Non-negative curvature and torus actions

Abstract

Let ${\mathcal{M}}_0^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M\in {\mathcal{M}}_0^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres of dimensions greater than or equal to three. As an immediate consequence, we prove the Maximal Symmetry Rank Conjecture for all $M\in {\mathcal{M}}_0^n$. Finally, we show the Maximal Symmetry Rank Conjecture for simply-connected, non-negatively curved manifolds holds for dimensions less than or equal to nine without assuming the torus action is almost isotropy-maximal or isotropy-maximal.

Improved exposition, statements of Theorems 3.2 and 3.7 modified slightly

References in corpus (4)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• ${\mathsf{S}}^1$-actions on $4$-manifolds and fixed point homogeneous manifolds of nonnegative curvature
• Non-negatively curved 6-manifolds with almost maximal symmetry rank

Cited by in corpus (3)

• Local symmetry rank bound for positive intermediate Ricci curvatures
• Upper bounds for the dimension of tori acting on GKM manifolds
• Non-negatively curved 6-manifolds with almost maximal symmetry rank