Clifford vectors
Curtis M. Fulton · Pacific Journal of Mathematics · 1964
In this paper we present a generalization of parallel vector fields in a Riemannian space.As it turns out, such fields exist in spaces of constant positive curvature.Restricting ourselves to a Riemannian 3-space throughout, we need the oriented third-order tensor [3, p. 249]whose covariant derivative vanishes [3, pp.251-252].The latter fact is best ascertained by the use of geodesic coordinates.If we write the determinant of the metric tensor with the aid of permutation symbols we also find without difficulty DEFINITION.Let the direction of a vector field at any point be that of the unit vector V.The field is said to consist of Clifford vectors ifTHEOREM.If the Riemannian curvature K is constant and equal to L 2 , the system of equations (2) is completely integrable.If, at any point, solutions of (2) exist in all directions, then K -L 2 -const.