Prescribing Ricci curvature on homogeneous manifolds

Abstract: Given a symmetric 2-tensor $T$ on a manifold $M$, it is a classical problem in Riemannian geometry to ask about the existence (and uniqueness) of a metric $g$ on $M$ such that $\textrm{Ric}(g) = T$ (see e.g. [Besse,Chap.5]). Assuming that $M$ is a homogeneous manifold, we will consider in the talk the $G$-invariant version of the problem, where $G$ is a (unimodular, not necessarily compact) Lie group acting transitively on $M$.

After an overview of results and questions, we will give a formula for the differential $d\textrm{Ric}$ of the function $\textrm{Ric}$ at a $G$-invariant metric $g$, which is precisely the Lichnerowicz Laplacian acting on $G$-invariant symmetric 2-tensors. The formula is in terms of the moment map for the variety of Lie algebras.

As an application, we will consider the concept of Ricci local invertibility for a metric $g$, i.e., when the kernel of $d\textrm{Ric}$ at $g$ consists only of the subspace generated by $g$. This is equivalent to the existence of a $G$-invariant solution $g'$ to the Prescribed Ricci Problem $\textrm{Ric}(g') = cT$ (for some $c>0$), for any $G$-invariant $T$ sufficiently close to $\textrm{Ric}(g)$. Our main result is that any irreducible naturally reductive metric on $M$ with respect to $G$ is Ricci locally invertible.

This is joint work in progress with Cynthia Will.

differential geometry

Audience: researchers in the topic

( video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Riemannian geometry.

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