On the existence of local quaternionic contact geometries

TL;DR

This paper proves the local existence of real analytic quaternionic contact structures with prescribed curvature data, demonstrating their dependence on a finite set of real analytic functions.

Contribution

It establishes the local existence of quaternionic contact geometries with specified curvature conditions using Cartan-Kähler theory, detailing their functional dependence.

Findings

Proves local existence of quaternionic contact structures with given curvature functions.

Shows dependence of geometries on a finite set of real analytic functions.

Utilizes Cartan-Kähler theory to achieve these results.

Abstract

We exploit the Cartan-K\"ahler theory to prove the local existence of real analytic quaternionic contact structures for any prescribed values of the respective curvature functions and their covariant derivatives at a given point on a manifold. We show that, in a certain sense, the different real analytic quaternionic contact geometries in $4n+3$ dimensions depend, modulo diffeomorphisms, on $2n+2$ real analytic functions of $2n+3$ variables.