Symmetric space, strongly isotropy irreducibility and equigeodesic properties

TL;DR

This paper classifies homogeneous manifolds based on their equigeodesic properties in Riemannian and Finsler geometries, linking these properties to space decompositions and symmetric space structures.

Contribution

It provides new classification theorems for Riemannian and Finsler equigeodesic spaces, connecting equigeodesic properties with space decompositions and symmetric space classifications.

Findings

Riemannian equigeodesic spaces decompose into Euclidean and strongly isotropy irreducible factors.

Finsler equigeodesic spaces are locally products of specific symmetric and homogeneous spaces.

Homogeneous manifolds with all invariant Finsler metrics being Berwald are classified.

Abstract

A smooth curve on a homogeneous manifold $G/H$ is called a Riemannian equigeo-desic if it is a homogeneous geodesic for any $G$-invariant Riemannian metric. The homogeneous manifold $G/H$ is called Riemannian equigeodesic, if for any $x\in G/H$ and any nonzero $y\in T_x(G/H)$, there exists a Riemannian equigeodesic $c(t)$ with $c(0)=x$ and $\dot{c}(0)=y$. These two notions can be naturally transferred to the Finsler setting, which provides the definitions for Finsler equigeodesic and Finsler equigeodesic space. We prove two classification theorems for Riemannian equigeodesic spaces and Finsler equigeodesic spaces respectively. Firstly, a homogeneous manifold $G/H$ with connected simply connected quasi compact $G$ and connected $H$ is Riemannian equigeodesic if and only if it can be decomposed as a product of Euclidean factors and compact strongly isotropy irreducible factors. Secondly, a homogeneous manifold $G/H$ with a compact semi simple $G$ is Finsler equigeodesic if and only if it can be locally decomposed as a product, in which each factor is $Spin(7)/G_2$, $G_2/SU(3)$ or a symmetric space of compact type. These results imply that symmetric space and strongly isotropy irreducible space of compact type can be interpreted by equigeodesic properties. As an application, we classify the homogeneous manifold $G/H$ with a compact semi simple $G$, such that all $G$-invariant Finsler metrics on $G/H$ are Berwald. It suggests a new project in homogeneous Finsler geometry, to systematically study the homogeneous manifold $G/H$ on which all $G$-invariant Finsler metrics satisfy certain geometric property.