Prescribing Ricci curvature on homogeneous spaces

TL;DR

This paper investigates the prescribed Ricci curvature problem on homogeneous spaces with G-invariant metrics, showing that in the compact case, the Ricci curvature map is generically locally injective and surjective, using a new formula for the Lichnerowicz Laplacian.

Contribution

It introduces a formula for the Lichnerowicz Laplacian in terms of the moment map, demonstrating the generic local invertibility of the Ricci curvature map on compact homogeneous spaces.

Findings

The Ricci curvature map is generically locally invertible in the compact case.

A new formula for the Lichnerowicz Laplacian is established.

The property of local injectivity and surjectivity is shown to be generic in the compact setting.

Abstract

The prescribed Ricci curvature problem in the context of G-invariant metrics on a homogeneous space M=G/K is studied. We focus on the metrics at which the Ricci curvature map is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.