2016/05/19 by Andreas Arvanitoyeorgos
Mathematics · #math.DG #msc:53C25 #msc:53C30 #msc:22C05 #msc:22E60 #msc:13P05 #msc:13P10 #msc:13P15
published as Bull. Greek Math. Soc. 58 (2010-2015) 75-97 · 25 pages. arXiv admin note: text overlap with arXiv:1311.1579
arxiv created 2016/05/19 · arxiv updated 2016/05/20
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 V5ℝn (n≥ 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 Gröbner bases techniques. Finally, we discuss some open problems.