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Nonconnected moduli spaces of nonnegative sectional curvature metrics on simply connected manifolds. (English) Zbl 1385.53022

Authors’ abstract: We show that in each dimension \(4n+3\), \(n\geq 1\), there exist infinite sequences of closed smooth simply connected manifolds \(M\) of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension 7, and our result also hold for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of I. Belegradek et al. [J. Differ. Geom. 89, No. 1, 49–85 (2011; Zbl 1242.53035)], we obtain that for each such \(M\) the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold \(M\times \mathbb{R}\) also has infinitely many components.

MSC:

53C20 Global Riemannian geometry, including pinching
58D17 Manifolds of metrics (especially Riemannian)
58D27 Moduli problems for differential geometric structures
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C30 Differential geometry of homogeneous manifolds
57R19 Algebraic topology on manifolds and differential topology
57R55 Differentiable structures in differential topology

Citations:

Zbl 1242.53035
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References:

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