# Covering maps and ideal embeddings of compact homogeneous spaces

## TL;DR

This paper investigates ideal embeddings of irreducible compact homogeneous spaces in Euclidean spaces, showing that differences in certain spectral properties prevent some spaces from admitting such embeddings.

## Contribution

It establishes a criterion involving covering maps and spectral invariants that determines the existence of ideal embeddings for these spaces.

## Key findings

- If bb:bc: M e2a8a0 N is a covering map and bb_1(M)
e bb_1(N), then N does not admit an ideal embedding.
- M could admit an ideal embedding even when N does not.
- The study links spectral properties with geometric embedding conditions.

## Abstract

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.