Parallel Transport in Riemannian Geometry
In (pseudo) Riemannian geometry, a metric connection is any connection whose parallel transport mappings preserve the metric tensor. Thus a metric connection is any connection Γ such that, for any two vectors X, Y ∈ Tγ(s)
Taking the derivative at t=0, the associated differential operator ∇ must satisfy a product rule with respect to the metric:
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