It expresses the condition that the torsion of ∇ is zero, and as such is also called torsion-freeness. The second condition is sometimes called symmetry of ∇. It is further equivalent to require that the connection is induced by a principal bundle connection on the orthonormal frame bundle. It may also be equivalently phrased as saying that the metric tensor is preserved by parallel transport, which is to say that the metric is parallel when considering the natural extension of ∇ to act on (0,2)-tensor fields: ∇ g = 0. It may be equivalently expressed by saying that, given any curve in M, the inner product of any two ∇–parallel vector fields along the curve is constant. ![]() The first condition is called metric-compatibility of ∇. ![]() ∇ X Y − ∇ Y X =, where denotes the Lie bracket of X and Y.
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