local coordinates where denotes the parameter along the integral curve of based at on . In these coordinates, the Lie derivative reduces to the coordinate derivative, that is, and by the definition of the Killing field the left-hand side vanishes. Properties A Killing field is determined uniquely by a vector at some point and its gradient (i.e. all covariant derivatives of the field at the point). The Lie bracket of two Killing fields is still a Killing field. The Killing fields on a manifold M thus form a Lie subalgebra of vector fields on M. This is the Lie algebra of the isometry group of the manifold if M is complete. A Riemannian manifold with a transitive group of isometries is a homogeneous space. For compact manifolds Negative Ricci curvature implies there are no nontrivial (nonzero) Killing fields. Nonpositive Ricci curvature implies that any Killing field is parallel. i.e. covariant derivative along any vector field is identically zero. If the sectional curvature is positive and the dimension of M is even, a Killing field must have a zero. The covariant divergence of every Killing vector field vanishes. If is a Killing vector field and is a harmonic vector field, then is a harmonic function. If is a Killing vector field and is a harmonic p-form, then Geodesics Each Killing vector corresponds to a quantity which is conserved along geodesics. This conserved quantity is the metric product between the Killing vector and the geodesic tangent vector. Along an affinely parametrized geodesic with tangent vector then given the Killing vector , the quantity is conserved: This aids in analytically studying motions in a spacetime with symmetries. Stress-energy tensor Given a conserved, symmetric tensor , that is, one satisfying and , which are properties typical of a stress-energy tensor, and a Killing vector , we can construct the conserved quantity satisfying Cartan decomposition As noted above, the Lie bracket of two Killing fields is still a Killing field. The Killing fields on a manifold thus form a Lie subalgebra of all vector fields on Selecting a point the algebra can be decomposed into two parts: and where is the covariant derivative. These two parts intersect trivially but do not in general split . For instance, if is a Riemannian homogeneous space, we have if and only if is a Riemannian symmetric space. Intuitively, the isometries of locally define a submanifold of the total