Levi-Civita Connection

Prerequisites: Connections, Manifold. Full path (7 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

also known as Christoffel Symbols; see also connections

Intuitive

The Levi-Civita connection is a mathematical tool that we use to parallel transport vectors around a manifold.

Parallel transport is just the simplest way to compare vectors at different points in the manifold.

Parallel is necessary, for example, to define the covariant derivative.

Concrete

Christoffel symbols $\Gamma^i_{jk}$ are a particular type of connection that every Riemannian or pseudo-Riemannian (for example Lorentzian) manifold has (called the Levi-Civita connection).

Examples

The diagram below shows three concrete examples for connections (Christoffel symbols) on simple 2-dimensional manifolds. For a more detailed explanation see Fun with Symmetry.

Abstract

Let $(M, g)$ be a pseudo-Riemannian manifold of any signature. The fundamental theorem of (pseudo-)Riemannian geometry states that there is exactly one affine connection $\nabla$ on $TM$ that is

• torsion-free: $T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y] = 0$,
• metric-compatible: $\nabla g = 0$, that is $X\big(g(Y,Z)\big) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$.

This is the Levi-Civita connection. Combining the metric-compatibility condition for the three cyclic orderings of $X, Y, Z$ and using the vanishing torsion gives the Koszul formula

$$2 g(\nabla_X Y, Z) = X g(Y,Z) + Y g(X,Z) - Z g(X,Y) + g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X) ,$$

which proves uniqueness and, since the right-hand side is $C^\infty(M)$-linear in $Z$ and $g$ is non-degenerate, existence. In a coordinate basis the brackets vanish and with $\nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k$ we get

$$\Gamma^k_{ij} = \tfrac{1}{2} g^{kl} \big( \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} \big),$$

symmetric in $i, j$. Lowering the index, $\Gamma_{lij} = g_{lk} \Gamma^k_{ij} = \tfrac{1}{2}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})$.

Frame bundle formulation. Metric compatibility says that parallel transport maps orthonormal frames to orthonormal frames, so the Levi-Civita connection is a principal connection on the orthonormal frame bundle $O(M)$, with structure group $O(p, q)$ for signature $(p, q)$. In terms of a local orthonormal coframe $e^a$ and the connection 1-forms $\omega^a{}_b$ (with $\omega_{ab} = -\omega_{ba}$ after lowering with the flat metric $\eta_{ab}$), the torsion-free condition is Cartan's first structure equation

$$de^a + \omega^a{}_b \wedge e^b = 0 ,$$

which determines $\omega^a{}_b$ uniquely. The curvature 2-form is $R^a{}_b = d\omega^a{}_b + \omega^a{}_c \wedge \omega^c{}_b$. This spin-connection form is the one needed to couple spinors to gravity.

Curvature and geodesics. The Riemann tensor is $R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$. For the Levi-Civita connection it has the symmetries $R_{abcd} = -R_{bacd} = -R_{abdc} = R_{cdab}$ and satisfies the first Bianchi identity $R_{a[bcd]} = 0$. Here $R_{abcd} = g\big(\partial_a, R(\partial_c, \partial_d)\partial_b\big)$. Antisymmetry in $c, d$ holds by definition, antisymmetry in $a, b$ follows from metric compatibility, the Bianchi identity from vanishing torsion, and the pair symmetry from the last two together. The geodesics of $\nabla$ (curves with $\nabla_{\dot\gamma} \dot\gamma = 0$) coincide with the critical points of the energy functional $\int g(\dot\gamma, \dot\gamma)\, dt$. The holonomy group of $\nabla$ is contained in $O(p, q)$ (in $SO(p,q)$ when $M$ is orientable). For Riemannian manifolds the possible holonomy groups of irreducible, non-locally-symmetric metrics are listed by Berger's classification.

Why is it interesting?

The Christoffel symbols appear in the most important equations of general relativity: the Einstein equation and the geodesic equation.

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Manifold
5. Fiber Bundles
6. Parallel Transport
7. Connections

Leads to: Einstein Equation, Geodesic Equation.