# Fundamental theorem of Riemannian geometry

**fundamental theorems of Riemannian geometry**

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.wikipedia

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### Riemannian geometry

**Riemannianlocal to global theoremsRiemann geometry**

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.

### Levi-Civita connection

**Christoffel symbolconnectionsLevi-Civita**

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric. Because a metric also defines the geodesics of a differential manifold, for some metric there is not only one connection defining the same geodesics (some examples can be found of a connection on leading to the straight lines as geodesics but having some torsion in contrary to the trivial connection on, i.e. the usual directional derivative), and given a metric, the only connection which defines the same geodesics (which leaves the metric unchanged by parallel transport) and which is torsion-free is the Levi-Civita connection (which is obtained from the metric by differentiation).

The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties.

### Pseudo-Riemannian manifold

**pseudo-Riemannianpseudo-Riemannian metricpseudo**

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.

In particular, the fundamental theorem of Riemannian geometry is true of pseudo-Riemannian manifolds as well.

### Torsion tensor

**torsiontorsion-freetorsion form**

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric. The first condition means that the metric tensor is preserved by parallel transport, while the second condition expresses the fact that the torsion of ∇ is zero. Because a metric also defines the geodesics of a differential manifold, for some metric there is not only one connection defining the same geodesics (some examples can be found of a connection on leading to the straight lines as geodesics but having some torsion in contrary to the trivial connection on, i.e. the usual directional derivative), and given a metric, the only connection which defines the same geodesics (which leaves the metric unchanged by parallel transport) and which is torsion-free is the Levi-Civita connection (which is obtained from the metric by differentiation).

### Riemannian manifold

**Riemannian metricRiemannianRiemannian manifolds**

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.

### Affine connection

**connectionaffineaffine connections**

### Metric tensor

**metricmetricsround metric**

Here a metric (or Riemannian) connection is a connection which preserves the metric tensor.

### Lie bracket of vector fields

**Lie bracketLie bracketscommutator of vector fields**

### Vector field

**vector fieldsvectorgradient flow**

### Parallel transport

**parallelparallel-transporttransported**

The first condition means that the metric tensor is preserved by parallel transport, while the second condition expresses the fact that the torsion of ∇ is zero. Because a metric also defines the geodesics of a differential manifold, for some metric there is not only one connection defining the same geodesics (some examples can be found of a connection on leading to the straight lines as geodesics but having some torsion in contrary to the trivial connection on, i.e. the usual directional derivative), and given a metric, the only connection which defines the same geodesics (which leaves the metric unchanged by parallel transport) and which is torsion-free is the Levi-Civita connection (which is obtained from the metric by differentiation).

### Contorsion tensor

**contorsion**

The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor.

### Covariant derivative

**covariant differentiationtensor derivativecovariant differential**

The following technical proof presents a formula for Christoffel symbols of the connection in a local coordinate system.

### Action (physics)

**actionaction principleaction integral**

There are quicker and simpler methods to obtain the Christoffel symbols for a given metric, e.g. using the action integral and the associated Euler-Lagrange equations.

### Geodesic

**geodesicsgeodesic flowgeodesic equation**

### Directional derivative

**normal derivativedirectionalderivative**

Because a metric also defines the geodesics of a differential manifold, for some metric there is not only one connection defining the same geodesics (some examples can be found of a connection on leading to the straight lines as geodesics but having some torsion in contrary to the trivial connection on, i.e. the usual directional derivative), and given a metric, the only connection which defines the same geodesics (which leaves the metric unchanged by parallel transport) and which is torsion-free is the Levi-Civita connection (which is obtained from the metric by differentiation).

### Subscript and superscript

**superscriptsubscriptsuperscripts**

That is, an index repeated subscript and superscript implies that it is summed over all values.

### Nash embedding theorem

**Nash-Kuiper theoremembedding theoremNash and Kuiper's C 1 embedding theorem**

* Nash embedding theorem

### Fundamental theorem

**fundamental equationFundamental lemmaList of fundamental theorems**

### List of things named after Bernhard Riemann

**List of topics named after Bernhard Riemannnamed after**

### Metric tensor (general relativity)

**metric tensormetricspacetime metric**

According to the fundamental theorem of Riemannian geometry, there is a unique connection ∇ on any semi-Riemannian manifold that is compatible with the metric and torsion-free.

### Metric connection

**Riemannian connectioncompatibility with the metriccompatible**

It is unique by the fundamental theorem of Riemannian geometry.