Riemannian connection
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In mathematics, a Riemannian connection is a connection 
on 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 Xg(Y,Z) denotes the derivative of the function g(Y,Z) along this vector field X.
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