Why are the Killing fields on a complete Riemannian manifold themselves complete (that is, the integral curves of the Killing fields are defined for all time)?



Geodesic completeness and complete Killing fields
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.


