Distributions and Legendrian foliations in dimension 3

TL;DR

This paper investigates the structure of distributions on 3-dimensional manifolds, demonstrating the existence of tangent vector fields sharing the same rational first integrals, and extends similar results to higher dimensions.

Contribution

It establishes the existence of tangent vector fields with identical rational first integrals for distributions on 3D manifolds and generalizes to integrable distributions in higher dimensions.

Findings

Existence of tangent vector fields with the same first integrals in 3D

Extension of results to integrable distributions in any dimension

Provides new insights into the structure of distributions and their integrals

Abstract

We study the field of rational first integrals of distributions. We show that for a distribution on 3 dimensional manifolds there exists a tangent vector field with the same field of first integrals. We also show a similar result for integrable distributions in any dimension.