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Positive intermediate Ricci curvature with maximal symmetry rank

Lee Kennard, Lawrence Mouillé

Abstract

Generalizing the foundational work of Grove and Searle, the second author proved upper bounds on the ranks of isometry groups of closed Riemannian manifolds with positive intermediate Ricci curvature and established some topological rigidity results in the case of maximal symmetry rank and positive second intermediate Ricci curvature. Here, we recover even stronger topological rigidity, including results for higher intermediate Ricci curvatures and for manifolds with nontrivial fundamental groups.

Positive intermediate Ricci curvature with maximal symmetry rank

Abstract

Generalizing the foundational work of Grove and Searle, the second author proved upper bounds on the ranks of isometry groups of closed Riemannian manifolds with positive intermediate Ricci curvature and established some topological rigidity results in the case of maximal symmetry rank and positive second intermediate Ricci curvature. Here, we recover even stronger topological rigidity, including results for higher intermediate Ricci curvatures and for manifolds with nontrivial fundamental groups.
Paper Structure (7 sections, 26 theorems, 7 equations, 2 figures, 1 table)

This paper contains 7 sections, 26 theorems, 7 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximal symmetry rank for $\mathbf{Ric_2 > 0}$ in dimension 4
  4. Maximal symmetry rank for $\mathbf{Ric_2 > 0}$ in dimension 6
  5. Maximal symmetry rank for $\mathbf{Ric_2 > 0}$ in dimensions $\mathbf{2n \geq 8}$
  6. Maximal symmetry rank for $\mathbf{Ric_k > 0}$ with $\mathbf{k \geq 3}$
  7. Fundamental groups for maximal symmetry rank

Key Result

Theorem 1.1

If $\mathsf{T}^n$ acts effectively by isometries on a closed, simply connected Riemannian $(2n-1)$-manifold $M$ with $\mathop{\mathrm{Ric}}\nolimits_{2} > 0$, then $M$ is diffeomorphic to $S^{2n-1}$.

Figures (2)

  • Figure 1: Weighted orbit space of closed simply connected $4$-dimensional $\mathsf{T}^2$-manifold.
  • Figure 2: Decomposing the orbit space of a simply connected $4$-dimensional $\mathsf{T}^2$-manifold with Euler characteristic $\geq 5$.

Theorems & Definitions (41)

  • Definition
  • Theorem 1.1: Mouille22b
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 2.1
  • Lemma 2.2: Betti Number Lemma
  • Lemma 2.3: Isotropy Rank Lemma
  • Lemma 2.4: Spherical Isotropy Rank Lemma
  • ...and 31 more