Suppose we have a vector field ##V## defined everywhere on a manifold ##M##. Consider now point ##p \in M##. As a consequence of the existence and uniqueness theorem of differential equations. this implies that ##V## gives rise to a unique local flow(adsbygoogle = window.adsbygoogle || []).push({});

$$\theta:(-\epsilon,\epsilon) \times U \to M$$

for ##(-\epsilon,\epsilon) \in \mathbb{R}## and ##p \in U## where ##U## is an open subset of M.

Now if ##\theta_t(p) = \theta(t,p)##, and if ##V## is a killing field, then ##\theta_t: U \to M## should be isometries. But do they belong to isometry group ##Isom(M)##? I.e. are they global as a consequence of ##V## being globally defined and a killing field?

Is there not a bijective correspondence between global isometries and globally defined Killing vector fields?

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# Killing vector field => global isomorphisms?

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