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 is a smooth manifold is a smooth vector field, one is interested in finding integral curves such that:

for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations is furthermore a complete vector field, is a diffeomorphism is in fact an action of the additive Lie group.

via the equation:

It is then a simple result.

is called the infinitesimal generator..

Definition

be a Lie group with corresponding Lie algebra be a smooth manifold endowed with a smooth action, called the . is given by any of the following equivalent definitions:

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

can then be shown to be a Lie algebra homomorphism.

Applications

Lie groups

. It is a well-known result act on itself via right-multiplication, the corresponding fundamental vector fields are precisely the left-invariant vector fields.

Hamiltonian group actions

In the motivation-actions and complete vector fields. Similarly, given a symplectic manifold, 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 is a Hamiltonian vector field if there exists a smooth function satisfying

is the interior product. This motivates the definition of a Hamiltonian group action is a Hamiltonian group action if there exists a moment map,

.

References