Nonconnected moduli spaces of positive sectional curvature metrics

Kreck, Matthias; Stolz, Stephan

doi:10.1090/S0894-0347-1993-1205446-4

Nonconnected moduli spaces of positive sectional curvature metrics
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by Matthias Kreck and Stephan Stolz

J. Amer. Math. Soc. 6 (1993), 825-850

DOI: https://doi.org/10.1090/S0894-0347-1993-1205446-4

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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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