Finsler geometry and we illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a question of M. Matsumoto about local conformal mapping between two Minkowski spaces, we describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds, solve a conjecture of S. Deng and Z. Hou on the Berwaldian character of locally symmetric Finsler spaces, and extend the classic result of H.C. Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions. 
Most proofs in this paper go along the following scheme: using the correspondence $F \mapsto g_F$we reduce the Finslerian problem to a similar problem for the Binet-Legendre metric, which is easier and is already solved in most cases we consider. The solution of the Riemannian problem provides us with the additional information that helps to solve the initial Finslerian problem. 
Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced by that of partial smoothness, a notion we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.

/pdf/the-binet-legendre-metric-in-finsler-geometry-3co92cyynq.pdf

Publisher Summary This chapter focuses on n-dimensional riemannian spaces admitting a group of motions of order n(n-1)/2+ 1. If an n-dimensional Rieinannian space admits a group of motions of the maximum order n(n+l)/2, then, the space is of constant curvature. An n-dimensional finslerian space for n>2, n ≠4, admits a group of motions of order greater than n(n-l)/2+ 1, then the space is Riemannian and of constant curvature(2). The largest group of motions in a space of constant curvature being of order n(n+1)/2, if it denote it by G, then Gr must be a subgroup of G. In an n-dimensional Euclidean space for n ≠4, n ≠8, any subgroup of the rotation group of order (n - 1) (n - 2)/2 fixes one and only one direction. If an n-dimensional Riemannian space for n ≠4 admits a group of motions of order n(n - 1)/2+ 1, then the group is transitive.

/pdf/on-n-dimensional-riemannian-spaces-admitting-a-group-of-1k86mfr4g6.pdf

On $n$-dimensional Riemannian spaces admitting a group of motions of order $n(n-1)/2+1$

Publisher Summary This chapter presents the detailed study of the symmetric affinely connected spaces Ԅn All the possible curvature tensors of Ԅn as well as the isotropic subgroup of G are determined In case G is transitive, all the simply connected Ԅn's are explicit They fall into three individual cases and two classes depending on a non-negative constant Among them, there are three affinely flat spaces, two of which are homeomorphic with the euclidean space while the other one homeomorphic with the product of a line and an (n - 1)-sphere, n > 2 As for the nonflat cases, they are all homeomorphic with the euclidean space The chapter presents the determination of Ԅn when G is transitive

https://www.ams.org/tran/1955-080-01/S0002-9947-1955-0073254-9/S0002-9947-1955-0073254-9.pdf

Kentaro Yano

Papers

On $n$-dimensional Riemannian spaces admitting a group of motions of order $n(n-1)/2+1$

A class of affinely connected spaces

A class of affinely connected spaces