Metric connection

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. Other common equivalent formulations of a metric connection include:

A special case of a metric connection is the Riemannian connection, of which the Levi-Civita connection is a particularly important special case. For both of these, the bundle E is the tangent bundle TM of a manifold. The Levi-Civita connection is the specific Riemannian connection that is torsion free.

A special case of the metric connection is the Yang-Mills connection. That is, most of the machinery of defining a connection on a vector bundle, and then defining a curvature tensor and the like, can go through without requiring any compatibility with the bundle metric. However, once one does require compatibility with the bundle metric, one is able to define an inner product, which can then be used to construct the Hodge star, the Hodge dual and the Laplacian. The Yang-Mills equations of motion are formulated in terms of the dual; a metric connection satisfying these may be called a Yang-Mills connection.

Let ${\displaystyle \sigma ,\tau }$ be two different sections of the vector bundle E, and let X be a vector field on the base space M of the bundle. Let ${\displaystyle \langle \cdot ,\cdot \rangle }$ define the bundle metric, that is, the metric on the vector bundle E. Then, a connection D on E is a metric connection if it satisfies the equation

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