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Fundamental vector field : ウィキペディア英語版
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.
==Motivation==

Important to applications in mathematics and physics is the notion of a flow on a manifold. In particular, if M is a smooth manifold and X is a smooth vector field, one is interested in finding integral curves to X . More precisely, given p \in M one is interested in curves \gamma_p: \mathbb R \to M such that
: \gamma_p'(t) = X_, \qquad \gamma_p(0) = p,
for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If X is furthermore a complete vector field, then the flow of X , defined as the collection of all integral curves for X , is a diffeomorphism of M. The flow \phi_X: \mathbb R \times M \to M given by \phi_X(t,p) = \gamma_p(t) is in fact an action of the additive Lie group (\mathbb R,+) on M.
Conversely, every smooth action A:\mathbb R \times M \to M defines a complete vector field X via the equation
: X_p = \left.\frac\right|_ A(t,p).
It is then a simple result that there is a bijective correspondence between \mathbb R actions on M and complete vector fields on M .
In the language of flow theory, the vector field X is called the ''infinitesimal generator''. Intuitively, the behaviour of the flow at each point corresponds to the "direction" indicated by the vector field. It is a natural question to ask whether one may establish a similar correspondence between vector fields and more arbitrary Lie group actions on M .

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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