Parallel vector fields
Curtis M. Fulton · Proceedings of the American Mathematical Society · 1965
We speak of a parallel vector field if parallel transport is independent of the path of propagation [1, p. 239]. Riemannian spaces which are flat offer a simple example for the existence of such fields [1, p. 239]. We wish to generalize the concept of parallel displacement of a vector in a Riemannian space. In our case, spaces of constant negative curvature are the most conspicuous to admit parallel vector fields. DEFINITION. Let the direction of a vector field at any point be that of the unit vector V. The field is said to be parallel if