Abstract
Let us call two metrics of positive Ricci (resp. nonnegative sectional) curvature on a closed manifold M geometrically distinct if they represent different components in the moduli space of all metrics of positive Ricci (resp. nonnegative sectional) curvature of M. Manifolds which carry infinitely many geometrically distinct metrics of positive Ricci or nonnegative curvature have been exhibited in all odd high dimensions and more recently also in some even dimensions. In my talk I will survey results in this direction and discuss invariants used to distinguish these metrics.
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