Levi-Civita connection - Wikipedia

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 (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. The components (structure coefficients) of this connection with respect to a system of local coordinates are called Christoffel symbols. The Levi-Civit

Levi-Civita connection - Wikipedia Jump to content From Wikipedia, the free encyclopedia Affine connection on the tangent bundle of a manifold A connection on the sphere rolls the tangent plane from one point to another. As it does so, the point of contact traces out a curve in the plane: the development . 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 th

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