Fundamental vector field
In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions. 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