A Riemannian manifold with skew-circulant structures and an associated locally conformal K\"{a}hler manifold

TL;DR

This paper studies a 4-dimensional Riemannian manifold with a skew-circulant tensor structure, exploring its curvature properties and establishing its relation to locally conformal Kähler manifolds.

Contribution

It introduces a new class of manifolds with skew-circulant structures, analyzes their curvature, and links them to locally conformal Kähler geometry.

Findings

Defined a fundamental tensor invariant under conformal transformations

Derived curvature properties of the manifold

Constructed examples including a Lie group and associated 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 K\"{a}hler manifold.