I (think I) know the difference between orientable and non-orientable topological surfaces. I don't know the difference between co-orientable and non-coorientable surfaces. I must admit that I am not really trained in topology but I am able to understand most intuitive pictures.
Can somebody please give a simple example of a coorientable and non-coorientable surface?
Somebody wrote that "a circle going round a Möbius strip" is orientable but not coorientable. This example is confusing: What is meant by "a circle going round a Möbius strip"?
The background of this question is that I try to understand whether or not in electromagnetism it would be possible to have a non-orientable surface with an electric charge. An orientable surface, infinitely thin, theoretically can carry an electric charge, but how about an infinitely thin Möbius strip or projective plane?
This question was inspired by an advanced mathematical article that I didn't understand:
It is highly significant that while the vanishing of magnetic charge requires Σ to be orientable, the vanishing of electric charge requires it to be coorientable. A simple (1 + 1)-dimensional example of a submanifold that is orientable but not coorientable is a circle going round a Möbius strip. The circle S1 is orientable, but on the Möbius strip a consistent normal vector cannot be defined along S1 . More generally: a surface Σ -> M of dimension n-1 is coorientable if there is a vector field along Σ which is everywhere transversal to TΣ (hence the boundary of a manifold is always coorientable). If the manifold M itself is orientable, then a k-dimensional Σ is coorientable if and only if Σ is orientable.
My ultimate goal is to understand:
Can a orientable surface carry electric charge? Yes.
Can a non-orientable surface carry electric charge?
Can a coorientable surface carry electric charge? No, according to the above.
Can a non-coorientable surface carry electric charge? Probably yes, according to the above.
Can a coorientable surface carry magnetic charge?
Can a non-coorientable surface carry magnetic charge?
Can a orientable surface carry magnetic charge? No, according to the above.
Can a non-orientable surface carry magnetic charge? Probably yes, according to the above.