Smooth actions of connected compact Lie groups with a free point are determined by two vector fields

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Abstract: Consider a smooth action

m a t h b f G i m e s M i g h t a r r o w M

of a compact connected Lie group

m a t h b f G

on a connected manifold

M

. Assume the existence of a point of

M

whose isotropy group has a single element (free point). Then we prove that there exist two complete vector field

X, X_{1}

such that their group of automorphisms equals

m a t h b f G

regarded as a group of diffeomorphisms of

M

(the existence of a free point implies that the action of

m a t h b f G

is effective). Besides, some examples of effective actions with no free point where this result fails are exhibited.

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