Abstract

In this paper we study non-negatively curved and rationally elliptic GKM_4 manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds. Moreover, we give a simplified proof of a characterisation of products of simplices among orbit spaces of locally standard torus manifolds. This characterisation was originally proved in Wiemeler (J Lond Math Soc 91(3): 667–692, 2015) and was used there to obtain a classification of non-negatively curved torus manifolds.

Highlights

  • The classification of Riemannian manifolds with positive or non-negative sectional curvature is one of the most prominent open problems in differential geometry

  • In this paper we continue our investigation [17] of isometric torus actions of GKM type on Riemannian manifolds with sectional curvature bounded from below

  • In [17] we showed that a positively curved Riemannian manifold admitting an isometric GKM3 torus action has the same real cohomology ring as a compact rank one symmetric space

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Summary

Introduction

The classification of Riemannian manifolds with positive or non-negative sectional curvature is one of the most prominent open problems in differential geometry. They are all combinatorially equivalent to one of the following: I 3, 3, 3, 2 × I , 2 × I Using these restrictions, we show that the combinatorial type of a neighborhood of a vertex in the GKM graph of a non-negatively curved GKM4 manifold is the same as that of a neighborhood of a vertex in the vertex edge graphof a finite product i ni × i mi. We show that the combinatorial type of a neighborhood of a vertex in the GKM graph of a non-negatively curved GKM4 manifold is the same as that of a neighborhood of a vertex in the vertex edge graphof a finite product i ni × i mi Extending this local result to all ofthen yields the covering described in Theorem 1.1. We would like to thank the anonymous referee for comments which helped to improve the presentation of this paper

GKM manifolds
GKM orbifolds
Torus manifolds and orbifolds
Coverings of GKM graphs
Extending GKM graphs
A model
Upper bound for the dimension of the acting torus
GKM manifolds with invariant almost complex structures
Torus manifolds revisited
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