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Covering maps and ideal embeddings of compact homogeneous spaces. (English) Zbl 1391.53063

The notion of ideal embeddings was introduced by the author in [in: The third Pacific Rim geometry conference. Proceedings of the conference, Seoul, Korea, December 16–19, 1996. Cambridge, MA: International Press. 7–60 (1998; Zbl 1009.53041)]. It should be pointed out that an ideal embedding is an isometric embedding which receives the least possible amount of tension from the surrounding space at each point.
In the paper under review, ideal embeddings of irreducible compact homogenous spaces in Euclidean spaces are studied. The main result states that if \(\pi:M \rightarrow N\) is a covering map between two irreducible compact homogeneous spaces with \(\lambda_1(M)\neq \lambda_1(N)\) (where \(\lambda_1\) is the first positive eigenvalue of the Laplacian \(\Delta\) on a given manifold), then \(N\) does not admit an ideal embedding in a Euclidean space, although \(M\) could.

MSC:

53C30 Differential geometry of homogeneous manifolds
53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 1009.53041