On the curvature on G-manifolds with finitely many non-principal orbits

Abstract

We investigate the cuvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature and non-negative sectional curvature, and discuss some families of examples to which these existence results apply.

$$G$$ G -manifolds with positive Ricci curvature and many isolated singular orbits

“…In [BW2] Theorems 5 and 6, it is shown that the families in Examples 1.6 and 1.7 below both contain infinitely many homotopy types.…”

“…In [BW2] a general existence result is established for positive Ricci curvature metrics on G-manifolds with finitely many non-principal orbits (see Proposition 1.11 below). Using this result, many Ricci positive examples were presented.…”

“…G-manifolds with finitely many non-principal orbits in cohomogeneities greater than one have been studied both topologically in [BW1] and geometrically in [BW2]. As it will be relevant later, we will first dicuss briefly the key topological features of these objects.…”

“…Examples 1.7. ( [BW2]) Given Aloff-Wallach spaces W p 1 ,p 2 and W q 1 ,q 2 with p 2 1 + p 1 p 2 + p 2 2 = q 2 1 + q 1 q 2 + q 2 2 , there is a 13-dimensional SU(3)-manifold M 13 p 1 p 2 q 1 q 2 of cohomogeneity 5, with orbit space a suspension of CP 2 and two singular orbits equal to the given Aloff-Wallach manifolds, which admits an invariant metric of positive Ricci curvature.…”

“…The problem with the Ricci positive examples appearing in [BW2] is that all examples contain either one or two non-principal orbits. A natural question ([BW2] Open Problem number 1) is thus: is it possible to find invariant Ricci positive metrics on manifolds having more than two non-principal orbits? This is an intriguing question as the obvious candidates just fail.…”