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In every odd dimension $n\geq 5$ we exhibit large classes of closed n-dimensional manifolds which admit infinitely many different geometries of positive Ricci curvature, i.e., manifolds for which their moduli space of metrics of positive Ricci curvature has infinitely many connected components.

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

In C. R. Math. Acad. Sci. Paris 348 (2010) pp. 283--285 (arXiv:0811.0840) we constructed examples of    -manifolds with finite second homotopy group and non-vanishing    -genus. The reasoning was based on an equivariant surgery lemma which only holds under additional assumptions. To remedy the situation we give a construction using explicit equivariant surgeries.

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

In each dimension $4k+1\geq 9$, we exhibit infinite families of closed manifolds with fundamental group $\mathbb Z_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\geq 7$.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy RP^5 has infinitely many path components. We also show that in each dimension 4k+1 there are at least 2^{2k} homotopy RP^{4k+1}s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4k+3≥7.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

We show that in each dimension 4n+3, n>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 holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, inconjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

Let M be a Milnor sphere or, more generally, the total space of a linear S^3-bundle over S^4 with H^4(M;Q) = 0. We show that the moduli space of metrics of nonnegative sectional curvature on M has infinitely many path components. The same holds true for the moduli space of m etrics of positive Ricci curvature on M.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds