Curvature homogeneous manifolds in dimension 4

TL;DR

This paper classifies complete four-dimensional curvature homogeneous manifolds with cohomogeneity one symmetry, revealing they are either symmetric spaces or specific examples related to the Veronese surface, and introduces a useful diagonalization technique.

Contribution

It provides a complete classification of such manifolds and introduces a method to partially diagonalize metric functions via equivariant diffeomorphisms.

Findings

Classified all complete curvature homogeneous 4D manifolds with cohomogeneity one symmetry.

Identified that these manifolds are either symmetric spaces or specific examples by Tsukada.

Established a general technique for partially diagonalizing metric functions in any dimension.

Abstract

We classify complete curvature homogeneous metrics on simply connected four dimensional manifolds which are invariant under a cohomogeneity one action. We show that they are either isometric to a symmetric space with one of its cohomogeneity one actions, or to a complete example by Tsukada on the normal bundle of the Veronese surface in CP^2. Along the way we show (in any dimension) that via an equivariant diffeomorphism the functions describing the metric can be partially diagonalized, a fact that may be useful for other problems as well