Spheres and circles with respect to an indefinite metric on a Riemanian manifold with a skew-circulant structure

TL;DR

This paper investigates the geometry of spheres and circles within a 4-dimensional manifold equipped with an indefinite metric and a skew-circulant tensor structure, revealing their properties under these complex geometric conditions.

Contribution

It introduces a novel study of geometric objects in a manifold with an indefinite metric and a skew-circulant tensor, expanding understanding of such structures in differential geometry.

Findings

Characterization of spheres and circles in the indefinite metric context

Analysis of the compatibility between the metric and tensor structure

Description of geometric properties influenced by the skew-circulant structure

Abstract

We study hyper-spheres, spheres and circles, with respect to an indefinite metric, in a tangent space on a 4-dimensional differentiable manifold. The manifold is equipped with a positive definite metric and an additional tensor structure of type (1, 1)11. The fourth power of the additional structure is minus identity and its components form a skew-circulant matrix in some local coordinate system. The both structures are compatible and they determine an associated indefinite metric on the manifold.