Levi-Civita connection

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In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free. The fundamental theorem of Riemannian geometry states that there is a unique connection that satisfies these properties. The connection formalizes and generalizes the "rolling without slipping or twisting" method of transporting tangent planes of a smooth surface embedded in R 3 {\displaystyle \mathbb {R} ^{3}} (or generally, any Riemannian manifold, by the Nash embedding theorems). The covariant derivative is defined given any affine connection. In the theory of Riemannian and pseudo-Riemannian manifolds, the "covariant derivative" by default refers to the one defined using the Levi-Civita connection....

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