# Levi-Civita transport

## Definition

Let be a differential manifold and a Riemannian metric on (thus 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.