The geometric Cauchy problem for rank-one submanifolds

TL;DR

This paper investigates the geometric Cauchy problem for rank-one submanifolds in Euclidean space, providing conditions for local solutions and a parametric description of these solutions.

Contribution

It introduces a framework for solving the geometric Cauchy problem for rank-one submanifolds and offers sufficient conditions for local well-posedness with explicit parametric solutions.

Findings

Established sufficient conditions for local well-posedness.

Provided a parametric description of solutions.

Analyzed the geometric structure of rank-one submanifolds.

Abstract

Given a smooth distribution $\mathscr{D}$ of $m$-dimensional planes along a smooth regular curve $\gamma$ in $\mathbb{R}^{m+n}$, we consider the following problem: to find an $m$-dimensional rank-one submanifold of $\mathbb{R}^{m+n}$, that is, an $(m-1)$-ruled submanifold with constant tangent space along the rulings, such that its tangent bundle along $\gamma$ coincides with $\mathscr{D}$. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution.