Spheres and circles with respect to an indefinite metric on a Riemannian manifold with circulant structures

TL;DR

This paper studies spheres and circles on a 3D manifold with circulant structures, exploring their equations under an indefinite metric related to a compatible Riemannian metric.

Contribution

It introduces equations of spheres and circles in a manifold with circulant structures and an indefinite metric, extending geometric analysis in this setting.

Findings

Derived equations of spheres and circles using the associated indefinite metric

Established compatibility conditions between the Riemannian and associated metrics

Analyzed geometric properties of these curves in the circulant structure context

Abstract

We consider a $3$-dimensional differentiable manifold with two circulant structures -- a Riemannian metric and an additional structure, whose third power is the identity. The structure is compatible with the metric such that an isometry is induced in any tangent space of the manifold. Further, we consider an associated metric with the Riemannian metric, which is necessary indefinite. We find equations of a sphere and of a circle, which are given in terms of the associated metric, with respect to the Riemannian metric.