Killing field

In military science,killing field isfieldfire, perhaps covered by machine guns; and/or,regionwhich artillery, cannon, and/or mortars have been registered. Suchterm may be useddescribeapproachesan "ideal" defensive fortification.

In mathematics,Killing field, named after Wilhelm Killing, isvector field that preservesRiemannian metric. In other words,flow diffeomorphisms act as isometries. A Killing fielddetermined uniquely byvector at some pointits gradient. Killing fields formLie algebradimension not greater than ((n + 1)n)/2.

On compact manifoldsnegative Ricci curvature thereno nontrivial (nonzero) Killing fields. Nonnegative Ricci curvature implies thatfieldparallel oncompact manifold. If Ricci curvaturepositive, than Killing field must havezero oncompact manifold.