basic question about the definition of orientability

Question

I am a non-math major student who just started learning differential geometries. Very hard for me but my research needs it. I have trouble understanding the definition of orientability. Why is it related to the determinant of the Jacobian matrix of the transformation of two charts? I just can't figure out what the determinant represents here. The definition is: We say that an atlas $(U_i,ϕ_i)$ is oriented if the jacobian of the transformation between two charts at the intersection is positive. (i.e. $\det\operatorname{Jac}ϕ_j∘ϕ^{−1}_i>0$). Thank you!

Answer 1

Is the definition of an orientation of a vector space clear to you? It just rests on the following observation: any two ordered basis of a vector space V are related by a linear transformation- wherein I send the first vector of the first basis to the first vector of the second basis, and so on... If we restrict ourselves to only linear transformations with positive determinant, then some particular ordered basis's cannot be transformed into eachother - the key idea is that the determinant is multiplicative: det(AB) = det(A)det(B).

We actually now have two classes of different kinds of basis - in the "standard coordinates" I can distinguish these classes by writing out the coordinates of each basis vector as the columns of a matrix using the ordering (the first vector in that basis becomes the first column, etc.). This matrix has positive determinant for those in the first class, and negative determinant for those in the second one. Because of the multiplicativity of the determinant, I can't get from one class to another by multiplying by a matrix with positive determinant. A choice of one of the two classes defines an orientation on V. Let's call a basis in that chosen class positively oriented.

(*Being an ordered basis is important here, since an orientation on the plane should mean instructions specifying what it means to turn left. The basis ((1,0),(0,1)) encodes this, I start at the first vector and the second one is to my "left". (In higher dimensions this is more subtle. The right hand rule is the method for three dimensions.))

Now for a manifold M, you need to associate a tangent space at each point. Then an orientation on M is a consistent choice of orientation on these tangent spaces. What does consistent mean? Well, one way to think about this is that if I took a chart, then my first choice of orientations gives rise to a choice of orientation on all the tangent spaces of R^n. We generally think of these as "being the same," which is just the observation that we can identify vectors in different tangent spaces by translating them around. Now the consistency idea here means that if a take a positively oriented basis of the tangent space at one point, then drag it to another nearby one in my chart, the resulting basis is in the class of positively oriented basis for the tangent space at that point.

Now we may have to sometimes pass from one chart to another in order to check this consistency on the entire manifold. When we do so, the Jacobian tells us how the numbers representing our basis changes. The Jacobian being positive just means that if a basis for the tangent space at a point is positively oriented in some chart, then it is in the other chart also. Therefore, we can continue to wander around our manifold, and check that every choice of orientations on the tangent spaces is consistent.

Hope that helps. It is a tricky notion. Of course, it is a good idea to think about how the sphere is oriented, and how the Mobious band is not.