Automorphism Supergroups Of Supermanifolds

TL;DR

This paper extends classical results on automorphism groups of manifolds to supermanifolds, introducing mixed supermanifolds and demonstrating that automorphism supergroups form Lie supergroups.

Contribution

It generalizes the classical theorem to the supermanifold category and introduces the concept of mixed supermanifolds, broadening the scope of supergeometry.

Findings

Automorphism supergroups of supermanifolds are Lie supergroups.

Introduction of mixed supermanifolds as a new supergeometric concept.

Examples illustrating classical and super-specific automorphism groups.

Abstract

A classical theorem states that the group of automorphisms of a manifold $M$ preserving a $G$-structure of finite type is a Lie group. We generalize this statement to the category of $cs$ manifolds and give some examples, some of which being generalizations of classical notions, others being particular to the super case. Notably, we have to introduce a new notion of supermanifolds which we call mixed supermanifolds.