Resumen de A necessary and sufficient condition for a vector field to be Killing
Lynge (see (4)), proved that X vector field on a compact and connected manifold M, is Killing in a Riemannian metric g, if and only if there exist k complete vector fields X1,....., Xk; such that: 1) the flows of the Xi are all periodic, 2) [Xi,Xj] = 0, i,j =1, ...,k, 3) X = Sik1 ci Xi, ci real numbers.
We study a generalisation of this result in no-compact manifolds, obtaining: If X is a complete vector field on M, oi its uniparametric group; then X is Killing in a Riemannian metric g, if and only if: a) oi acts freely and properly, or b) there exist k complete vector fields in Lynge-conditions. We use some results of proper actions, contained in (3) and (4).
We generalise this result to proper and free actions of Lie groups. Finally we study the Riemannian metrics, in which a vector field in a)-conditions can be Killing.
