# 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 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 ${\displaystyle \nabla }$ on the tangent bundle of a pseudo-Riemannian manifold (M, g) such that ${\displaystyle \nabla _{X}g=0}$ for all vector fields X on M. Equivalently, ${\displaystyle \nabla }$ is Riemannian if the parallel transport it defines preserves the metric g.

A given connection ${\displaystyle \nabla }$ is Riemannian if and only if

${\displaystyle Xg(Y,Z)=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)}$

for all vector fields X, Y and Z on M, where ${\displaystyle Xg(Y,Z)}$ denotes the derivative of the function ${\displaystyle g(Y,Z)}$ along this vector field ${\displaystyle X}$.

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 ${\displaystyle g_{ab}}$, a covariant derivative is said to be compatible with the metric if the following condition is satisfied:

${\displaystyle \nabla _{c}\,g_{ab}=0.}$

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, ${\displaystyle \nabla }$ and ${\displaystyle \nabla '}$, there exists a tensor for transforming from one to the other:

${\displaystyle \nabla _{a}x_{b}=\nabla _{a}'x_{b}-{C_{ab}}^{c}x_{c}.}$

If the space is also torsion-free, then the tensor ${\displaystyle {C_{ab}}^{c}}$ is symmetric in its first two indices.

## References

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