Christoffel symbols

Let $(𝑀, 𝑔 𝑎 𝑏)$ be a semi-Riemannian manifold and $𝑥 : 𝑈 \to ℝ 𝑚$ be local coördinates. Then the Christoffel symbols of the first kind are

Γ 𝛼 𝜇 𝜈 = 12 (𝜕 𝜇 𝑔 𝛼 𝜈 + 𝜕 𝜈 𝑔 𝛼 𝜇 - 𝜕 𝛼 𝑔 𝜇 𝜈)

which are the components of

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

These may either be introduced

Properties

$Γ 𝛼 𝛼 𝛽 = 𝜕 𝛽 l n \sqrt | 𝑔 |$ , i.e. $\sqrt | 𝑔 | Γ 𝛼 𝛼 𝛽 = 𝜕 𝛽 \sqrt | 𝑔 |$ .

Proof

By ^I1 we have

Γ 𝛼 𝛼 𝛽 = 𝑔 𝛿 𝛼 Γ 𝛿 𝛼 𝛽 = 12 𝑔 𝛿 𝛼 (𝜕 𝛼 𝑔 𝛿 𝛽 + 𝜕 𝛽 𝑔 𝛿 𝛼 - 𝜕 𝛿 𝑔 𝛼 𝛽) = 12 𝑔 𝛿 𝛼 (𝜕 𝛿 𝑔 𝛼 𝛽 + 𝜕 𝛽 𝑔 𝛿 𝛼 - 𝜕 𝛿 𝑔 𝛼 𝛽) = 12 𝑔 𝛿 𝛼 𝜕 𝛽 𝑔 𝛿 𝛼 = 12 t r (𝐠 - 1 𝜕 𝛽 𝐠)! = 12 𝜕 𝛽 l n | d e t 𝐠 | = 𝜕 𝛽 l n \sqrt | 𝑔 |

proving ^P1.

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