Curvature properties of Riemannian manifolds with skew-circulant structures

TL;DR

This paper studies 4-dimensional Riemannian manifolds with a skew-circulant tensor structure, analyzing their curvature properties, Ricci tensor characteristics, and sectional curvatures for special 2-planes, revealing new geometric invariants.

Contribution

It characterizes manifolds with skew-circulant structures where curvature tensors are invariant under the structure, providing explicit curvature and Ricci tensor properties.

Findings

Curvature tensors invariant under the skew-circulant structure.

Explicit expressions for sectional curvatures of special 2-planes.

Properties of the Ricci tensor in these manifolds.

Abstract

We consider a 4-dimensional Riemannian manifold M endowed with a right skew-circulant tensor structure S, which is an isometry with respect to the metric g and the fourth power of S is minus identity. We determine a class of manifolds (M, g, S), whose curvature tensors are invariant under S. For such manifolds we obtain properties of the Ricci tensor. Also we get expressions of the sectional curvatures of some special 2-planes in a tangent space of (M, g, S).