where is the volume element, the Hodge dual of the constant 1. Note that three different inner products are required to construct this action: the metric connection on E, an inner product on End(E), equivalent to the quadratic Casimir operator (the trace of a pair of matricies), and the Hodge dual. Riemannian connection 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. For every Riemannian connection, one may write a (unique) corresponding Levi-Civita connection. The difference between the two is given by the contorsion tensor. In component notation, the covariant derivative is compatible with the metric tensor if Although other covariant derivatives may be defined, usually one only 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. A word about notation It is conventional to change notation and use the nabla symbol ∇ in place of D in this setting; in other respects, these two are the same thing. That is, ∇ = D from the previous sections above. Likewise, the inner product on E is replaced by the metric tensor g on TM. This is consistent with historic usage, but also avoids confusion: for the general case of a vector bundle E, the underlying manifold M is not assumed to be endowed with a metric. The special case of manifolds with both a metric g on TM in addition to a bundle metric on E leads to Kaluza–Klein theory. See also Vertical and horizontal bundles References Connection (mathematics) Riemannian geometry