On a Class of Special Riemannian Manifolds

TL;DR

This paper studies a special class of four-dimensional Riemannian manifolds with circulant metric and affinor structure, deriving conditions for their geometric properties and symmetries.

Contribution

It characterizes conditions under which the affinor structure is covariantly constant and explores the implications for the manifold's geometry.

Findings

q^4 = id, indicating a cyclic structure

g(qx, qy) = g(x, y), showing invariance under q

Conditions for nabla q = 0 involving gradients of A and C

Abstract

We consider a four dimensional Riemannian manifold M with a metric g and an affinor structure q. We note the local coordinates of g and q are circulant matrices. Their first orders are (A, B, C, B)(A, B, C are smooth functions on M) and (0, 1, 0, 0), respectively. Let nabla be the connection of g. Then we obtain: 1) q^{4}=id; g(qx, qy)=g(x,y), x, y are arbitrary vector fields on M, 2) nabla q =0 if and only if grad A=(grad C)q^{2}; 2.grad B= (grad C)(q+q^{3}),