Fundamental vector field

In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes 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.

Important to applications in mathematics and physics is the notion of a flow on a manifold. In particular, if ${\displaystyle M}$ is a smooth manifold and ${\displaystyle X}$ is a smooth vector field, one is interested in finding integral curves to ${\displaystyle X}$. More precisely, given ${\displaystyle p\in M}$ one is interested in curves ${\displaystyle \gamma _{p}:\mathbb {R} \to M}$ such that

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