Basic automorphism of Cartan foliations covered by fibrations

TL;DR

This paper investigates the structure and size of the basic automorphism groups of Cartan foliations, especially those covered by fibrations, providing conditions for their Lie group structure and methods for their determination.

Contribution

It establishes sufficient conditions for basic automorphism groups to form finite-dimensional Lie groups and offers estimates of their dimensions for certain Cartan foliations.

Findings

Conditions for Lie group structure of automorphism groups.

Dimension estimates for these groups.

Method for finding automorphism groups in specific cases.

Abstract

The basic automorphism group ${A}_B(M,F)$ of a Cartan foliation $(M, F)$ is the quotient group of the automorphism group of $(M, F)$ by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism groups. Estimates of the dimension of these groups are obtained. For some class of Cartan foliations with integrable an Ehresmann connection, a method for finding of basic automorphism groups is specified.