Transplanting geometrical structures

TL;DR

This paper demonstrates that geometric structures can be transplanted from germs to entire manifolds, enabling the extension of local curvature identities and realization problems to global compact manifolds.

Contribution

It proves that transplantation of geometric structures from germs to manifolds is always possible under certain conditions, extending local results to global settings.

Findings

Transplantation of geometric structures is always possible if the manifold admits such a structure.

Curvature identities valid for compact manifolds also hold for germs.

Local realization problems can be extended to the global compact case.

Abstract

We say that a germ G of a geometric structure can be transplanted into a manifold M if there is a suitable geometric structure on M which agrees with G on a neighborhood of some point P of M. We show for a wide variety of geometric structures that this transplantation is always possible provided that M does in fact admit some such structure of this type. We use this result to show that a curvature identity which holds in the category of compact manifolds admitting such a structure holds for germs as well and we present examples illustrating this result. We also use this result to show geometrical realization problems which can be solved for germs of structures can in fact be solved in the compact setting as well.