Levi-Civita transport

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Let M be a differential manifold and g a Riemannian metric on M (thus (M,g) is a Riemannian manifold). Parallel transport or Levi-Civita transport on the tangent bundle is the rule that associates, to any smooth curve, the transport along that curve as per the Levi-Civita connection.

We can also define Levi-Civita transport on tensor powers of the tangent bundle, tensor powers of the cotangent bundle, and tensor products of these, because we can define the Levi-Civita connection on each of these.