{{Short description|Instrument in differential geometry}} {{Use British English|date=June 2025}} 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<ref name="HouBook">{{Citation | last1=Hou | first1=Bo-Yu | title=Differential Geometry for Physicists | journal=Advanced Series on Theoretical Physical Science | publisher=World Scientific Publishing Company | isbn=978-9810231057 | year=1997| volume=6 | doi=10.1142/3448 | bibcode=1997ASTPS...6.....H }}</ref> is the notion of a flow on a manifold. In particular, if <math> M </math> is a smooth manifold and <math> X</math> is a smooth vector field, one is interested in finding integral curves to <math> X </math>. More precisely, given <math> p \in M </math> one is interested in curves <math> \gamma_p: \mathbb R \to M </math> such that: :<math> \gamma_p'(t) = X_{\gamma_p(t)}, \qquad \gamma_p(0) = p, </math> for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If <math> X </math> is furthermore a complete vector field, then the flow of <math> X </math>, defined as the collection of all integral curves for <math> X </math>, is a diffeomorphism of <math> M</math>. The flow <math> \phi_X: \mathbb R \times M \to M </math> given by <math> \phi_X(t,p) = \gamma_p(t) </math> is in fact an action of the additive Lie group <math> (\mathbb R,+) </math> on <math> M</math>.
Conversely, every smooth action <math> A:\mathbb R \times M \to M </math> defines a complete vector field <math> X </math> via the equation: : <math> X_p = \left.\frac{d}{dt}\right|_{t=0} A(t,p). </math> It is then a simple result<ref name="da Silva">{{Cite book | author = Ana Cannas da Silva | author-link = Ana Cannas da Silva | title=Lectures on Symplectic Geometry | publisher=Springer | isbn=978-3540421955| year=2008}}</ref> that there is a bijective correspondence between <math> \mathbb R </math>-actions on <math> M </math> and complete vector fields on <math> M </math>.
In the language of flow theory, the vector field <math> X </math> is called the ''infinitesimal generator''.<ref name="Lee">{{Cite book | last1 = Lee | first1 = John | title=Introduction to Smooth Manifolds | publisher=Springer | isbn= 0-387-95448-1 | year=2003}}</ref> 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 <math> M </math>.
==Definition== Let <math> G </math> be a Lie group with corresponding Lie algebra <math> \mathfrak g </math>. Furthermore, let <math> M </math> be a smooth manifold endowed with a smooth action <math> A : G \times M \to M </math>. Denote the map <math> A_p: G \to M </math> such that <math> A_p(g) = A(g,p) </math>, called the ''orbit map of <math> A</math> corresponding to <math> p </math>''.<ref name="Audin">{{Cite book | last1 = Audin | first1 = Michèle | title=Torus Actions on Symplectic manifolds | url = https://archive.org/details/springer_10.1007-978-3-0348-7960-6 | publisher=Birkhäuser | isbn= 3-7643-2176-8 | year=2004}}</ref> For <math> X \in \mathfrak g </math>, the fundamental vector field <math> X^\# </math> corresponding to <math> X </math> is given by any of the following equivalent definitions:<ref name="da Silva"/><ref name="Audin"/><ref name="Libermann">{{Cite book | last1 = Libermann | first1 = Paulette | author1-link = Paulette Libermann | last2 = Marle | first2 = Charles-Michel | title = Symplectic Geometry and Analytical Mechanics | publisher = Springer | isbn = 978-9027724380 | year = 1987 | url-access = registration | url = https://archive.org/details/symplecticgeomet0000libe }}</ref> *<math> X^\#_p = d_e A_p(X) </math> *<math> X^\#_p = d_{(e,p)}A\left(X,0_{T_p M}\right) </math> *<math> X^\#_p = \left. \frac{d}{dt} \right|_{t=0} A\left( \exp(tX), p \right)</math> where <math> d </math> is the differential of a smooth map and <math> 0_{T_pM} </math> is the zero vector in the vector space <math> T_p M</math>.
The map <math> \mathfrak g \to \Gamma(TM), X \mapsto -X^\# </math> can then be shown to be a Lie algebra homomorphism.<ref name="Libermann"/>
==Applications==
===Lie groups=== The Lie algebra of a Lie group <math> G </math> may be identified with either the left- or right-invariant vector fields on <math> G </math>. It is a well-known result<ref name="Lee"/> that such vector fields are isomorphic to <math> T_e G </math>, the tangent space at identity. In fact, if we let <math> G </math> 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 <math> \mathbb R </math>-actions and complete vector fields. Similarly, given a symplectic manifold <math> (M,\omega) </math>, 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 <math> (M,\omega) </math>, we say that <math> X_H</math> is a Hamiltonian vector field if there exists a smooth function <math> H: M \to \mathbb R </math> satisfying :<math> dH = \iota_{X_H}\omega </math> where the map <math> \iota </math> is the interior product. This motivates the definition of a ''Hamiltonian group action'' as follows: If <math> G </math> is a Lie group with Lie algebra <math> \mathfrak g </math> and <math> A: G\times M \to M </math> is a group action of <math> G </math> on a smooth manifold <math> M </math>, then we say that <math> A </math> is a Hamiltonian group action if there exists a moment map <math> \mu: M \to \mathfrak g^* </math> such that for each: <math> X \in \mathfrak g </math>, : <math> d\mu^X = \iota_{X^\#}\omega, </math> where <math> \mu^X:M \to \mathbb R, p \mapsto \langle \mu(p),X \rangle </math> and <math> X^\# </math> is the fundamental vector field of <math> X </math>.
==References== {{Reflist}}
Category:Lie groups Category:Symplectic geometry Category:Hamiltonian mechanics Category:Smooth manifolds