Levi-Civita transport

Definition

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.