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Singular curves of hyperbolic $(4, 7)$-distributions of type $C_3$

Goo Ishikawa, Yoshinori Machida

Abstract

A distribution of rank $4$ on a $7$-dimensional manifold is called a $(4, 7)$-distribution if its local sections generate the whole tangent space by taking Lie brackets once. Singular curves of $(4, 7)$-distributions are studied in this paper. In particular the class of hyperbolic $(4, 7)$-distributions of type $C_3$ is introduced and singular curves are completely described via prolongations for them.

Singular curves of hyperbolic $(4, 7)$-distributions of type $C_3$

Abstract

A distribution of rank on a -dimensional manifold is called a -distribution if its local sections generate the whole tangent space by taking Lie brackets once. Singular curves of -distributions are studied in this paper. In particular the class of hyperbolic -distributions of type is introduced and singular curves are completely described via prolongations for them.
Paper Structure (6 sections, 8 theorems, 40 equations)

This paper contains 6 sections, 8 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Singular curves of distributions
  3. $(4, 7)$-distributions and their singular curves
  4. Prolongations of hyperbolic $(4, 7)$-distributions
  5. Proof of main theorems
  6. Isotropic Grassmannian on $6$-dimensional symplectic space

Key Result

Theorem 1.2

Let $D \subset TX$ be a hyperbolic $(4, 7)$-distribution. Then (1) If the direction $[u] \in P(C_{x})$ is of type $C_3$, then there exists a $D$-singular path $\gamma : (\mathbf{R}, 0) \to X$ with $\gamma(0) = x, [\gamma'(0)] = [u]$. In particular $u \in {\hbox{\it{SVC}}}$. (2) If $D$ is of type $C_

Theorems & Definitions (14)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Definition 4.1
  • Lemma 4.2
  • Remark 4.3
  • ...and 4 more