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

A given connection $\nabla$ is Riemannian if and only if

\nabla_Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ)

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

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

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

\nabla_a x_b = \nabla_a' x_b - {C_{ab}}^c x_c.

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

References

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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