The geometric Cauchy problem for constant-rank submanifolds

TL;DR

This paper addresses the geometric Cauchy problem for constant-rank submanifolds, establishing existence and uniqueness of solutions near a given submanifold under certain conditions.

Contribution

It provides a constructive method to solve the geometric Cauchy problem for constant-rank submanifolds, proving local existence and uniqueness.

Findings

Existence of solutions under certain assumptions.

Uniqueness of solutions in a neighborhood of the initial submanifold.

Constructive approach to solving the geometric Cauchy problem.

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 $S$.