# 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
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.

### 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

### 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.