Communications in Analysis and Geometry

Volume 30 (2022)

Number 8

Homogeneous metrics with prescribed Ricci curvature on spaces with non-maximal isotropy

Pages: 1849 – 1893

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n8.a8

Authors

Mark Gould (School of Mathematics and Physics, University of Queensland, St Lucia, QLD, Australia)

Artem Pulemotov (School of Mathematics and Physics, University of Queensland, St Lucia, QLD, Australia)

Abstract

Consider a compact Lie group $G$ and a closed subgroup $H \lt G$. Suppose $\mathcal{M}$ is the set of $G$-invariant Riemannian metrics on the homogeneous space $M = G/H$. We obtain a sufficient condition for the existence of $g \in \mathcal{M}$ and $c \gt 0$ such that the Ricci curvature of $g$ equals $cT$ for a given $T \in \mathcal{M}$. This condition is also necessary if the isotropy representation of $M$ splits into two inequivalent irreducible summands.

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Mark Gould’s research is supported under Australian Research Council’s Discovery Projects funding scheme (DP140101492 and DP160101376).

Artem Pulemotov is a recipient of the Australian Research Council Discovery Early-Career Researcher Award (DE150101548).

Received 30 July 2019

Accepted 5 March 2020

Published 13 July 2023