Local Equivalence of Riemannian Submersions via Differential Invariants
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Sharipov, Xurshid Sharipova, Sadoqat Salimov, Esanjon Mardiev, Islomkhon |
| author_facet | Sharipov, Xurshid Sharipova, Sadoqat Salimov, Esanjon Mardiev, Islomkhon |
| contents | 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 $φ=|K|$ and the curvature $2$-form $Ω=dθ|_{\mathcal H}$ of the associated connection $1$-form $θ$, and we prove an equivalence criterion phrased purely in terms of the base data $(\bar g,φ,Ω)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14820 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Equivalence of Riemannian Submersions via Differential Invariants Sharipov, Xurshid Sharipova, Sadoqat Salimov, Esanjon Mardiev, Islomkhon Differential Geometry 53C20, 53C12, 58A15, 22E60 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 $φ=|K|$ and the curvature $2$-form $Ω=dθ|_{\mathcal H}$ of the associated connection $1$-form $θ$, and we prove an equivalence criterion phrased purely in terms of the base data $(\bar g,φ,Ω)$. 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. |
| title | Local Equivalence of Riemannian Submersions via Differential Invariants |
| topic | Differential Geometry 53C20, 53C12, 58A15, 22E60 |
| url | https://arxiv.org/abs/2603.14820 |