  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.
