nLab fundamental vector field

Definitions

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}\cong T_e G$, $M$ a $C^1$-differentiable manifold and

$$\nu\colon G \times M\to M$$

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

$$(\chi_A)_m = (T_{1_G}\nu_m)(A),\,\,\,\,m\in M,$$

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

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

There is a dual notion as well. Given a right $C^1$-differentiable $G$-manifold $E$, a $C^1$-differentiable $1$-form $\omega$ with values in $\mathfrak{g}$,

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

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

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”).