LeviCivitakopplingen
LeviCivitakopplingen, commonly known as the Levi-Civita connection, is the canonical affine connection on a Riemannian manifold (M, g). It is defined by two fundamental properties: it is torsion-free and metric-compatible (∇ g = 0). Among all connections on M, there is a unique one that satisfies these conditions; existence and uniqueness are guaranteed by the Koszul formula.
The Koszul formula states that for vector fields X, Y, Z on M, 2g(∇_X Y, Z) = X
Geodesics are curves γ whose tangent vector is parallel transported along itself: ∇_{\dot γ} \dot γ = 0. In coordinates,
Parallel transport defined by the Levi-Civita connection preserves the inner product, so the length and angles
The curvature of the manifold is captured by the Riemann curvature tensor R(X,Y)Z = ∇_X ∇_Y Z −