Cohomogeneity one manifolds with quasipositive curvature

Abstract: Let $G$ be a Lie group acting by isometries on a Riemannian manifold $(M,g)$. The action is of cohomogeneity one, if the orbit space $M/G$ is one-dimensional. In this sense cohomogeneity one manifolds are the most symmetric manifolds after homogeneous spaces, which have a $0$-dimensional orbit space. In this talk we will give a classification of cohomogeneity one manifolds admitting an invariant metric with quasipositive sectional curvature, except for two infinite families in dimension $7$. A Riemannian manifold has quasipositive sectional curvature, if it has non-negative sectional curvature and contains one point, where all tangent planes have positive sectional curvature. A similar classification in positive curvature has already been obtained by Verdiani in even dimensions and Grove, Wilking and Ziller in odd dimensions. Surprisingly, our result only adds two more examples to their list: an Eschenburg space and a Bazaikin space, which were previously known to admit metrics with quasipositive curvature.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

( paper | video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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