An extension of a theorem of Nagano on transitive Lie algebras

Abstract:

Let $M$ be a real analytic manifold, and let $L$ be a transitive Lie algebra of real analytic vector fields on $M$. A concept of completeness is introduced for such Lie algebras. Roughly speaking, $L$ is said to be complete if the integral trajectories of vector fields in $L$ are defined “as far as $L$ permits". Examples of situations where this assumption is satisfied: (i) $L$ = a transitive Lie algebra all of whose elements are complete vector fields, and (ii) $L$ = the set $V(M)$ of all real analytic vector fields on $M$. Our main result is: if $M,M’$ are connected manifolds, then every Lie algebra isomorphism $F:L \to L’$ between complete transitive Lie algebras of real analytic vector fields on $M,M’$ which carries the isotropy subalgebra ${L_m}$ of a point $m$ of $M$ to the isotropy subalgebra ${L_{m’}}$ of $m’\in M’$ is induced by a (unique) real analytic diffeomorphism $f:M \to M’$ such that $f(m) = m’$, provided that one of the following two conditions is satisfied: (l) $M$ and $M’$ are simply connected, or (2) the Lie algebras $L$ and $L’$ separate points. Nagano had proved this result for the case $L = V(M),L’ = V(M’),M$ and $M’$ compact.