LeviCivitaConnection - crowlogic/arb4j GitHub Wiki

The Levi-Civita connection, named after the Italian mathematician Tullio Levi-Civita, is a mathematical concept in differential geometry and general relativity. It is a specific type of affine connection, which is a geometric object that describes how vectors can be parallel transported along curves on a manifold. The Levi-Civita connection is a fundamental tool for studying the curvature, geodesics, and other properties of curved spaces or spacetimes.

The Levi-Civita connection is unique in that it is torsion-free and metric-compatible. "Torsion-free" means that the connection has no intrinsic twist, while "metric-compatible" means that the connection preserves the inner product (the metric) of tangent vectors when they are parallel transported along a curve.

The Levi-Civita connection is defined using the Christoffel symbols of the second kind, which are given by:

{ₖᵢⱼ} = (1/2)g^(ₖₗ)(∂ᵢg_ₗⱼ + ∂ⱼg_ₗᵢ - ∂ₗg_ᵢⱼ)

Here, {ₖᵢⱼ} are the Christoffel symbols, g^(ₖₗ) is the inverse of the metric tensor g_ₖᵢ, and ∂ᵢ, ∂ⱼ, and ∂ₗ are partial derivatives with respect to the coordinates xᵢ, xⱼ, and xₗ, respectively.

The Levi-Civita connection is used to define covariant derivatives, which allow for the differentiation of tensor fields in a manner that is consistent with the curvature of the underlying manifold. It also plays a central role in the study of the Riemann curvature tensor, which is a key object in general relativity, describing the curvature of spacetime and its effect on the motion of particles and the propagation of light.