Fundamental vector field explained
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:
\gammap'(t)=
, \gammap(0)=p,
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:
Xp=\left.
\right|t=0A(t,p).
It is then a simple result
[1] 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.
[2] 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
.
[3] For
, the fundamental vector field
corresponding to
is any of the following equivalent definitions:
[1] [3] [4]
=\left.
\right|t=0A\left(\exp(tX),p\right)
where
is the
differential of a smooth map and
is the
zero vector in the
vector space
.
The map
akg\to\Gamma(TM),X\mapsto-X\#
can then be shown to be a Lie algebra homomorphism.
[4] 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
[2] 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
\muX:M\toR,p\mapsto\langle\mu(p),X\rangle
and
is the fundamental vector field of
Notes and References
- Book: Ana Cannas da Silva . Ana Cannas da Silva . Lectures on Symplectic Geometry . Springer . 978-3540421955. 2008.
- Book: Lee . John . Introduction to Smooth Manifolds . Springer . 0-387-95448-1 . 2003.
- Book: Audin . Michèle . Torus Actions on Symplectic manifolds . Birkhäuser . 3-7643-2176-8 . 2004.
- Book: Libermann . Paulette . Paulette Libermann . Marle . Charles-Michel . Symplectic Geometry and Analytical Mechanics . Springer . 978-9027724380 . 1987 . registration .