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On Eta-Einstein Sasakian Geometry

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Abstract

A compact quasi-regular Sasakian manifold M is foliated by one-dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact Kähler orbifold . In the case when the transverse space is also Einstein the corresponding Sasakian manifold M is said to be Sasakian η-Einstein. In this article we study η-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem in the context of Sasakian geometry to prove the existence of η-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl and Lorenzian Sasakian-Einstein structures.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, University of New Mexico, Albuquerque, NM, 87131, USA

    Charles P. Boyer

  2. Max-Planck-Institut für Mathematik, 53111, Bonn, Germany

    Krzysztof Galicki

  3. Dipartimento di Matematica e Informatica, Universitá di Cagliari, Cagliari, Italy

    Paola Matzeu

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  1. Charles P. Boyer
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  2. Krzysztof Galicki
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  3. Paola Matzeu
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Correspondence to Charles P. Boyer.

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Communicated by G.W. Gibbons

Krzysztof Galicki: On leave from Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA. E-mail: galicki@math.unm.edu

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Boyer, C., Galicki, K. & Matzeu, P. On Eta-Einstein Sasakian Geometry. Commun. Math. Phys. 262, 177–208 (2006). https://doi.org/10.1007/s00220-005-1459-6

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