# Parallel transport in discrete differential geometry - programming a game

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..

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Maybe I just got that wrong, but I'm not quite sure whether you have the correct grasp of what parallel transport does. In particular, parallel transport does not lead to a contradiction to the hairy ball theorem. This is due to the fact that parallel transport from $A$ to $B$ may depend on the path chosen from A to B, which is in particular true for the sphere. See, e.g., en.wikipedia.org/wiki/Holonomy. –  user20266 Apr 20 '12 at 18:06
The "constant direction" is an additional piece of information that is given by an extra structure which may be called the "parallel transport", the "connection", the "Christoffel symbols" etc. It may be obtained from the ambient space (in your example you may think your sphere to be immersed into the Euclidean space). If your manifold is "abstract" than you need to provide this structure. Of course, you may choose a Riemannian structure as underlying. –  Yuri Vyatkin Apr 21 '12 at 0:34
@Thomas, thanks for the comment, I've encountered holonomy already and I'll have to digest it a bit more to fully understand how it will be useful. I was trying to point out a semantic, not necessarily mathematical contradiction between the terms: direction and non-parallelizable manifold.If I only have a naive parametric map (u,v)->(cos(u)sin(v), sin(u)sin(v), cos(v)) for the sphere and c(t) = (u(t),v(t)), then sphere(c(t)) is a curve. Do I still need to delve into the Christoffel symbols in order to carry out a simulation? Isn't it possible to do without them? What about the discrete case? –  teodron Apr 21 '12 at 6:56
@teodron have you already seen this geometry.caltech.edu/pubs/GSD06.pdf ? These lectures seem to address your question. –  Yuri Vyatkin Apr 22 '12 at 13:56
@YuriVyatkin - thank you for the resources, I knew most of those papers. There's a discrete differential geometry paper by Polthier that discusses the concept of straightest geodesics. I will investigate whether his unfolding of a discretized manifold in a vertex neighborhood might help decide how to simulate parallel transport. I was hoping to somehow avoid unfolding as it's a bit "crude" and not quite the solution I wanted. I was hoping for a more "Euler like integration" to simulate parallel displacement. Thanks for the valuable papers(in a single url :) ). –  teodron Apr 23 '12 at 15:24