Metric connection

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.

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).

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.

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.

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.

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.

The notation of A for the connection form comes from physics, in historical reference to the vector potential field of electromagnetism and gauge theory.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

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

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

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.

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.

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.

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.

Here d is the ordinary differential of a scalar function.

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

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.

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.

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

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.

The notation of A for the connection form comes from physics, in historical reference to the vector potential field of electromagnetism and gauge theory.