G-manifolds with positive Ricci curvature and many isolated singular orbits

TL;DR

This paper constructs G-manifolds with positive Ricci curvature in specific cohomogeneity classes, allowing for arbitrary numbers of isolated singular orbits, expanding the understanding of curvature and symmetry in geometric structures.

Contribution

It demonstrates the existence of G-manifolds with positive Ricci curvature having any specified number of isolated singular orbits in cohomogeneity 3 and even numbers in cohomogeneity 5.

Findings

Existence of G-manifolds with positive Ricci curvature and arbitrary isolated singular orbits in cohomogeneity 3.

Extension of the result to cohomogeneity 5 with even number of singular orbits.

Construction methods for invariant metrics with positive Ricci curvature on these manifolds.

Abstract

We show that in cohomogeneity 3 there are G-manifolds with any given number of isolated singular orbits and an invariant metric of positive Ricci curvature. We show that the corresponding result is also true in cohomogeneity 5 provided the number of singular orbits is even.