The Gauss-Bonnet and Chern-Lashof Theorems in a Simply Connected Symmetric Space of Compact Type

Abstract

In this paper, we prove the theorems of the Gauss-Bonnet and Chern-Lashof types for low dimensional compact submanifolds in a simply connected symmetric space of compact type. In particular, in the case where the ambient space is a sphere, we need not to give the restriction for the dimension of the submanifold. Those proofs are performed by applying the Morse theory to squared distance functions.