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What is torsion in differential geometry intuitively?

Then $d_\nabla$ is the exterior derivative if and only if $\nabla$ is torsion free. Moreover $d_\nabla^2=0$ if and only if $\nabla$ is torsion free. … .$ The torsion of a connection is the obstruction to the induced calculus of the connection to be the usual/natural calculus on a manifold. …
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