A Riemannian manifold with skew-circulant structures and an associated locally conformal Kähler manifold

Abstract

A 4-dimensional Riemannian manifold M , equipped with an additional tensor structure S, whose fourth power is minus identity, is considered. The structure S has a skew-circulant matrix with respect to some basis of the tangent space at a point on M . Moreover, S acts as an isometry with respect to the metric g. A fundamental tensor is defined on such a manifold (M, g, S) by g and by the covariant derivative of S. This tensor satisfies a characteristic identity which is invariant to the usual conformal transformation. Some curvature properties of (M, g, S) are obtained. A Lie group as a manifold of the considered type is constructed. A Hermitian manifold associated with (M, g, S) is also considered. It turns out that it is a locally conformal Kähler manifold.