Common
Lie bracket of vector fields
Operator in differential topology
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 {\displaystyle X} and Y {\displaystyle Y} on a smooth manifold M {\displaystyle M} a third vector field denoted [ X , Y ] {\displaystyle [X,Y]} . Conceptually, the Lie bracket [ X , Y ] {\displaystyle [X,Y]} is the derivative of Y {\displaystyle Y} along the flow generated by X {\displaystyle X} , and is sometimes denoted L...
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