Local Equivalence of Riemannian Submersions via Differential Invariants

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.