Locally 2-fold symmetric manifolds are locally symmetric

TL;DR

This paper proves that manifolds with local 2-fold symmetry, across Riemannian, pseudoriemannian, and Finslerian geometries, are necessarily locally symmetric, extending the understanding of symmetry properties in differential geometry.

Contribution

It establishes that for any k ≥ 2, locally k-fold symmetric manifolds are inherently locally symmetric, unifying symmetry concepts across multiple geometric frameworks.

Findings

Locally 2-fold symmetric manifolds are locally symmetric.

The result applies to Riemannian, pseudoriemannian, and Finslerian manifolds.

Symmetry properties are unified for k ≥ 2 across different geometries.

Abstract

A manifold is locally \emph{$k$-fold symmetric}, if for any point and any $k$-dimensional vector subspace tangent to this point there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that $k$-dimensional vector subspace is minus the identity. We show that for $k \ge 2$, Riemannian, pseudoriemannian and Finslerian locally $k$-fold symmetric manifolds are locally symmetric.