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On page 3 of the book before exercise 1.2, is written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free condition results compatibility with differentiable structure? Please help me understand it.

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"Torsion-free" for a connection is an equation where one side, the Lie bracket of two vector fields, depends only on the differentiable structure of the manifold (and not the Riemannian metric) and the other side is defined in terms of the connection. So it is a condition that says that the connection is "compatible" with the differentiable structure. – Deane Yang Apr 4 '13 at 13:22
This is a cross-post:… – Ryan Budney Apr 4 '13 at 16:15

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