Mathematical Research Letters
Volume 1, Issue 1, January 1994 pp. 115-121.
New Examples of Inhomogeneous Einstein Manifolds of Positive Scalar CurvatureAuthors: Charles P. Boyer, Krzysztof Galicki, and Benjamin M. Mann
Author institution: University of New Mexico
Summary: 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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