# Local symmetry rank bound for positive intermediate Ricci curvatures

## TL;DR

This paper establishes a new upper bound on the symmetry rank of manifolds with positive intermediate Ricci curvature, extending classical results and confirming the Maximal Symmetry Rank Conjecture in this context.

## Contribution

It introduces a local argument to derive a symmetry rank bound for manifolds with positive intermediate Ricci curvature, generalizing previous curvature-specific results.

## Key findings

- Proves a symmetry rank bound: r ≤ ⌊(n+k)/2⌋ for manifolds with positive k-th intermediate Ricci curvature.
- Confirms the Maximal Symmetry Rank Conjecture for manifolds with positive intermediate Ricci curvature.
- Provides an optimal dimensional restriction on isometric immersions into manifolds with positive intermediate Ricci curvature.

## Abstract

We use a local argument to prove if an $r$-dimensional torus acts isometrically and effectively on a connected $n$-dimensional manifold which has positive $k^\mathrm{th}$-intermediate Ricci curvature at some point, then $r \leq \lfloor \frac{n+k}{2} \rfloor$. This symmetry rank bound generalizes those established by Grove and Searle for positive sectional curvature and Wilking for quasipositive curvature. As a consequence, we show that the symmetry rank bound in the Maximal Symmetry Rank Conjecture for manifolds of non-negative sectional curvature holds for those which also have positive intermediate Ricci curvature at some point. In the process of proving our symmetry rank bound, we also obtain an optimal dimensional restriction on isometric immersions of manifolds with non-positive intermediate Ricci curvature into manifolds with positive intermediate Ricci curvature, generalizing a result by Otsuki.