it is clear (geometrically) that is parallelizable because we can consider a vector field of unit tangent vectors at each point (say, in anti-clockwise direction).
I am having trouble understanding how to write down the above argument mathematically ....we need to come up with a smooth vector field , so that .
what is the choice for a smooth function to have the vector field as above??
can we consider the following (coordinate) vector field ??
....I think we have to be carefull here, we need to be smooth in any chart. will it be the case here?
For example, if we consider then I don't think the field will do the same trick...even though it's coordinate functions are constant=1 in some chart....but it is known that any smoth (or even continuous) field on a 2-sphere has at least one point where the tangent is zero.
For an explicit proof, you can say that the unit vector field defined by is continuous.
now, why defines a smooth vector field on the whole circle?
Also, regarding your last example of a vector field , ...I think I have a problem understanding how a vector field on the ambient space (in this case ) translates to a vector field of a submanifold.
I am thinking of tangent vectors as derivations (or germs, in case of complex/analytic manifolds).
Now, can I argue as follows to show that the above vector field defines a smooth vector field on :
Consider the following (smooth) vector field on defined as .
I want to show that this defines a smooth vector field on . In other words, I need to show that this defines a derivation on .
So, let be a smooth function on some open set , so that .
Therefore (by the chain rule) .
So that is a tangent vector to .
Now, why can I just, instead of the above formal argument, say that this defines a tangent vector to simply because is orthogonal to ?
Also, is there a general result that will guarantee that this restricted smooth vector field (restricted to a submanifold) will be smooth again?
Sorry about too many questions, but I feel that there is a gap in my understanding of the subject...
Thanks a lot!
I won't have time for a lengthy answer; I hope the following will be of some interest nevertheless:
- the notation is to be taken with care: the definition of at depends on the choice of a chart at ; for instance refers implicitly to a parametrization of by (or with cos/sin) on the open subset or on . These two charts cover the circle, and the map is the "same" in both charts (we need two charts nevertheless because of the lack of injectivity), so that the derivation , defined locally (on each chart), is in fact global (coherent definition for both charts).
- there is a (nontrivial) equivalence between vector fields (i.e. indefinitely differentiable functions ) and derivations.
- in the case of submanifolds of , the tangent subspace can be thought in various equivalent ways: either it is defined intrinsically, like for a general manifold (no need of the surrounding manifold), or it can be viewed as a subspace of tangent to the submanifold (no wonder it is called "tangent" space). For the circle, as a submanifold of , the tangent space at a point can be identified with the line that is tangent to the circle at this point.
- if is a submanifold of , and if is a vector field of such that, for , , then (restricted to ) is a vector field on .