In the study of mathematics, and especially of differential geometry, fundamental vector fields are instruments that describe 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 M {\displaystyle M} is a smooth manifold and X {\displaystyle X} is a smooth vector field, one is interested in finding integral curves to X {\displaystyle X} . More precisely, given p ∈ M {\displaystyle p\in M} one is interested in curves γ p : R → M {\displaystyle \gamma _{p}:\mathbb {R} \to M} such that:
γ p ′ ( t ) = X γ p ( t ) , γ p ( 0 ) = p , {\displaystyle \gamma _{p}'(t)=X_{\gamma _{p}(t)},\qquad \gamma _{p}(0)=p,}
for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If X {\displaystyle X} is furthermore a complete vector field, then the flow of X {\displaystyle X} , defined as the collection of all integral curves for X {\displaystyle X} , is a diffeomorphism of M {\displaystyle M} . The flow ϕ X : R × M → M {\displaystyle \phi _{X}:\mathbb {R} \times M\to M} given by ϕ X ( t , p ) = γ p ( t ) {\displaystyle \phi _{X}(t,p)=\gamma _{p}(t)} is in fact an action of the additive Lie group ( R , + ) {\displaystyle (\mathbb {R} ,+)} on M {\displaystyle M} . Conversely, every smooth action A : R × M → M {\displaystyle A:\mathbb {R} \times M\to M} defines a complete vector field X {\displaystyle X} via the equation:
X p = d d t | t = 0 A ( t , p ) . {\displaystyle X_{p}=\left.{\frac {d}{dt}}\right|_{t=0}A(t,p).}
It is then a simple result that there is a bijective correspondence between R {\displaystyle \mathbb {R} } -actions on M {\displaystyle M} and complete vector fields on M {\displaystyle M} . In the language of flow theory, the vector field X {\displaystyle X} 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 M {\displaystyle M} .
Definition Let G {\displaystyle G} be a Lie group with corresponding Lie algebra g {\displaystyle {\mathfrak {g}}} . Furthermore, let M {\displaystyle M} be a smooth manifold endowed with a smooth action A : G × M → M {\displaystyle A:G\times M\to M} . Denote the map A p : G → M {\displaystyle A_{p}:G\to M} such that A p ( g ) = A ( g , p ) {\displaystyle A_{p}(g)=A(g,p)} , called the orbit map of A {\displaystyle A} corresponding to p {\displaystyle p} . For X ∈ g {\displaystyle X\in {\mathfrak {g}}} , the fundamental vector field X # {\displaystyle X^{\#}} corresponding to X {\displaystyle X} is given by any of the following equivalent definitions:
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