Parallel Transport Explained

Introduction: How parallel transport works.
What? Make the inconsistent parallelism over two different coordinate patches go away!
Illustrating parallel transport along a curve on the sphere.
There’s a fun fact about parallel transporting on the globe!
Then, see the details of parallel transport!!!
Next, a footage shows how to ‘connect’ nearby tangent planes:
Compare the exterior derivative with the gradient, which vanishes in the direction of tilt of the slope of the hill.
Consider using functions of covectors for finding dual vectors as a different perspective.
The problem: Find the departure from a parallel transported vector along some curve.
The solution: Lift the path for calculating departure in the direction of the curve.
Resources.
The book “The Road to Reality” by Roger Penrose:
10.3. Vector fields and 1-forms
10.4. Components, scalar products
12.3. Scalars, vectors and covectors
14.2. Parallel transport
14.3. Covariant derivative
15.7. Non-triviality in a bundle connection
15.8. Bundle curvature

Inspired by episode #020 of The Portal with host Eric Weinstein