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Nonconnected moduli spaces of positive sectional curvature metrics


Authors: Matthias Kreck and Stephan Stolz
Journal: J. Amer. Math. Soc. 6 (1993), 825-850
MSC: Primary 53C20; Secondary 53C21, 57R20, 58D27, 58G10
DOI: https://doi.org/10.1090/S0894-0347-1993-1205446-4
MathSciNet review: 1205446
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Abstract: For a closed manifold $ M$ let $ \Re _{{\text{sec}}}^ + (M)$ (resp. $ \Re _{{\text{Ric}}}^ + (M)$) be the space of Riemannian metrics on $ M$ with positive sectional (resp. Ricci) curvature and let $ {\text{Diff}}(M)$ be the diffeomorphism group of $ M$, which acts on these spaces. We construct examples of $ 7$-dimensional manifolds for which the moduli space $ \Re _{{\text{sec}}}^ + (M)/{\text{Diff}}(M)$ is not connected and others for which $ \Re _{{\text{Ric}}}^ + (M)/{\text{Diff}}(M)$ has infinitely many connected components. The examples are obtained by analyzing a family of positive sectional curvature metrics on homogeneous spaces constructed by Aloff and Wallach, on which $ SU(3)$ acts transitively, respectively a family of positive Einstein metrics constructed by Wang and Ziller on homogeneous spaces, on which $ SU(3) \times SU(2) \times U(1)$ acts transitively.


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Additional Information

DOI: https://doi.org/10.1090/S0894-0347-1993-1205446-4
Keywords: Positive sectional curvature, positive Ricci curvature, index of the Dirac operator
Article copyright: © Copyright 1993 American Mathematical Society

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