Mathematical Research Letters

Volume 1 (1994)

Number 1

New Examples of Inhomogeneous Einstein Manifolds of Positive Scalar Curvature

Pages: 115 – 121



Charles P. Boyer (University of New Mexico)

Krzysztof Galicki (University of New Mexico)

Benjamin M. Mann (University of New Mexico)


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.

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