Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions

Abstract

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.

Generalized surgery on Riemannian manifolds of positive Ricci curvature

“…Recall that the symmetry rank of a Riemannian manifold, which was introduced by Grove and Searle [20], is the rank of its isometry group. It was asked in [11] and [27] what the maximal symmetry rank for a closed, simply-connected n-dimensional Riemannian manifold of positive Ricci curvature is. By [11,Corollary D], see also [27, p. 23] and [28, p. 3796], it is given by (n − 2) in dimensions n = 4, 5, 6, and in dimension n ≥ 7 it lies between (n − 4) and (n − 2).…”

“…It was asked in [11] and [27] what the maximal symmetry rank for a closed, simply-connected n-dimensional Riemannian manifold of positive Ricci curvature is. By [11,Corollary D], see also [27, p. 23] and [28, p. 3796], it is given by (n − 2) in dimensions n = 4, 5, 6, and in dimension n ≥ 7 it lies between (n − 4) and (n − 2). We can now use the twisted suspension to show that it is given by (n − 2) in all dimensions n ≥ 4.…”

Generalized Surgery on Riemannian Manifolds of Positive Ricci Curvature

“…From the classification results it also follows that none of these manifolds have non-trivial third Betti number and all have second Betti number bounded from above by 3. Finally, Sha and Yang [33], by using a surgery theorem similar to Theorem 1.1, and Corro and Galaz-García [13], by using a lifting result of Gilkey, Park and Tuschmann [18], obtained metrics of positive Ricci curvature on connected sums of sphere bundles, which in dimension 6 are given as follows:…”

Torus actions on manifolds with positive intermediate Ricci curvature

“…In dimensions $$n\leqslant 6$$, spheres or their Riemannian products provide examples of Ricci positive manifolds with symmetry rank $$n-2$$. Corro and Galaz‐García show in [8] that for dimensions $$n\geqslant 7$$, there exist closed, simply connected $$n$$‐dimensional manifolds that admit metrics of positive Ricci curvature with symmetry rank $$n-4$$. It is still unknown whether it is possible to find $$n$$‐dimensional manifolds that admit metrics of positive Ricci curvature with symmetry rank $$n-2$$ for $$n\geqslant 7$$.…”