This generalizes to the Lie derivative of any tensor field along the flow generated by
The Lie bracket is an R-bilinear operation and turns the set of all smooth vector fields on the manifold
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.
[1] V. I. Arnold refers to this as the "fisherman derivative", as one can imagine being a fisherman, holding a fishing rod, sitting in a boat.
Both the boat and the float are flowing according to vector field
, and the fisherman lengthens/shrinks and turns the fishing rod according to vector field
The Lie bracket is the amount of dragging on the fishing float relative to the surrounding water.
[2] There are three conceptually different but equivalent approaches to defining the Lie bracket: Each smooth vector field
may be regarded as a differential operator acting on smooth functions
arises from a unique smooth vector field
This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation: Let
denote the tangent map derivative operator.
can be defined as the Lie derivative: This also measures the failure of the flow in the successive directions
: Though the above definitions of Lie bracket are intrinsic (independent of the choice of coordinates on the manifold
), in practice one often wants to compute the bracket in terms of a specific coordinate system
for the associated local basis of the tangent bundle, so that general vector fields can be written
can be written as smooth maps of the form
The Lie bracket of vector fields equips the real vector space
(i.e., smooth sections of the tangent bundle
) with the structure of a Lie algebra, which means [ • , • ] is a map
Furthermore, there is a "product rule" for Lie brackets.
means that following the flows in these directions defines a surface embedded in
as coordinate vector fields: Theorem:
This is a special case of the Frobenius integrability theorem.
The Lie bracket of two left invariant vector fields is also left invariant, which defines the Jacobi–Lie bracket operation
For a matrix Lie group, whose elements are matrices
, each tangent space can be represented as matrices:
The invariant vector field corresponding to
, and a computation shows the Lie bracket on
corresponds to the usual commutator of matrices: As mentioned above, the Lie derivative can be seen as a generalization of the Lie bracket.