Covering maps and ideal embeddings of compact homogeneous spaces

The notion of ideal embeddings was introduced in [B.-Y. Chen, {Strings of Riemannian invariants, inequalities, ideal immersions and their applications.} The Third Pacific Rim Geometry Conference (Seoul, 1996), 7-60, Int. Press, Cambridge, MA, 1998]. Roughly speaking, an ideal embedding (or a best of living) 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 via covering maps. Our main result states that $\pi: M\to N$ is a covering map between two irreducible compact homogeneous spaces and if $\lambda_1(M)\ne \lambda_1(N)$, then $N$ doesn't admit an ideal embedding in any Euclidean space; although $M$ could.