Progress on homogeneous Einstein manifolds and some open probrems

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 V5R n (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.

Ricci iteration on homogeneous spaces

“…As noted earlier, α 0 ∈ (α − , α + ). Therefore, applying Lemma 4.12 to the triple g α− , g 0 , g α+ shows that g −1 := Ric g 0 ∈ M, i.e., {T ∈ M : α − ≤ z 1 /z 2 ≤ α + } ⊂ M (3) . By induction, it follows that {T ∈ M :…”

“…It follows that α 0 < α + . Thus, we may apply Lemma 4.12 to the triple g α− , g 0 , g α+ to conclude that g −1 := Ric g 0 ∈ M, i.e., {T ∈ M : z 1 /z 2 ∈ [α − , α + ]} ⊂ M (3) .…”

“…Applying Lemma 4.12 to the triple g 1 , g 0 , g α− shows that g −1 := Ric g 0 ∈ M since, by assumption, Ric g α− = g α− and Ric g 1 are positive-definite. Thus, {T ∈ M (2) : z 1 /z 2 < α − } ⊂ M (3) .…”

“…The manifolds we investigate here are compact homogeneous spaces. The study of Einstein metrics and the Ricci flow on such spaces is an active field, see, e.g., [6,Chapter 7], [28,12,24,9,8,42,30,4,7,1,3].…”