Non-negative curvature and torus actions

2015/06/29 by Escher, Christine, Searle, Catherine
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1506.08685

Abstract

Let M₀ⁿ 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∈ M₀ⁿ, 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∈ M₀ⁿ. 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.

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