I would like to get a better intuitive grip on how parallel transport works. I once saw a video a German guy made with a little car having a gyroscope. That car was dragged on a big beach ball and the wheels moved. Simultaneously, a small arrow on top of that car was "parallel transported" via the modified gyro mechanism.
What I aim to construct is an algorithm that doesn't make use of any parametric equations, nor of the Christoffel Symbols, connections, etc. on manifolds as they'd introduce a vicious circle. I'd like to start with a discrete polyhedral manifold and move an object that "lives" in the tangent space of that manifold along a given curve. This may sound too general, so, for starters, consider a smooth manifold, that's not discrete as I initially proposed.
Say my entity is at a point A on that manifold on knows that it has to hold an arrow/vector in a given direction while moving towards another point B. A and B are connected via a very small curve. If v was the direction in A, how would it look like when it reaches B? That's the goal of the algorithm. So, to rephrase the question, how can one know how to rotate and translate the tangent space from A to B if the displacement is small, but it does actually take place on a portion of a manifold, not in the normal Euclidian space? In case of a sphere, there is no theoretical notion of "keep the arrow pointing towards the East" - due to the Hairy ball theorem.. but all intuitive explanations of parallel transport do make use of such a notion, a notion of keeping a vector pointing in a certain "constant" direction. That "constant" direction is a notion that makes sense only when referring to what happens in the tangent space at a point, right?
So, how would one correctly approach this: a small vehicle on a manifold, in a virtual/game environment. The vehicle moves (and it has wheels! - these might help gather rotation/distortion/etc information from the curved manifold it lies on). On top of the vehicle, a vector should me moved like it was parallel transported. To cut it short, please analyze the already classic applet. Sorry for the very long description..