Lie bracket of vector fields

Each smooth vector field ${\displaystyle X:M\rightarrow TM}$ on a manifold M may be regarded as a differential operator acting on smooth functions ${\displaystyle f(p)}$ (where ${\displaystyle p\in M}$ and ${\displaystyle f}$ of class ${\displaystyle C^{\infty }(M)}$) when we define X(f) to be another function whose value at a point ${\displaystyle p}$ 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 ${\displaystyle \delta _{1}\circ \delta _{2}-\delta _{2}\circ \delta _{1}}$ of any two derivations ${\displaystyle \delta _{1}}$ and ${\displaystyle \delta _{2}}$ is again a derivation, where ${\displaystyle \circ }$ denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:

${\displaystyle [X,Y](f)=X(Y(f))-Y(X(f))\;\;{\text{ for all }}f\in C^{\infty }(M).}$

Let ${\displaystyle \Phi _{t}^{X}}$ 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:

${\displaystyle [X,Y]_{x}\ =\ ({\mathcal {L}}_{X}Y)_{x}\$ :=\ \lim _{t\to 0}{\frac {(\mathrm {D} \Phi _{-t}^{X})Y_{\Phi _{t}^{X}(x)}\,-\,Y_{x}}{t}}\ =\ \left.{\tfrac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}(\mathrm {D} \Phi _{-t}^{X})Y_{\Phi _{t}^{X}(x)}.}

This also measures the failure of the flow in the successive directions ${\displaystyle X,Y,-X,-Y}$ to return to the point x:

${\displaystyle [X,Y]_{x}\ =\ \left.{\tfrac {1}{2}}{\tfrac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}\right|_{t=0}(\Phi _{-t}^{Y}\circ \Phi _{-t}^{X}\circ \Phi _{t}^{Y}\circ \Phi _{t}^{X})(x)\ =\ \left.{\tfrac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}(\Phi _{\!-{\sqrt {t}}}^{Y}\circ \Phi _{\!-{\sqrt {t}}}^{X}\circ \Phi _{\!{\sqrt {t}}}^{Y}\circ \Phi _{\!{\sqrt {t}}}^{X})(x).}$

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 ${\displaystyle \{x^{i}\}}$. We write ${\displaystyle \partial _{i}={\tfrac {\partial }{\partial x^{i}}}}$ for the associated local basis of the tangent bundle, so that general vector fields can be written ${\displaystyle \textstyle X=\sum _{i=1}^{n}X^{i}\partial _{i}}$and ${\displaystyle \textstyle Y=\sum _{i=1}^{n}Y^{i}\partial _{i}}$for smooth functions ${\displaystyle X^{i},Y^{i}:M\to \mathbb {R} }$. Then the Lie bracket can be computed as:

${\displaystyle [X,Y]:=\sum _{i=1}^{n}\left(X(Y^{i})-Y(X^{i})\right)\partial _{i}=\sum _{i=1}^{n}\sum _{j=1}^{n}\left(X^{j}\partial _{j}Y^{i}-Y^{j}\partial _{j}X^{i}\right)\partial _{i}.}$

If M is (an open subset of) Rⁿ, then the vector fields X and Y can be written as smooth maps of the form ${\displaystyle X:M\to \mathbb {R} ^{n}}$ and ${\displaystyle Y:M\to \mathbb {R} ^{n}}$, and the Lie bracket ${\displaystyle [X,Y]:M\to \mathbb {R} ^{n}}$ is given by:

${\displaystyle [X,Y]:=J_{Y}X-J_{X}Y}$

where ${\displaystyle J_{Y}}$ and ${\displaystyle J_{X}}$ are n × n Jacobian matrices (${\displaystyle \partial _{j}Y^{i}}$ and ${\displaystyle \partial _{j}X^{i}}$ respectively using index notation) multiplying the n × 1 column vectors X and Y.