A note on maximal symmetry rank, quasipositive curvature, and low dimensional manifolds

TL;DR

This paper classifies manifolds with positive or quasipositive curvature that admit maximal symmetry rank torus actions, showing they are diffeomorphic to standard spaces like spheres or projective spaces.

Contribution

It proves that manifolds with maximal symmetry rank and positive or quasipositive curvature are diffeomorphic to well-known standard manifolds, extending classification results.

Findings

Maximal symmetry rank torus actions on positively curved spheres and projective spaces are linear.

4- and 5-manifolds with quasipositive curvature and maximal symmetry rank are diffeomorphic to spheres or complex projective planes.

Abstract

We show that any effective isometric torus action of maximal rank on a compact Riemannian manifold with positive (sectional) curvature and maximal symmetry rank, that is, on a positively curved sphere, lens space, complex or real projective space, is equivariantaly diffeomorphic to a linear action. We show that a compact, simply connected Riemannian 4- or 5-manifold of quasipositive curvature and maximal symmetry rank must be diffeomorphic to the 4-sphere, complex projective plane or the 5-sphere.