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 connection for which the covariant derivatives of the metric on E vanish.
- A principal connection on the bundle of orthonormal frames of E.
A special case of a metric connection is the Riemannian connection, of which the Levi-Civita connection is a particularly important special case.
- The "gauge covariant" version of a gauge theory accounts for this effect by introducing a gauge field (in mathematical language, an Ehresmann connection) and formulating all rates of change in terms of the covariant derivative with respect to this connection