In mathematics, a metric connection is a connection in a vector bundle E equipped with 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 Levi-Civita connection. Here the bundle E is the tangent bundle of a manifold. In addition to being a metric connection, the Levi-Civita connection is required to be torsion free.
An important special case of a metric connection is a Riemannian connection. This is a connection on the tangent bundle of a pseudo-Riemannian manifold (M, g) such that for all vector fields X on M. Equivalently, is Riemannian if the parallel transport it defines preserves the metric g.
A given connection is Riemannian if and only if
for all vector fields X, Y and Z on M, where denotes the derivative of the function along this vector field .
Although other covariant derivatives may be supported within the metric, usually one only ever considers the metric-compatible one. This is because given two covariant derivatives, and , there exists a tensor for transforming from one to the other:
If the space is also torsion-free, then the tensor is symmetric in its first two indices.
- Rodrigues, W. A.; Fernández, V. V.; Moya, A. M. (2005). "Metric compatible covariant derivatives". arXiv:math/0501561.
- Wald, Robert M. (1984), General Relativity, University of Chicago Press, ISBN 0-226-87033-2
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