The geometric Cauchy problem for constant-rank submanifolds

Matteo Raffaelli

Vol. 340 (2026), No. 2, 399–408

DOI: 10.2140/pjm.2026.340.399

Abstract

Given a smooth $s$-dimensional submanifold $S$ of ${ℝ}^{m+c}$ and a smooth distribution $\mathcal{𝒟}\supset TS$ of rank $m$ along $S$, we study the following geometric Cauchy problem: to find an $m$-dimensional rank-$s$ submanifold $M$ of ${ℝ}^{m+c}$ (that is, an $m$-submanifold with constant index of relative nullity $m-s$) such that $M\supset S$ and $TM{|}_S=\mathcal{𝒟}$. In particular, under some reasonable assumption and using a constructive approach, we show that a solution exists and is unique in a neighborhood of $S$.

Keywords

constant nullity, distributions along submanifolds, index of relative nullity, ruled submanifold, vector cross product

Mathematical Subject Classification

Primary: 53A07

Secondary: 53B25, 58A30

Milestones

Received: 9 September 2024

Revised: 9 September 2025

Accepted: 2 October 2025

Published: 26 November 2025