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 . For, the fundamental vector field corresponding to is any of the following equivalent definitions:
where is the exterior derivative and is the zero vector in the vector space .
The map can then be shown to be a Lie algebra homomorphism.
Read more about this topic: Fundamental Vector Field
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