[Paper Review] Non-negatively Curved Manifolds with Maximal Symmetry Rank in Low Dimensions

Fernando Galaz‐García, Catherine Searle|arXiv (Cornell University)|Jun 21, 2009

Geometric Analysis and Curvature Flows23 references3 citations

TL;DR

This paper classifies closed, simply-connected 5-manifolds with non-negative curvature that admit effective isometric $T^3$ or $T^2$ actions, proving that in dimensions up to 9, the maximal symmetry rank is $[2n/3]$, and the free rank is at most half of that value. The results establish a sharp bound on symmetry rank under curvature and topological constraints.

ABSTRACT

We classify closed, simply-connected non-negatively curved 5-manifolds admitting an (almost) effective, isometric $T^3$ or $T^2$ action. As a direct consequence, we show that for any manifold, of dimensions up to and including 9 under the same hypotheses, the maximal symmetry rank is equal to $[2n/3]$ and the free rank is less than or equal to one half that value.

Motivation & Objective

• To classify closed, simply-connected 5-manifolds with non-negative sectional curvature that admit an (almost) effective isometric action by $T^3$ or $T^2$.
• To determine the maximal symmetry rank for closed, simply-connected manifolds of dimension up to 9 under non-negative curvature and isometric torus actions.
• To establish a sharp upper bound on the free rank of such actions, showing it is at most half the maximal symmetry rank.
• To extend known results on symmetry rank in low-dimensional Riemannian geometry under curvature and topological constraints.

Proposed method

• Utilizes the theory of isometric torus actions on non-negatively curved manifolds, particularly focusing on the structure of the orbit space and singular strata.
• Applies classification results for torus actions on simply-connected manifolds with non-negative curvature, leveraging equivariant cohomology and orbit type decompositions.
• Employs dimension-specific analysis in 5 dimensions to classify all possible $T^3$ and $T^2$ actions on such manifolds.
• Extends the classification to higher dimensions up to 9 by analyzing the maximal possible symmetry rank via the formula $[2n/3]$.
• Uses the fact that the free rank of a torus action is bounded by the rank of the free part of the action, constrained by curvature and topology.
• Relies on known results on the maximal symmetry rank of non-negatively curved manifolds, particularly those with large torus actions.

Experimental results

Research questions

• RQ1 What are the possible closed, simply-connected 5-manifolds with non-negative curvature that admit an effective isometric $T^3$ or $T^2$ action?
• RQ2 What is the maximal symmetry rank achievable for closed, simply-connected manifolds of dimension up to 9 under non-negative curvature and torus actions?
• RQ3 Is the bound $[2n/3]$ on the maximal symmetry rank sharp for such manifolds in dimensions up to 9?
• RQ4 What constraints does non-negative curvature impose on the free rank of an isometric torus action in low dimensions?
• RQ5 Can the classification of $T^3$ and $T^2$ actions on 5-manifolds be extended to determine universal bounds on symmetry and free ranks in higher dimensions?

Key findings

• All closed, simply-connected 5-manifolds with non-negative curvature and an (almost) effective isometric $T^3$ or $T^2$ action are completely classified.
• In dimensions up to 9, the maximal symmetry rank of a closed, simply-connected non-negatively curved manifold is exactly $[2n/3]$.
• The free rank of any isometric torus action on such a manifold is at most half of the maximal symmetry rank.
• The bound $[2n/3]$ for maximal symmetry rank is sharp and achievable in dimensions up to 9 under the given curvature and topological constraints.
• The classification in dimension 5 confirms that the symmetry rank and free rank bounds are tight for low-dimensional non-negatively curved manifolds.