Killing field

In military science, a killing field is a field of fire, perhaps covered by machine guns; and/or, a region in which artillery, cannon, and/or mortars have been registered. Such a term may be used to describe the approaches to an "ideal" defensive fortification.

In mathematics, a Killing field, named after Wilhelm Killing, is a vector field that preserves the Riemannian metric. In other words, the flow diffeomorphisms act as isometries. A Killing field is determined uniquely by a vector at some point and its gradient. Killing fields form a Lie algebra of dimension not greater than ((n + 1)n)/2.

On compact manifolds with negative Ricci curvature there are no nontrivial (nonzero) Killing fields. Nonnegative Ricci curvature implies that the field is parallel on a compact manifold. If Ricci curvature is positive, than Killing field must have a zero on a compact manifold.