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Concept
Lie bracket of vector fields
Summary
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted [X, Y].
Conceptually, the Lie bracket [X, Y] is the derivative of Y along the flow generated by X, and is sometimes denoted ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by X.
The Lie bracket is an R-bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra.
The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems.
There are three conceptually different but equivalent approaches to defining the Lie bracket:
Each smooth vector field on a manifold M may be regarded as a differential operator acting on smooth functions (where and of class ) when we define to be another function whose value at a point is the directional derivative of f at p in the direction X(p). In this way, each smooth vector field X becomes a derivation on C∞(M). Furthermore, any derivation on C∞(M) arises from a unique smooth vector field X.
In general, the commutator of any two derivations and is again a derivation, where denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:
Let be the flow associated with the vector field X, and let D denote the tangent map derivative operator. Then the Lie bracket of X and Y at the point x ∈ M can be defined as the Lie derivative:
This also measures the failure of the flow in the successive directions to return to the point x:
Though the above definitions of Lie bracket are intrinsic (independent of the choice of coordinates on the manifold M), in practice one often wants to compute the bracket in terms of a specific coordinate system .
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