We show that, for each n 1, there exist infinitely many spin and non-spin diffeomorphism types of closed, smooth, simply-connected (n + 4)-manifolds with a smooth, effective action of a torus T n+2 and a metric of positive Ricci curvature invariant under a T n -subgroup of T n+2 . As an application, we show that every closed, smooth, simplyconnected 5-and 6-manifold admitting a smooth, effective torus action of cohomogeneity two supports metrics with positive Ricci curvature.
We show that a singular Riemannian foliation of codimension 2 on a compact simply-connected Riemannian (n + 2)-manifold, with regular leaves homeomorphic to the n-torus, is given by a smooth effective n-torus action.
We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more concise proof for the existence of slices.
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