Compatibility with Riemannian metrics refers to the relationship between a linear connection on a manifold and the inner product structure induced by a Riemannian metric. A connection is said to be compatible with a Riemannian metric if it preserves the metric under parallel transport, meaning that the inner product of any two tangent vectors remains unchanged as they are transported along curves on the manifold. This concept is crucial for understanding how curvature and geometric properties interact within the context of differential geometry.
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