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Fat distributions with Reeb directions need not be complex contact

Javier Martínez-Aguinaga

Abstract

It is well known that every complex contact $3$-manifold, when regarded as a real manifold, gives rise to a fat $(4,6)$-distribution that admits two Reeb directions. Nonetheless, it was an open question whether the converse was true. This was not known even at the level of germs. The present work completely answers this question in the negative. We construct the first example of a fat distribution with two Reeb directions that does not support a complex contact structure anywhere, not even locally nor up to diffeomorphism. This result answers an open question by A. Bhowmick. In the second part of this work we prove a stronger result. By applying suitable $C^\infty$-perturbations to our construction, we show that the space of complex-contact germs has infinite codimension within the space of fat $(4,6)$-distribution germs with Reeb directions.

Fat distributions with Reeb directions need not be complex contact

Abstract

It is well known that every complex contact -manifold, when regarded as a real manifold, gives rise to a fat -distribution that admits two Reeb directions. Nonetheless, it was an open question whether the converse was true. This was not known even at the level of germs. The present work completely answers this question in the negative. We construct the first example of a fat distribution with two Reeb directions that does not support a complex contact structure anywhere, not even locally nor up to diffeomorphism. This result answers an open question by A. Bhowmick. In the second part of this work we prove a stronger result. By applying suitable -perturbations to our construction, we show that the space of complex-contact germs has infinite codimension within the space of fat -distribution germs with Reeb directions.
Paper Structure (7 sections, 15 theorems, 65 equations, 1 figure)

This paper contains 7 sections, 15 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometry of corank-$2$ distributions in dimension $6$
  3. Description of the almost complex structure $J:TM\to TM$.
  4. A fat distribution with Reeb directions that is not complex contact
  5. Almost complex structure $J:Q\to Q$ on $Q=TM/{\mathcal{D}}$
  6. Almost complex structure on ${\mathcal{D}}$
  7. Infinite codimension within the space of fat germs with Reeb directions

Key Result

Theorem 1.2

There exists a global fat $(4,6)$-distribution $({\mathbb{R}}^6,{\mathcal{D}})$ with Reeb directions that does not support a complex contact structure anywhere, not even locally nor up to diffeomorphism.

Figures (1)

  • Figure 1: Hierarchy of corank-$2$ distribution-germs in dimension $6$ defined by $\operatorname{Diff}$-invariant conditions. Each vertical arrow in the tree indicates that the class on top contains the class below. By Theorem \ref{['thm:global']} we know that Complex-contact germs and Fat-germs with Reeb directions constitute distinct classes. Furthermore, Theorem \ref{['InfiniteCodimension']} shows that the former has infinite codimension within the latter. Note that the subdivision is not exhaustive. We just depict the main germ-types discussed in the article.

Theorems & Definitions (43)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 33 more