# Metric connection

**Riemannian connectioncompatibility with the metriccompatibleconnection one-formmetric connectionsmetric-compatibilityYang-Mills connectionYang-Mills connections**

In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve.wikipedia

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### Levi-Civita connection

**Christoffel symbolconnectionsLevi-Civita**

A special case of a metric connection is a Riemannian connection; there is a unique such which is torsion free, the Levi-Civita connection.

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.

### Connection (vector bundle)

**connectionKoszul connectionlinear connection**

Then, a connection D on E is a metric connection if:

The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).

### Parallel transport

**parallelparallel-transporttransported**

Equivalently, \nabla is Riemannian if the parallel transport it defines preserves the metric g.

In (pseudo) Riemannian geometry, a metric connection is any connection whose parallel transport mappings preserve the metric tensor.

### Connection form

**connectionconnection one-formvertical subspace**

Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity.

One may then define a connection that is compatible with this bundle metric, this is the metric connection.

### Christoffel symbols

**Christoffel symbolChristoffel coefficientsChristoffel connection**

Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity.

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.

### Gluon field strength tensor

**gluon field tensor**

The notation presented here is that which is commonly used in physics; for example, it can be immediately recognizable as the gluon field strength tensor.

A more mathematically formal derivation of these same ideas (but a slightly altered setting) can be found in the article on metric connections.

### Gauge theory

**gauge groupgauge theoriesgauge symmetry**

The notation of A for the connection form comes from physics, in historical reference to the vector potential field of electromagnetism and gauge theory.

For example, it is sufficient to ask that a vector bundle have a metric connection; when one does so, one finds that the metric connection satisfies the Yang-Mills equations of motion.

### Vector bundle

**vector bundlesWhitney sumdirect sum**

### Bundle metric

**Riemannian bundle metricfibre metricmetric**

Let define a bundle metric, that is, a metric on the vector fibers of E.

### Inner product space

**inner productinner-product spaceinner products**

### Torsion tensor

**torsiontorsion-freetorsion form**

A special case of a metric connection is a Riemannian connection; there is a unique such which is torsion free, the Levi-Civita connection.

### Tangent bundle

**Canonical vector fieldrelative tangent bundletangent vector bundle**

In this case, the bundle E is the tangent bundle TM of a manifold, and the metric on E is induced by a Riemannian metric on M.

### Hodge star operator

**Hodge dualHodge starcodifferential**

However, once one does require compatibility, this metric connection defines an inner product, Hodge star, Hodge dual, and Laplacian, which are required to formulate the Yang-Mills equations.

### Laplace operator

**LaplacianLaplacian operatorscalar Laplacian**

However, once one does require compatibility, this metric connection defines an inner product, Hodge star, Hodge dual, and Laplacian, which are required to formulate the Yang-Mills equations.

### Connection (principal bundle)

**principal connectionconnectionconnections**

### Section (fiber bundle)

**sectionsectionsLocal section**

Let \sigma,\tau be any local sections of the vector bundle E, and let X be a vector field on the base space M of the bundle.

### Differential form

**differential forms2-formtwo-form**

Here d is the ordinary differential of a scalar function.

### Endomorphism

**endomorphismsEndofunctionendomorphism monoid**

The latter is a function on the bundle of endomorphisms so that

### Curvature of Riemannian manifolds

**curvatureabstract definition of curvaturecurvature of a Riemannian manifold**

Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity.

### Curvature form

**Bianchi identityBianchi identitiescurvature**

### Atlas (topology)

**atlascharttransition map**

The above definitions also apply to local smooth frames as well as local sections.

### Exterior algebra

**exterior productexterior powerwedge product**

Let \Gamma(E) denote the space of differentiable sections on E, let \Omega^p(M) denote the space of p-forms on M, and let be the endomorphisms on E.