New examples of inhomogeneous Einstein manifolds of positive scalar curvature.(English)Zbl 0842.53033

It is one of the most interesting subjects in Riemannian geometry to study compact Einstein manifolds. Classically, the first examples of such manifolds are compact symmetric spaces and certain compact Riemannian homogeneous spaces. The first examples of compact simply connected inhomogeneous Einstein manifolds with positive scalar curvature were found by D. Page in [Phys. Lett. B 79, 235-238 (1978)]. Then, further examples have been constructed by many authors.
The purpose of the present article is to announce the explicit construction of a new infinite family of compact inhomogeneous Einstein manifolds of positive scalar curvature. Each individual manifold has a Sasakian 3-structure and all of these examples are bi-quotients of unitary groups of the form $$U(1)\backslash U(n)/U(n- 2)$$ (for details, see the authors’ recent article [University of New Mexico and Max-Planck-Institute preprints, June 1993]).
In particular, the authors consider the case where $$n= 3$$ and show that there are infinite families of mutually distinct homotopy types, where each member of the subfamily is strongly inhomogeneous (namely, it is not homotopic equivalent to any compact Riemannian homogeneous space).

MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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