A torsion-free connection is a type of connection on a differentiable manifold that allows for the parallel transport of vectors without introducing any twisting or rotation. This means that when you parallel transport a vector along a curve, it maintains its direction without being altered by the curvature of the manifold. This property is crucial for defining how derivatives of tensor fields behave under coordinate transformations and contributes to the broader understanding of covariant derivatives and their properties.
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