Parallelltransport

Parallelltransport is a concept in differential geometry that describes how to move a vector along a curve in a differentiable manifold. It is a way to compare vectors at different points on the manifold by "transporting" them in a way that preserves their directional properties relative to the manifold's structure. Imagine having a vector at one point on a curved surface. Parallelltransport allows you to define a corresponding vector at another point on that surface, ensuring that it remains "parallel" to the original vector in the context of the surface's curvature.

The process of parallelltransport is defined using a connection, which is a mathematical tool that specifies how to differentiate vector fields along curves. Different connections can lead to different notions of parallelltransport. For instance, on a sphere, transporting a vector around a closed loop might not result in the original vector; it could be rotated. This phenomenon is related to the curvature of the manifold and is a key aspect studied through parallelltransport.

In essence, parallelltransport provides a way to define a "straight line" or geodesic path on a curved space by ensuring that the tangent vector to the path undergoes parallelltransport along itself. This concept is fundamental in understanding geometric properties of manifolds and has applications in various fields of physics, such as general relativity, where it is used to describe the motion of objects in curved spacetime.