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We consider a $3$-dimensional differentiable manifold with two circulantstructures -- a Riemannian metric and an additional structure, whose thirdpower is the identity. The structure is compatible with the metric such that anisometry is induced in any tangent space of the manifold. Further, we consideran 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 theassociated metric, with respect to the Riemannian metric.

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