Spheres and circles in a tangent space of a 4-dimensional Riemannian manifold with respect to an indefinite metric

Abstract

Our study is in the tangent space at an arbitrary point on a 4-dimensional Riemannian manifold. The manifold is equipped with an additional tensor structure of type (1, 1), whose fourth power is the identity and the second power is an almost product structure. The metric and the additional structure are compatible, such that an isometry is induced in every tangent space. They determine an associated metric, which is necessarily indefinite. We study spheres and circles, which are given with respect to the associated metric, in some special subspaces of a single tangent space of the manifold.