Regularity of infinitesimal automorphisms of involutive structures

TL;DR

This paper establishes the smoothness of infinitesimal automorphisms in involutive structures by developing a regularity theory for sections of vector bundles with a compatible connection, extending classical results to a broader geometric context.

Contribution

It introduces a novel regularity framework for sections of vector bundles over involutive structures using $V$-connections, generalizing Lewy's theorem for CR functions.

Findings

Infinitesimal automorphisms are smooth under certain conditions.

Develops a regularity theory for $V$-sections.

Identifies nondegeneracy conditions for smoothness.

Abstract

In this paper, we prove that infinitesimal automorphisms of an involutive structure are smooth. For this, we build a regularity theory for sections of vector bundles over an involutive structure $(M,V)$ endowed with a connection compatible with $V$, which we call $V$-connection. We show that $V$-sections, i.e. sections which are parallel with respect to $V$ under the $V$-connection, satisfy an analogue of Hans Lewy's theorem as formulated for CR functions on an abstract CR manifold by Berhanu and Xiao, and introduce certain (generically satisfied) nondegeneracy conditions ensuring their smoothness.