Orbifolds, geometric structures and foliations. Applications to harmonic maps

TL;DR

This paper explores the extension of classical geometric structures to orbifolds, emphasizing their relation to foliations and proposing a foliated approach to harmonic maps between Riemannian orbifolds.

Contribution

It demonstrates that geometric structures on manifolds can be adapted to orbifolds and introduces a foliated framework for studying harmonic maps in this context.

Findings

Classical geometric theories extend to orbifolds.

Foliated structures provide new insights into orbifold geometry.

A novel approach to harmonic maps between Riemannian orbifolds is proposed.

Abstract

In recent years a lot of attention has been paid to topological spaces which are a bit more general than smooth manifolds - orbifolds. Orbifolds are intuitively speaking manifolds with some singularities. The formal definition is also modelled on that of manifolds, an orbifold is a topological space which locally is homeomorphic to the orbit space of a finite group acting on $R^n$. Orbifolds were defined by Satake, as V-manifolds, then studied by W. Thurston, who introduced the term "orbifold". Due to their importance in physics, and in particular in the string theory, orbifolds have been drawing more and more attention. In this paper we propose to show that the classical theory of geometrical structures, easily translates itself to the context of orbifolds and is closely related to the theory of foliated geometrical structures, cf. \cite{Wo0}. Finally, we propose a foliated approach to the study of harmonic maps between Riemannian orbifolds based on our previous research into transversely harmonic maps.