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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 7th 2023

    Created:

    Definition

    Given a smooth manifold XX, the Lie bracket of vector fields uu and vv can be defined in several ways.

    As commutator of derivations

    Since derivations of smooth functions are vector fields, we can identify uu and vv with the corresponding derivations C (X)C (X)C^\infty(X)\to C^\infty(X).

    Taking the commutator uvvuuv-vu of these derivations produces another derivation, which is denoted by [u,v][u,v], and which can be identified with a vector field on XX.

    As a Lie derivative

    Alternatively, we can set

    [u,v]= uv= vu,[u,v]=\mathcal{L}_u v=\mathcal{L}_v u,

    where \mathcal{L} denotes the Lie derivative of a vector field.

    Properties

    The real vector space of vector fields on XX equipped with the Lie bracket forms a Lie algebra.

    Related concepts

    v1, current