Fundamental vector field

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In mathematics, a fundamental vector field is a vector field on a manifold which is induced by a group action.

[edit] Definition

Let M be a manifold and G a Lie group with left action l on M, $l : G \times M \to M$ . Now define

$\zeta : \mathfrak{g} \to \mathcal{X}(M), X \mapsto \zeta_X : M \to \mathrm{T}M, m \mapsto \zeta_X(m) := \mathrm{T}_{(e, m)}l(X, 0_m) = \mathrm{T}_e(l_m)X$

which maps from the Lie algebra of G to vector fields on M. The images of ζ are the fundamental vector fields. ζ is a linear map and commutes with the Lie bracket, so it is a Lie algebra homomorphism.