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Immersions of cohomogeneity one Riemannian manifolds. (English) Zbl 0880.53040

A compact, connected Lie group \(G\) is said to act by cohomogeneity \(k\) on a Riemannian manifold if \(G\) acts effectively and isometrically with a principal orbit of codimension \(k\). In this situation the Riemannian manifold is said to be of cohomogeneity \(k\).
This paper paper deals with compact cohomogeneity one Riemannian manifolds \(M^n\) with positive curvature. In this case C. Searle [Math. Z. 214, 491-498 (1993; Zbl 0804.53057)] proved that if \(2 \leq n=\dim M\leq 6\), then \(M^n\) is diffeomorphic to either \(S^n\) or \(\mathbb{C} P^{n/2}\). The author considers the case that \(M^n\) is acted (isometrically and with cohomogeneity one) by a compact connected Lie group \(G\) whose principal orbits are isotropy irreducible. It is shown that the universal covering \(\widetilde M\) of \(M\) is acted by \(G\) and there is a \(G\)-equivariant isometric embedding \(\widetilde M \) into Euclidean space \(\mathbb{R}^{n+1}\) realizing \(\widetilde M\) as a hypersurface of revolution.
Reviewer: Anna Fino (Torino)

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds

Citations:

Zbl 0804.53057
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References:

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