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arXiv · 2409.04358

The geometric Cauchy problem for constant-rank submanifolds

Matteo Raffaelli
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Abstract

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

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Matteo Raffaelli. 2024-09-06. The geometric Cauchy problem for constant-rank submanifolds. https://doi.org/10.2140/pjm.2026.340.399

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