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 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 .

Definition
Let  be a Lie group with corresponding Lie algebra . Furthermore, let  be a smooth manifold endowed with a smooth action . Denote the map  such that , called the orbit map of  corresponding to . For , the fundamental vector field  corresponding to  is any of the following equivalent definitions:

where  is the differential of a smooth map and  is the zero vector in the vector space .

The map  can then be shown to be a Lie algebra homomorphism.

Applications

Lie groups
The Lie algebra of a Lie group  may be identified with either the left- or right-invariant vector fields on . It is a well-known result that such vector fields are isomorphic to , the tangent space at identity. In fact, if we let  act on itself via right-multiplication, the corresponding fundamental vector fields are precisely the left-invariant vector fields.

Hamiltonian group actions

In the motivation, it was shown that there is a bijective correspondence between smooth  actions and complete vector fields. Similarly, there is a bijective correspondence between symplectic actions (the induced diffeomorphisms are all symplectomorphisms) and complete symplectic vector fields.

A closely related idea is that of Hamiltonian vector fields. Given a symplectic manifold , we say that  is a Hamiltonian vector field if there exists a smooth function  satisfying:

where the map  is the interior product. This motivatives the definition of a Hamiltonian group action as follows: If  is a Lie group with Lie algebra  and  is a group action of  on a smooth manifold , then we say that  is a Hamiltonian group action if there exists a moment map  such that for each: , 
 
where  and  is the fundamental vector field of

References

Lie groups
Symplectic geometry
Hamiltonian mechanics
Smooth manifolds