# Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.wikipedia
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### Affine connection

connectionaffineaffine connections
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.
If the manifold is further endowed with a Riemannian metric then there is a natural choice of affine connection, called the Levi-Civita connection.

### Connection (mathematics)

connectionconnectionsconnected
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric. In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.
This latter means of parallel transport is the Levi-Civita connection on the sphere.

### Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.

### Holonomy

holonomy groupholonomiesRiemannian holonomy
Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
Important examples include: holonomy of the Levi-Civita connection in Riemannian geometry (called Riemannian holonomy), holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles.

### Fundamental theorem of Riemannian geometry

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

### Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection
The components of this connection with respect to a system of local coordinates are called Christoffel symbols.
In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection.

### Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
This curvature tensor is given in terms of the Levi-Civita connection \nabla by the following formula:

### Covariant derivative

covariant differentiationtensor derivativecovariant differential
In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
This new derivative – the Levi-Civita connection – was covariant in the sense that it satisfied Riemann's requirement that objects in geometry should be independent of their description in a particular coordinate system.

### Elwin Bruno Christoffel

ChristoffelElwin ChristoffelChristoffel, Elwin Bruno
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.
In the same paper he introduced the Christoffel symbols and which express the components of the Levi-Civita connection with respect to a system of local coordinates.

### Metric connection

Riemannian connectioncompatibility with the metriccompatible
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.
A special case of a metric connection is a Riemannian connection; there is a unique such which is torsion free, the Levi-Civita connection.

### Torsion tensor

torsiontorsion-freetorsion form
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.
There is a unique connection which absorbs the torsion, generalizing the Levi-Civita connection to other, possibly non-metric situations (such as Finsler geometry).

### Geodesic

geodesicsgeodesic flowgeodesic equation
If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to their arc length.
Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.

### Tullio Levi-Civita

Levi-CivitaTullio Levi CivitaLevi-Civita, Tullio
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. In 1917, Levi-Civita pointed out its importance for the case of a hypersurface immersed in a Euclidean space, i.e., for the case of a Riemannian manifold embedded in a "larger" ambient space.

### Pseudo-Riemannian manifold

pseudo-Riemannianpseudo-Riemannian metricpseudo
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric. In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
This allows one to speak of the Levi-Civita connection on a pseudo-Riemannian manifold along with the associated curvature tensor.

### Polar coordinate system

polar coordinatespolarpolar coordinate
The images below show parallel transport of the Levi-Civita connection associated to two different Riemannian metrics on the plane, expressed in polar coordinates.
An orthonormal frame with respect to this metric is given bywith dual coframeThe connection form relative to this frame and the Levi-Civita connection is given by the skew-symmetric matrix of 1-formsand hence the curvature form Ω = dω + ω∧ω vanishes identically.

### Manifold

manifoldsboundarymanifold with boundary
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.

### Tangent bundle

Canonical vector fieldrelative tangent bundletangent vector bundle
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.

### Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric. In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. In 1917, Levi-Civita pointed out its importance for the case of a hypersurface immersed in a Euclidean space, i.e., for the case of a Riemannian manifold embedded in a "larger" ambient space.

### Gregorio Ricci-Curbastro

RicciGregorio Ricci CurbastroRicci-Curbastro
Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.

### Parallel transport

parallelparallel-transporttransported
Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy. In 1906, L. E. J. Brouwer was the first mathematician to consider the parallel transport of a vector for the case of a space of constant curvature.

### L. E. J. Brouwer

BrouwerLuitzen Egbertus Jan BrouwerL.E.J. Brouwer
In 1906, L. E. J. Brouwer was the first mathematician to consider the parallel transport of a vector for the case of a space of constant curvature.

### Mathematician

mathematiciansapplied mathematicianMathematics
In 1906, L. E. J. Brouwer was the first mathematician to consider the parallel transport of a vector for the case of a space of constant curvature.