# While doing parallel transport, what are we actually doing?

The Covariant Derivative which is used in General Relativity is obtained by considering the concept of Parallel transport of a vector.

A vector is transported parallelly from one point on the manifold to the other because we cannot do algebraic operation on objects at two different points on a manifold.

My question is while doing a parallel transport what are we doing actually? What is the action of Parallel transport means mathematically? Mathematically, what operation on a vector makes to go under a parallel transport?

• Wold Mathematics be a better home for this question? Dec 22 '17 at 17:03
• Parallel transport is defined by adding additional structure to a manifold, a think called a connection, such that you can define a covariant derivative as a derivative along a vector( an infinitesimal displacement) that keeps the "orientation" of the derivative of the field constant. Dec 22 '17 at 18:00
• Have you seen en.wikipedia.org/wiki/Covariant_derivative ? It' s pretty good, and if you have some basic knowledge of differential geometry you can understand the formal description. Dec 22 '17 at 18:01
• Also, in the same spirit: mathoverflow.net/q/75220 Dec 22 '17 at 18:11
• Maybe you should explain why the name "parallel transport" is not enough. Dec 22 '17 at 18:18