# Difference between revisions of "Levi-Civita connection"

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{{further|[[Christoffel symbols of a connection]]}} | {{further|[[Christoffel symbols of a connection]]}} | ||

− | 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 | + | 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 \Gamma(TM) \to \Gamma(TM)</math>, which roughly ''differentiates'' one tangent vector along another. |

Let <math>\partial_1, \partial_2, \ldots, \partial_n</math> form a basis for the tangent space <math>TM</math>. Then, the Christoffel symbol <math>\Gamma_{ij}^k</math> is the component along <math>e_k</math> of the vector <math>\nabla_{\partial_i}\partial_j</math>. | Let <math>\partial_1, \partial_2, \ldots, \partial_n</math> form a basis for the tangent space <math>TM</math>. Then, the Christoffel symbol <math>\Gamma_{ij}^k</math> is the component along <math>e_k</math> of the vector <math>\nabla_{\partial_i}\partial_j</math>. | ||

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. | 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. |

## Latest revision as of 17:59, 6 January 2012

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.