Semi-Riemannian manifold

Christoffel symbols

Let be a semi-Riemannian manifold and be local coördinates. Then the Christoffel symbols of the first kind are

which are the components of

These may either be introduced

1. as the connexion coëfficients of the Levi-Civita connexion;
2. as the coëfficients of the geodesic equation.

Properties

See also properties of the Levi-Civita connexion.

1. , i.e. .

Proof

By ^I1 we have

proving ^P1.

---

tidy | en | sembr