Affine connexion

Levi-Civita connexion

Let be a semi-Riemannian manifold. The Levi-Civita connexion is the unique affine connexion on which is torsion-free and compatible with the metric tensor in the sense that

i.e. is covariantly constant.

Proof of existence and uniqueness

Let 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 of with so that the former is Levi-Civita. By Covariant derivative disagreement on tensor fields we have

or after lowering indices

By ^P1 we have

which fully determines .¹

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

so that

In particular this gives the Christoffel symbols as the connexion coëfficients.

Properties

Fundamental

Let . We take local coördinates .

1. for any vector field .

Proof

Expressing in terms of Christoffel symbols,

where is by ^P1. This proves ^P1.

Curvature

Consider the Riemannian curvature associated to , along with the Ricci curvature and scalar curvature .

1. , i.e. .
2. .
3. , i.e. .
4. Bianchi identity II. .
5. The number of independent components in is for a manifold of dimension . In particular we have for respectively.
6. If vanishes, then there exist local coördinate systems with the metric .

We take local coördinates .

1. .
2. .

Proof

proof

---

tidy | en | sembr

Footnotes

1. 2009. General relativity, theorem 3.1.1, pp. 35–36. ↩