Elementary Relative Tractor Calculus for Legendrean contact structures

TL;DR

This paper develops an elementary approach to relative tractor calculus for Legendrean contact structures, introducing an invariant partial connection and a second order invariant differential operator, advancing geometric analysis on such manifolds.

Contribution

It provides a new elementary construction of a partial connection and a relative BGG operator for Legendrean contact structures, simplifying previous methods.

Findings

Constructed an invariant partial connection on the quotient bundle TM/F.

Developed a naive relative tractor calculus framework.

Built a second order invariant differential operator as the first relative BGG operator.

Abstract

For a manifold $M$ endowed with a Legendrean (or Lagrangean) contact structure $E\oplus F \subset TM$, we give an elementary construction of an invariant partial connection on the quotient bundle $TM/F$. This permits us to develop a na\"{i}ve version of relative tractor calculus and to construct a second order invariant differential operator, which turns out to be the first relative BGG operator induced by the partial connection.