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?