# Levi-Civita connection

This lives as an element of: the space of all linear connections, which in turn sits inside the space of all -bilinear maps

## Contents

## Definition

### Given data

A Riemannian manifold (here, is a differential manifold and is the additional structure of a Riemannian metric on it).

More generally, we can also look at a pseudo-Riemannian manifold, or a manifold with a pseudo-Riemannian metric: a smoothly varying nondegenerate (not necessarily positive definite) symmetric bilinear form in each tangent space.

### Definition part

A **Levi-Civita connection** on is a linear connection on satisfying the following two conditions:

- The connection is metric, viz
- The connection is torsion-free, viz .

### Definition by formula

The formula for the Levi-Civita connection requires us to use the fact that is nondegenerate. So, instead of directly specifying for vector fields and , the formula specifies for vector fields, as follows:

.

### Additional use

The term **Levi-Civita connection** is sometimes also used for the connection induced on any tensor product involving the tangent and cotangent bundle, using the rule for tensor product of connections and the dual connection.

## Facts

### The Levi-Civita connection is unique

`Further information: Levi-Civita connection exists and is unique`

The proof combines the metric condition, the torsion-free condition, and the nondegeneracy of .

Note that the nondegeneracy of is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of

### Christoffel symbols

`Further information: Christoffel symbols of a connection`

The Levi-Civita connection on a manifold is a map . This means that at any point , it gives a map , which roughly *differentiates* one tangent vector along another.

Let form a basis for the tangent space . Then, the Christoffel symbol is the component along of the vector .

The Christoffel symbols thus give an *explicit description* of the Levi-Civita connection. Namely, the Levi-Civita connection can be expressed using the Christoffel symbols.