Transplanting geometrical structures

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. MSC 2010: 53B20 and 53B35.

A curvature identity on a~6-dimensional Riemannian manifold and its applications

“…10) g kl g uv S(g ij g kl g uv ) = 48(m + 2)(m + 4)g ij , where A ij = R abcd;i R abcd;j and B ij = R ibcd;a R jbcd;a . Thus, from H Multiplying (5.11) by m, we have the following equation:…”

Affine projective Osserman structures

“…If (M, ∇ ) is an affine manifold, then (T P M, R P ) is an affine curvature model for any P ∈ M. Conversely, given an affine curvature model (V, A), then there exists a complete affine manifold (M, ∇ ) and a point P of M so that (V, A) is isomorphic to (T P M, R P ), i.e. every affine curvature model can be geometrically realized by a complete affine manifold (see Euh et al [2]).…”

“…Even in signature (2,2), the situation is far from clear although much progress has been made recently by Calviño-Louzao et al [6] in examining these questions and similar questions related to the skew-symmetric curvature operator and by Díaz-Ramos et al [14] in examining non-diagonalize Jacobi operators. Derdzinski [11] has examined questions concerning type III Jordan-Osserman metrics raised by Diaz-Ramos et al [12].…”

“…Conversely, given an affine curvature model (V, A), then there exists a complete affine manifold (M, ∇) and a point P of M so that (V, A) is isomorphic to (T P M, R P ), i.e. every affine curvature model can be geometrically realized by a complete affine manifold (see Y. Euh et al [2]). Let (V, A) be an affine curvature model.…”