Interpolation of geometric structures compatible with a pseudo Riemannian metric

TL;DR

This paper introduces a method to interpolate between various geometric structures compatible with a pseudo Riemannian metric, expanding the understanding of their relationships and providing explicit examples and fiber computations.

Contribution

It defines a new notion of interpolation for compatible geometric structures using generalized (para)complex structures and computes fiber structures for these interpolations.

Findings

Defined interpolation of compatible structures using generalized (para)complex structures

Computed fibers of twistor bundles for the new structures

Provided examples on Lie groups with invariant metrics

Abstract

Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a product structure and symmetric with respect to g, then r induces a pseudo Riemannian product structure on M. Sometimes the integrability condition is expressed by the closedness of an associated two-form: if j is almost complex on M and {\omega}(x, y) = g(jx, y) is symplectic, then M is almost pseudo K\"ahler. Now, product, complex and symplectic structures on M are trivial examples of generalized (para)complex structures in the sense of Hitchin. We use the latter in order to define the notion of interpolation of geometric structures compatible with g. We also compute the typical fibers of the twistor bundles of the new structures and give examples for M a Lie group with a left invariant metric.