Locally homogeneous aspherical Sasaki manifolds

Abstract

Let G/H be a contractible homogeneous Sasaki manifold.

A note on vanishing of equivariant differentiable cohomology of proper actions and application to CR-automorphism and conformal groups

“…Since the Kähler metric g is conformal to h, this map is also locally conformal for g, so g is locally holomorphically conformal to the standard flat Kähler space C n . By the above remark following Theorem 30, g is actually locally homothetic to the standard complex space C n , which also implies that g flat and locally holomorphically isometric to C n , proving (2). The remark also implies (3).…”

“…For a compact strictly pseudo-convex CR-manifold M , this result is originally due to Webster [41], see also [21], and [5,Proposition 4.4] for a related result in the context of compact Sasaki manifolds, compare also [2]. For further background on spherical CR-manifolds see [10,23].…”

“…Then η ′′ j is a G j -invariant nonnegative smooth function. Since W 1 is G jinvariant, η ′′ j also satisfies (1), (2). In particular, note that supp ǫ ′ j ⊆ supp η ′′ j .…”

“…Suppose that G acts properly, then by Theorem 14, (3) is satisfied. Now (3) clearly implies(2).Finally, if (2) is satisfied, the canonical class µ ∈ H 1 d G, C ∞ (M, R >0 )associated to either the CR-or conformal structure on M vanishes. In the first case, as is implied by Proposition 26, G preserves an associated contact Riemannian metric on M , respectively in the case of Proposition 27, the group G preserves a Riemannian metric in the given conformal class.…”