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ianhoolihan

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I've been wondering about this for some time. While I am only familiar with the basics of differential geometry, I have come across the Lie bracket commutator in a few places.

Firstly, what is the intuitive explanation of the Lie bracket [X,Y] of two vectors, if there is one? In terms of vectors being differential linear operators along specific curves?

Another specific example I came across is the invariant formula for the exterior derivative: http://en.wikipedia.org/wiki/Exterior_derivative#Invariant_formula

Is the Lie bracket here just a useful notation, or something deeper? I'm assuming deeper, so could someone explain why? What does it mean to have a different algebra (structure constants)? Do they relate to the curvature of the space?

Any other general comments on how the Lie bracket is related to differential geometry would be much appreciated!

Ianhoolihan