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fundamental vector field

Fundamental vector fields and differential forms

Definitions

Let GG be a Lie group with Lie algebra 𝔤T eG\mathfrak{g}\cong T_e G, MM a C 1C^1-differentiable manifold and

ν:G×MM \nu\colon G \times M\to M

a left C 1C^1-differentiable action. For every mMm \in M, denoted by ν m:GM\nu_m\colon G\to M the C 1C^1-differentiable map ν m:gν(g,m)\nu_m\colon g\mapsto \nu(g,m). If A𝔤A \in \mathfrak{g}, the C 1C^1-differentiable vector field on MM given by

(χ A) m=(T 1 Gν m)(A),mM, (\chi_A)_m = (T_{1_G}\nu_m)(A),\,\,\,\,m\in M,

sometimes also denoted A A^\sharp, is called the fundamental vector field corresponding to AA. If s:IGs\colon I\to G is a curve around s(0)=1 Gs(0) = 1_G representing AA, then (χ A) m(\chi_A)_m is represented by ts(t)mt \mapsto s(t) m.

Analogously, one can define the fundamental vector field for the right actions.

There is a dual notion as well. Given a right C 1C^1-differentiable GG-manifold EE, a C 1C^1-differentiable 11-form ω\omega with values in 𝔤\mathfrak{g},

ωΓ(T *E𝔤) \omega \in \Gamma (T^* E\otimes\mathfrak{g})

is called the fundamental differential form corresponding to AA, if for all pEp\in E, ω p(χ A)=A\omega_p(\chi_A) = A.

Applications

The fundamental vector fields are important in study of differentiable actions and particularly useful in the basic study of Ehresmann connections. Indeed, the connection form is a fundamental form (“1st Ehresmann condition”), and there is a characterization of those fundamental forms which are connection forms (“2nd Ehresmann condition”).

Revised on December 7, 2011 02:12:32 by Toby Bartels (64.89.53.157)