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The geometric Cauchy problem for constant-rank submanifolds

Matteo Raffaelli

TL;DR

The paper addresses the problem of extending a connected $s$-dimensional submanifold $S\subset \mathbb{R}^{m+c}$ along with a rank-$m$ distribution $\mathcal{D}\supset TS$ to an $m$-dimensional submanifold $M$ with constant index of relative nullity $m-s$. It reframes this as a geometric Cauchy problem for constant-rank submanifolds and develops a constructive approach based on $(m-s)$-ruled foliations: a local extension exists and is unique provided there is a normal direction $N^{\ast}\in\mathcal{D}^{\perp}$ with nonsingular shape operator $A^{\ast}$ and the compatibility $\mathrm{Im}\,\phi_p=\mathrm{Im}\,\phi_p(T_pS,N^{\ast}_p)$ for all $p\in S$. The method yields an explicit local parametrization $\sigma(a,b)=\xi(a)+\sum_{j=1}^{m-s} b^{j}X_j(a)$, with $X_j$ built from a frame of $\mathcal{D}$ and a cross product, providing a constructive alternative to the Gauss parametrization. These results extend prior work to arbitrary $s$ and codimensions, offering a practical tool for constructing constant-rank submanifolds in a neighborhood of $S$.

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 $S$.

The geometric Cauchy problem for constant-rank submanifolds

TL;DR

The paper addresses the problem of extending a connected -dimensional submanifold along with a rank- distribution to an -dimensional submanifold with constant index of relative nullity . It reframes this as a geometric Cauchy problem for constant-rank submanifolds and develops a constructive approach based on -ruled foliations: a local extension exists and is unique provided there is a normal direction with nonsingular shape operator and the compatibility for all . The method yields an explicit local parametrization , with built from a frame of and a cross product, providing a constructive alternative to the Gauss parametrization. These results extend prior work to arbitrary and codimensions, offering a practical tool for constructing constant-rank submanifolds in a neighborhood of .

Abstract

Given a smooth -dimensional submanifold of and a smooth distribution of rank along , we study the following geometric Cauchy problem: to find an -dimensional rank- submanifold of (that is, an -submanifold with constant index of relative nullity ) such that and . In particular, under some reasonable assumption and using a constructive approach, we show that a solution exists and is unique in a neighborhood of .
Paper Structure (3 sections, 7 theorems, 32 equations)

This paper contains 3 sections, 7 theorems, 32 equations.

Table of Contents

  1. Introduction and main result
  2. Constant-rank submanifolds
  3. Proof of the main result

Key Result

Theorem 1.2

Suppose that there exists a section $N^{\ast}$ of $\mathcal{D}^{\perp}$ such that the shape operator $A^{\ast} \coloneqq A_{N^{\ast}}$ of $S$ in direction $N^{\ast}$ is nonsingular, i.e., The geometric Cauchy problem for rank-$s$ submanifolds of $\mathbb{R}^{m+c}$ has a solution if and only if $\phi_{p}(T_{p}S, N^{\ast}_{p}) = \mathop{\mathrm{Im}}\nolimits \phi_{p}$ for all $p \in S$. Moreover:

Theorems & Definitions (21)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Corollary 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 11 more