# Difference between revisions of "Levi-Civita connection"

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The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times T_p(M) \to T_p(M)</math>, which roughly ''differentiates'' one tangent vector along another. | The Levi-Civita connection on a manifold <math>M</math> is a map <math>\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>. This means that at any point <math>p \in M</math>, it gives a map <math>T_p(M) \times T_p(M) \to T_p(M)</math>, which roughly ''differentiates'' one tangent vector along another. |

## Revision as of 20:54, 24 July 2009

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:

.

## 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

To show that the Levi-Civita connection exists, it suffices to check that the map sending to what we propose for is actually a linear map.

### 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.