Positively curved 7-dimensional manifolds

TL;DR

This paper classifies certain 7-dimensional positively curved manifolds with symmetry, showing they are diffeomorphic to spheres under specific group action conditions.

Contribution

It proves that 7-dimensional compact positively curved manifolds with high-symmetry group actions are diffeomorphic to spheres.

Findings

Manifolds are diffeomorphic to spheres when the semisimple part of the symmetry group has dimension greater than 6.

Provides classification results for positively curved 7-manifolds with symmetry.

Establishes conditions under which such manifolds are topologically spheres.

Abstract

We deal with seven dimensional compact Riemannian manifolds of positive curvature which admit a cohomogeneity one action by a compact Lie group G. We prove that the manifold is diffeomorphic to a sphere if the dimension of the semisimple part of G is bigger than 6.