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. 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. 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 pin M} one is interested in curves γ p : R → M {displaystyle gamma _{p}:mathbb {R} o M} such that 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} imes M o 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} imes M o M} defines a complete vector field X {displaystyle X} via the equation 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} . 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 imes M o M} . Denote the map A p : G → M {displaystyle A_{p}:G o 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 Xin {mathfrak {g}}} , the fundamental vector field X # {displaystyle X^{#}} corresponding to X {displaystyle X} is any of the following equivalent definitions: where d {displaystyle d} is the differential of a smooth map and 0 T p M {displaystyle 0_{T_{p}M}} is the zero vector in the vector space T p M {displaystyle T_{p}M} . The map g → Γ ( T M ) , X ↦ X # {displaystyle {mathfrak {g}} o Gamma (TM),Xmapsto X^{#}} can then be shown to be a Lie algebra homomorphism.