Lie bracket of vector fields

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 .
Conceptually, the Lie bracket is the derivative of "Y" along the flow generated by "X", and is sometimes denoted "formula_1" ("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.
Definitions.
There are three conceptually different but equivalent approaches to defining the Lie bracket:
Vector fields as derivations.
Each smooth vector field formula_2 on a manifold "M" may be regarded as a differential operator acting on smooth functions formula_3 (where formula_4 and formula_5 of class formula_6) when we define formula_7 to be another function whose value at a point formula_8 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 formula_9 of any two derivations formula_10 and formula_11 is again a derivation, where formula_12 denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:
Flows and limits.
Let formula_14 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 can be defined as the Lie derivative:
This also measures the failure of the flow in the successive directions formula_16 to return to the point "x":
In coordinates.
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 formula_18. We write formula_19 for the associated local basis of the tangent bundle, so that general vector fields can be written formula_20and formula_21for smooth functions formula_22. Then the Lie bracket can be computed as:
If "M" is (an open subset of) R"n", then the vector fields "X" and "Y" can be written as smooth maps of the form formula_24 and formula_25, and the Lie bracket formula_26 is given by:
where formula_28 and formula_29 are Jacobian matrices (formula_30 and formula_31 respectively using index notation) multiplying the column vectors "X" and "Y".
Properties.
The Lie bracket of vector fields equips the real vector space formula_32 of all vector fields on "M" (i.e., smooth sections of the tangent bundle formula_33) with the structure of a Lie algebra, which means [ • , • ] is a map formula_34 with:
An immediate consequence of the second property is that formula_37 for any formula_38.
Furthermore, there is a "product rule" for Lie brackets. Given a smooth (scalar-valued) function "f" on "M" and a vector field "Y" on "M", we get a new vector field "fY" by multiplying the vector "Yx" by the scalar "f"("x") at each point . Then:
where we multiply the scalar function "X"("f") with the vector field "Y", and the scalar function "f" with the vector field .
This turns the vector fields with the Lie bracket into a Lie algebroid.
Vanishing of the Lie bracket of "X" and "Y" means that following the flows in these directions defines a surface embedded in "M", with "X" and "Y" as coordinate vector fields:
Theorem: formula_40 iff the flows of "X" and "Y" commute locally, meaning formula_41 for all and sufficiently small "s", "t".
This is a special case of the Frobenius integrability theorem.
Examples.
For a Lie group "G", the corresponding Lie algebra formula_42 is the tangent space at the identity formula_43, which can be identified with the vector space of left invariant vector fields on "G". The Lie bracket of two left invariant vector fields is also left invariant, which defines the Jacobi–Lie bracket operation formula_44.
For a matrix Lie group, whose elements are matrices formula_45, each tangent space can be represented as matrices: formula_46, where formula_47 means matrix multiplication and "I" is the identity matrix. The invariant vector field corresponding to formula_48is given by formula_49, and a computation shows the Lie bracket on formula_50 corresponds to the usual commutator of matrices:
Applications.
The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.
Generalizations.
As mentioned above, the Lie derivative can be seen as a generalization of the Lie bracket. Another generalization of the Lie bracket (to vector-valued differential forms) is the Frölicher–Nijenhuis bracket.