fields span the entire tangent space of a symmetric space. Equivalently, the curvature tensor is covariantly constant on locally symmetric spaces, and so these are locally parallelizable; this is the Cartan–Ambrose–Hicks theorem. Generalizations Killing vector fields can be generalized to conformal Killing vector fields defined by for some scalar The derivatives of one parameter families of conformal maps are conformal Killing fields. Killing tensor fields are symmetric tensor fields T such that the trace-free part of the symmetrization of vanishes. Examples of manifolds with Killing tensors include the rotating black hole and the FRW cosmology. Killing vector fields can also be defined on any manifold M (possibly without a metric) if we take any Lie group G acting on it instead of the group of isometries. In this broader sense, a Killing vector field is the pushforward of a right invariant vector field on G by the group action. If the group action is effective, then the space of the Killing vector fields is isomorphic to the Lie algebra of G. See also Affine vector field Curvature collineation Homothetic vector field Killing form Killing horizon Killing spinor Matter collineation Spacetime symmetries References Riemannian geometry