# Metric connection

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.

## Riemannian connections

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 .

The Levi-Civita connection is the torsion-free Riemannian connection on a manifold. It is unique by the fundamental theorem of Riemannian geometry.

## Metric compatibility

In mathematics, given a metric tensor , a covariant derivative is said to be compatible with the metric if the following condition is satisfied:

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.

## References

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