On a Class of Special Riemannian Manifolds

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}),