Transplanting geometrical structures

• Published in: Differential geometry and its applications Vol. 31; no. 3; pp. 374 - 387
• Main Authors: Euh, Y., Gilkey, P., Park, J.H., Sekigawa, K.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-06-2013
• Subjects: (Almost)-(para)-complex; (Almost)-(para)-Hermitian; (Para)-Kähler; (Para)-Kähler–Weyl; Affine; Categories; Pseudo-Riemannian; 53B20; (Almost)-(para)-Hermitian; Pseudo-Riemannian; 53B35; (Almost)-(para)-complex; Affine; (Para)-Kähler–Weyl; (Para)-Kähler
• ISSN: 0926-2245; 1872-6984
• DOI: 10.1016/j.difgeo.2013.03.006

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.