Hi,
I would like to know why the Killings fields on on a complete riemannian manifold are themselves complete (that is, the integral curves of the Killing fields are defined for all time).
Thanks
Hi, I would like to know why the Killings fields on on a complete riemannian manifold are themselves complete (that is, the integral curves of the Killing fields are defined for all time). Thanks 


The corresponding flow, say $\Phi^t: M\to M$ preserves the metric and the field. Thus, for any $x\in M$, the curve $\alpha_x\colon t\mapsto \Phi^t(x)$ has constant speed. Therefore it can not escape to infinity in finite time. More precisely: if $\alpha_x$ is defined on a bounded interval $(a,b)$ then the restriction $\alpha_x(a,b)$ has finite length, and from completeness it can be extended to a neighborhood of $[a,b]$. This implies that $\alpha_x$ is defined on whole $\mathbb R$; i.e., the vector field is complete. 

