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.
Important to applications in mathematics and physics[1] is the notion of a flow on a manifold.
is a smooth vector field, one is interested in finding integral curves to
such that: for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations.
is furthermore a complete vector field, then the flow of
, defined as the collection of all integral curves for
is in fact an action of the additive Lie group
Conversely, every smooth action
defines a complete vector field
via the equation: It is then a simple result[2] that there is a bijective correspondence between
and complete vector fields on
In the language of flow theory, the vector field
is called the infinitesimal generator.
[3] 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
, the fundamental vector field
can then be shown to be a Lie algebra homomorphism.
may be identified with either the left- or right-invariant vector fields on
It is a well-known result[3] that such vector fields are isomorphic to
, the tangent space at identity.
act on itself via right-multiplication, the corresponding fundamental vector fields are precisely the left-invariant vector fields.
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.
is a Hamiltonian vector field if there exists a smooth function
This motivates the definition of a Hamiltonian group action as follows: If
is a Hamiltonian group action if there exists a moment map
is the fundamental vector field of