Levi-Civita connexion

Let $(𝑀, 𝑔 𝑎 𝑏)$ be a semi-Riemannian manifold. The Levi-Civita connexion $\nabla$ is the unique affine connexion on $𝑀$ which is torsion-free and compatible with the metric tensor in the sense that

\nabla 𝑐 𝑔 𝑎 𝑏 = 0,

i.e. $𝑔 𝑎 𝑏$ is covariantly constant.

Proof of existence and uniqueness

Let $˜ \nabla$ be any torsion-free affine connexion, which must exist at least locally since we may consider partial derivative as a local affine connexion. We solve for the connexion disagreement tensor $𝐶 𝑐 𝑎 𝑏$ of $\nabla$ with $˜ \nabla$ so that the former is Levi-Civita. By Covariant derivative disagreement on tensor fields we have

0 = \nabla 𝑐 𝑔 𝑎 𝑏 = ˜ \nabla 𝑐 𝑔 𝑎 𝑏 - 𝐶 𝑑 𝑐 𝑎 𝑔 𝑑 𝑏 - 𝐶 𝑑 𝑐 𝑏 𝑔 𝑎 𝑑

𝐶 𝑏 𝑐 𝑎 + 𝐶 𝑎 𝑐 𝑏 = ˜ \nabla 𝑐 𝑔 𝑎 𝑏 .

By ^P1 we have

2 𝐶 𝑐 𝑎 𝑏 = ˜ \nabla 𝑎 𝑔 𝑏 𝑐 + ˜ \nabla 𝑏 𝑔 𝑎 𝑐 - ˜ \nabla 𝑐 𝑔 𝑎 𝑏 .

which fully determines $\nabla$ .¹

From the above proof we see that $\nabla$ is related to any other affine connexion $˜ \nabla$ by the connexion disagreement tensor

𝐶 𝑐 𝑎 𝑏 = 12 (˜ \nabla 𝑎 𝑔 𝑏 𝑐 + ˜ \nabla 𝑏 𝑔 𝑎 𝑐 - ˜ \nabla 𝑐 𝑔 𝑎 𝑏)

so that

\nabla 𝑎 𝑋 𝑏 = ˜ \nabla 𝑎 𝑋 𝑏 + 𝐶 𝑏 𝑎 𝑐 𝑋 𝑐 .

Properties

Let $𝑔 = d e t (𝐠)$ . We take local coördinates $𝑥 : 𝑈 \to ℝ 𝑚$ .

$\sqrt | 𝑔 | \nabla 𝜇 𝑣 𝜇 = 𝜕 𝜇 \sqrt | 𝑔 | 𝑣 𝜇$ for any vector field $𝑣 𝑎 \in 𝔛 (𝑀)$ .

Proof

Expressing in terms of Christoffel symbols,

\sqrt | 𝑔 | \nabla 𝜇 𝑣 𝜇 = \sqrt | 𝑔 | 𝜕 𝜇 𝑣 𝜇 + \sqrt | 𝑔 | Γ 𝜇 𝜇 𝛿 𝑣 𝛿! = \sqrt | 𝑔 | 𝜕 𝜇 𝑣 𝜇 + 𝑣 𝛿 𝜕 𝛿 \sqrt | 𝑔 | = 𝜕 𝜇 \sqrt | 𝑔 | 𝑣 𝜇

Consider the Riemannian curvature $𝑅 𝑐 𝑑 𝑎 𝑏$ associated to $\nabla$ , along with the Ricci curvature $𝑅 𝑎 𝑏$ and scalar curvature $𝑅$ .

$𝑅 𝑎 𝑏 𝑐 𝑑 = 𝑅 [𝑎 𝑏] 𝑐 𝑑$ , i.e. $𝑅 𝑎 𝑏 𝑐 𝑑 = - 𝑅 𝑏 𝑎 𝑐 𝑑$ .
$𝑅 𝑎 𝑏 𝑐 𝑑 = 𝑅 𝑐 𝑑 𝑎 𝑏$ .
$𝑅 𝑎 𝑏 = 𝑅 (𝑎 𝑏)$ , i.e. $𝑅 𝑎 𝑏 = 𝑅 𝑏 𝑎$ .
Bianchi identity II. $\nabla 𝑎 𝑅 𝑑 𝑒 𝑏 𝑐 + \nabla 𝑏 𝑅 𝑑 𝑒 𝑐 𝑎 + \nabla 𝑐 𝑅 𝑑 𝑒 𝑎 𝑏 = 0$ .
The number of independent components in $𝑅 𝑐 𝑑 𝑎 𝑏$ is $112 𝑚 2 (𝑚 2 - 1)$ for a manifold of dimension $𝑚$ . In particular we have $0, 1, 6, 20$ for $𝑚 = 1, 2, 3, 4$ respectively.
If $𝑅 𝑐 𝑑 𝑎 𝑏$ vanishes, then there exist local coördinate systems with the metric $𝑔 𝜇 𝜈 = 𝜂 𝜇 𝜈$ .

We take local coördinates $𝑥 : 𝑈 \to ℝ 𝑚$ .

$𝑅 𝜌 𝜎 𝜇 𝜈 = 𝜕 𝜇 Γ 𝜌 𝜈 𝜎 - 𝜕 𝜈 Γ 𝜌 𝜇 𝜎 + Γ 𝜌 𝜇 𝜆 Γ 𝜆 𝜈 𝜎 - Γ 𝜌 𝜇 𝜆 Γ 𝜆 𝜇 𝜎$ .
$𝑅 𝛼 𝛽 = 𝜕 𝜇 Γ 𝜇 𝛼 𝛽 - Γ 𝜇 𝜎 𝛽 Γ 𝜎 𝛼 𝜇 - 𝜕 𝛼 𝜕 𝛽 l n \sqrt | 𝑔 | + Γ 𝜇 𝛼 𝛽 𝜕 𝜇 l n \sqrt | 𝑔 |$ .

Proof

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