Progress on homogeneous Einstein manifolds and some open probrems

TL;DR

This paper reviews progress on homogeneous Einstein metrics across various classes of homogeneous manifolds, introduces new invariant metrics on Stiefel manifolds, and discusses open problems in the field.

Contribution

It provides new invariant Einstein metrics on Stiefel manifolds and demonstrates the use of Gr"obner bases to prove their existence, advancing methods in the field.

Findings

New Einstein metrics on Stiefel manifolds $V_5\mathbb{R}^n$ for $n\ge 7$

Application of Gr"obner bases to solve polynomial systems in geometry

Discussion of open problems in homogeneous Einstein manifolds

Abstract

We give an overview of progress on homogeneous Einstein metrics on large classes of homogeneous manifolds, such as generalized flag manifolds and Stiefel manifolds. The main difference between these two classes of homogeneous spaces is that their isotropy representation does not contain/contain equivalent summands. We also discuss a third class of homogeneous spaces that falls into the intersection of such dichotomy, namely the generalized Wallach spaces. We give new invariant Einstein metrics on the Stiefel manifold $V_5\mathbb{R}^n$ ($n\ge 7$) and through this example we show how to prove existence of invariant Einstein metrics by manipulating parametric systems of polynomial equations. This is done by using Gr\"obner bases techniques. Finally, we discuss some open problems.