On a Riemannian manifold with a circulant structure whose third power is the identity

Abstract

We study a 3-dimensional Riemannian manifold equipped with a circulant tensor structure of type (1,1), whose third power is the identity. This structure and the metric are compatible such that an isometry is induced in any tangent space at an arbitrary point of the manifold. On such a manifold we define a fundamental tensor by the metric and by the covariant derivative of the circulant tensor structure. We obtain an important characteristic identity for this tensor. We establish that the image of the fundamental tensor with respect to the usual conformal transformation satisfies the same identity, i.e. the conformal manifold is of the same class.
We construct a Lie group as a manifold of the considered type and we find some of its geometrical characteristics.