Motivation
Important to applications in mathematics and physics is the notion of a flow on a manifold. In particular, if is a smooth manifold and is a smooth vector field, one is interested in finding integral curves to . More precisely, given one is interested in curves such that
for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If is furthermore a complete vector field, then the flow of, defined as the collection of all integral curves for, is a diffeomorphism of . The flow given by is in fact an action of the additive Lie group on .
Conversely, every smooth action defines a complete vector field via the equation
It is then a simple result that there is a bijective correspondence between actions on and complete vector fields on .
In the language of flow theory, the vector field is called the infinitesimal generator. Intuitively, the behaviour of the flow at each point corresponds to the "direction" indicated by the vector field. It is a natural question to ask whether one may establish a similar correspondence between vector fields and more arbitrary Lie group actions on
Read more about this topic: Fundamental Vector Field
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