Cohomogeneity one Riemannian manifolds of even dimension with strictly positive curvature. I. (English) Zbl 1028.53035
This paper deals with an even dimensional simply connected compact Riemannian manifold \(M\) with strictly positive sectional curvature. Let \(G\) be a compact Lie group acting by isometries on \(M\) with a codimension one orbit. Then, it is well-known that \(M\) is called a cohomogeneity one \(G\)-manifold.
In this paper, the author has proved the following main result: If \(G\) is a compact semisimple (but not simple) Lie group, then the cohomogeniety one \(G\)-manifold \(M\) is equivariantly diffeomorphic to a compact rank one symmetric space.
In this paper, the author has proved the following main result: If \(G\) is a compact semisimple (but not simple) Lie group, then the cohomogeniety one \(G\)-manifold \(M\) is equivariantly diffeomorphic to a compact rank one symmetric space.
Reviewer: Krishan Lal Duggal (Windsor / Ontario)
MSC:
53C20 | Global Riemannian geometry, including pinching |
53C35 | Differential geometry of symmetric spaces |