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.

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.

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

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.