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 g(Y,Z) + Y g(X,Z) − Z g(X,Y) + g([X,Y],Z) − g([X,Z],Y) − g([Y,Z],X). In local coordinates with g_{ij} and its inverse g^{ij}, the connection is expressed through the Christoffel symbols Γ^k_{ij} = (1/2) g^{kl} (∂_i g_{jl} + ∂_j g_{il} − ∂_l g_{ij}).

Geodesics are curves γ whose tangent vector is parallel transported along itself: ∇_{\dot γ} \dot γ = 0. In coordinates, this yields γ''^k + Γ^k_{ij} γ'^i γ'^j = 0. Geodesics generalize shortest paths: in a space with a positive-definite metric, they locally minimize length.

Parallel transport defined by the Levi-Civita connection preserves the inner product, so the length and angles of vectors transported along a curve are unchanged.

The curvature of the manifold is captured by the Riemann curvature tensor R(X,Y)Z = ∇_X ∇_Y Z − ∇_Y ∇_X Z − ∇_{[X,Y]} Z, whose components depend on the Christoffel symbols and their derivatives. The Levi-Civita connection extends to pseudo-Riemannian manifolds and plays a central role in general relativity, where it describes the motion of free-falling bodies in curved spacetime.