# Geodesic completeness and complete Killing fields

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

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I don't understand this question. What does it mean for a Killing field to exist for all time? A Killing field is a vector field on the manifold and has no dependence on time. – Deane Yang Apr 13 '11 at 22:08

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.
@Ken, I assume we have completeness. You might use Hopf–Rinow to show that "geodesic completeness" $\Leftrightarrow$ "completeness". – Anton Petrunin Apr 14 '11 at 0:50