The Prescribed Ricci Curvature Problem for Homogeneous Metrics

Abstract

The prescribed Ricci curvature problem consists in finding a Riemannian metric g on a manifold M such that the Ricci curvature of g equals a given (0, 2)-tensor field T . We survey the recent progress on this problem in the case where M is a homogeneous space.

The prescribed Ricci curvature problem for naturally reductive metrics on non-compact simple Lie groups

“…Particularly extensive contributions were made at that time by DeTurck and his collaborators. For a discussion of the subsequent advances, including the recent progress in the framework on homogeneous spaces, see the survey [BP19].…”

“…As it turns out, (1.2) arises in applications, such as the construction of the Ricci iteration; see [PR19,BPRZ19] and also [BP19,Section 3.10]. On the other hand, if M is open, it may be possible to obtain compelling existence theorems for (1.1) without the additional constant c. We refer to [Delay02,Delay18] for examples of such theorems.…”

“…In recent years, the third-named author and his collaborators produced a series of results, surveyed in [BP19], on the prescribed Ricci curvature problem in the class of homogeneous metrics. More precisely, suppose G is a connected Lie group.…”

The prescribed Ricci curvature problem for naturally reductive metrics on compact Lie groups

“…In general, however, complete solvability results are only available when the isotropy representation of G/H has two irreducible summands; see [Pul16b]. We refer to [BP19] for a survey.…”