论文标题
对称空间，强烈的各向同性不可约性和公平特性
Symmetric space, strongly isotropy irreducibility and equigeodesic properties
论文作者
论文摘要
如果对于任何$ g $ invariant Riemannian指标，则在同质歧管上的平滑曲线称为Riemannian Equigeo-Desic。同质歧管$ g/h $称为Riemannian EquigeOdesic，如果在g/h $中进行任何$ x \，而t_x（g/h）$中的任何非零$ y \，则存在riemannian equigeodesic $ c（t）$，带有$ c（0）= x $和$ \ dot x $ and $ \ dot，这两个概念可以自然地转移到Finsler设置上，该设置为Finsler EquigeEdesic和Finsler EquigeEdesic Space提供了定义。我们分别证明了Riemannian公平空间和Finsler EquigeEdesic空间的两个分类定理。首先，当且只有可以将其分解为欧几里得因素的产物和紧凑型等同素不可避免的因素时，只有连接的准紧凑型$ g $与连接的均匀连接的准紧凑型$ g $和连接的$ g $和连接的$ g/h $的同质流形$ g/h $。其次，当且仅当它可以本地分解为产品时，每个因素为$ spin（7）/g_2 $，$ g_2 $，$ g_2/su（3）$或压缩类型的对称空间时，每个因素为$ spin（7）/g_2 $，$ spin（7）/g_2 $，$ spin（7）/g_2 $（7）/g_2 $，$ g $ g/h $是Finsler equodesic。这些结果表明，紧凑型类型的对称空间和强烈的各向同性不可还原空间可以通过公平特性来解释。作为一个应用程序，我们将同质流形$ g/h $与紧凑的半简单$ g $分类，因此，$ g/h $上的所有$ g $ invariant Finsler指标都是Berwald。它提出了一个均匀的Finsler几何形状的新项目，以系统地研究同质流形$ g/h $，所有$ g $ invariant Finsler指标都满足某些几何属性。
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.
