An extension of a theorem of Nagano on transitive Lie algebras

Author:
Héctor J. Sussmann

Journal:
Proc. Amer. Math. Soc. **45** (1974), 349-356

MSC:
Primary 58A15; Secondary 57D25

DOI:
https://doi.org/10.1090/S0002-9939-1974-0356116-6

MathSciNet review:
0356116

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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 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 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 and compact.

**[1]**Tadashi Nagano,*Linear differential systems with singularities and an application to transitive Lie algebras*, J. Math. Soc. Japan**18**(1966), 398–404. MR**0199865**, https://doi.org/10.2969/jmsj/01840398**[2]**K. Shiga,*Some aspects of real-analytic manifolds and differentiable manifolds*, J. Math. Soc. Japan**16**(1964), 128–142. MR**0172298**, https://doi.org/10.2969/jmsj/01620128

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0356116-6

Article copyright:
© Copyright 1974
American Mathematical Society