Local Equivalence of Riemannian Submersions via Differential Invariants

TL;DR

This paper develops a practical method using differential invariants to determine local equivalence of Riemannian submersions, focusing on orbit submersions induced by Killing fields, with explicit formulas and decision procedures.

Contribution

It introduces explicit formulas and an invariant-based criterion for orbit submersions induced by Killing fields, along with a finite-order decision procedure and computational insights.

Findings

Derived explicit formulas for A and H in terms of Killing data.

Established an equivalence criterion based on base data (ar g,,).

Presented a finite-order invariant decision procedure with practical stopping rules.

Abstract

We study the local equivalence problem for Riemannian submersions under fiber-preserving isometries using differential invariants. After briefly recalling the vertical--horizontal splitting, the O'Neill tensors $A$ and $T$, and the mean curvature $H$ of the fibers, we outline a practical invariant-based equivalence workflow. Our main contribution is the analysis of a concrete model class: orbit submersions induced by a nowhere-vanishing Killing field $K$. In this class we derive explicit formulas for $A$ and $H$ in terms of the Killing data, namely the length function $\varphi=|K|$ and the curvature $2$-form $\Omega=d\theta|_{\mathcal H}$ of the associated connection $1$-form $\theta$, and we prove an equivalence criterion phrased purely in terms of the base data $(\bar g,\varphi,\Omega)$. We further present a finite-order invariant decision procedure under a stated genericity assumption (with a practical stopping rule), together with a compact low-order invariant profile and computational remarks. Benchmark examples (product submersions, warped products, and the Hopf fibration) illustrate both the strengths and limitations of low-order scalar signatures.