New Examples of Inhomogeneous Einstein Manifolds of Positive Scalar Curvature

The purpose of this note is to announce the explicit construction of a new infinite family of compact inhomogeneous Einstein manifolds of positive scalar curvature in every dimension of the form $\scriptstyle{4n-5}$ for $\scriptstyle{n >2.}$ In fact, each manifold has two, non-homothetic, Einstein metrics of positive scalar curvature. Moreover, in every fixed dimension, these families each contain infinitely many distinct homotopy types. Each individual manifold has a Sasakian 3-structure and all of these examples are bi-quotients of unitary groups of the form $\scriptstyle{U(1)\backslash U(n)/U(n-2).}$ In particular, when $\scriptstyle{n=3,}$ we obtain infinite subfamilies of mutually distinct homotopy types where each member of the subfamily is strongly inhomogeneous; that is, these Einstein manifolds are not even homotopy equivalent to any compact Riemannian homogeneous space.

Full Text (PDF format)

Published 1 January 1994