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# Mathematical Research Letters

## Volume 1 (1994)

### Number 1

### New Examples of Inhomogeneous Einstein Manifolds of Positive Scalar Curvature

Pages: 115 – 121

DOI: http://dx.doi.org/10.4310/MRL.1994.v1.n1.a14

#### Authors

#### Abstract

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.