An Extension of a Theorem of Nagano on Transitive Lie Algebras

Abstract

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