Covering Maps and Ideal Embeddings of Compact Homogeneous Spaces

Abstract

The notion of ideal embeddings was introduced by the author at the Third Pacific Rim Geometry Conference held at Seoul in 1996. Roughly speaking, an ideal embedding is an isometrical embedding which receives the least possible amount of tension from the surrounding space at each point. In this article, we study ideal embeddings of irreducible compact homogenous spaces in Euclidean spaces. Our main result states that if $\pi: M\to N$ is a covering map between two irreducible compact homogeneous spaces with $\lambda_1(M)\ne \lambda_1(N)$, then $N$ does not admit an ideal embedding in a Euclidean space, although $M$ could.