2014_04_20 

Mike Polioudakis 

Modeling manifolds 

I take the term “manifold” from topology.  The answer to the question in this note might lie in texts of differential geometry and-or differential topology but I am not good at looking through those texts.  If the answer does lie there, please go ahead and suggest such a text but please guide me to the right parts and please give me some indication why that is a good treatment.  

I would like to see more formal models of manifolds which can be used for modeling other stuff and that might be used to test questions in disciplines such as biology and physics.  For instance, the surface of a sphere is a manifold.  There should be equations for the surface of a sphere although I don’t recall them now off the top of my head.  With those equations, we could pose questions about the sum of angles in a triangle, and answer them.  

I would like to know if there are similar equations for other manifolds such as a torus, a Moebius strip, a Klein bottle, a saddle, etc.  

Since I am not highly mathematically adept, I seek simple, clear, logical, step-by-step derivations of the equations, explanations of the equations, and applications of the equations.  Think about a curious young undergraduate with college algebra and a little calculus.