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Nonconnected Moduli Spaces of Positive Sectional Curvature Metrics

Matthias Kreck and Stephan Stolz
Journal of the American Mathematical Society
Vol. 6, No. 4 (Oct., 1993), pp. 825-850
DOI: 10.2307/2152742
Stable URL: http://www.jstor.org/stable/2152742
Page Count: 26
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Nonconnected Moduli Spaces of Positive Sectional Curvature Metrics
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Abstract

For a closed manifold M let $\mathfrak{R}^+_{\sec}(M)$ (resp. $\mathfrak{R}^+_{\operatorname {Ric}}(M))$ be the space of Riemannian metrics on M with positive sectional (resp. Ricci) curvature and let $\operatorname{Diff}(M)$ be the diffeomorphism group of M, which acts on these spaces. We construct examples of 7-dimensional manifolds for which the moduli space $\mathfrak{R}^+_{\sec}(M)/\operatorname{Diff}(M)$ is not connected and others for which $\mathfrak{R}^+_{\operatorname{Ric}}(M)/\operatorname{Diff}(M)$ 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) × SU(2) × U(1) acts transitively.

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