Smooth paths and homotopies In applications of the fundamental group(oid) to smooth manifolds it is sometimes useful to have paths which are smooth, rather than merely continuous. For example, if we consider the local system of solutions of some linear partial differential equation, we can pull it back along a path, where it can be viewed as the solutions of an ordinary differential equation. The "nicer" the path is, the "nicer" will be the coefficients of the resulting ordinary differential equation.
It seems impossible to simply define this issue away by only allowing smooth paths in the first place, since a concatenation of smooth paths need not be smooth. Another possible problem: if we want to also restrict to smooth homotopies, we have to figure out what this means at the corners of the square.
Here's a couple of specific questions, since the previous paragraphs are so vague. Does every homotopy class of paths have a smooth representative? If two smooth paths are homotopic, can the homotopy be chosen to be smooth? How does one deal with the corners of the square? I imagine that the answers to these questions are written down somewhere.
 A: I recommend you to organize your questions. It becomes easier to answer just by adding numbers.

Your Questions:
  Let $N$ be smooth manifold and $\partial M \neq \phi$.
  
  
*
  
*(1)Does every homotopy class of paths have a smooth representative? 
  
*(2)If two smooth paths on $N$ are homotopic, can the homotopy be chosen to
  be smooth?  
  
*(3)How does one deal with the corners of the square?
  

For (1), it will be obvious if (2) is correct. As will be described later, since (2) will be Yes, so, (1) will also be Yes.
For (2), The first part of the following proposition seems to be the answer to your problem. See the P258 of ref 1　(
The second edition seems to be published now. I don't know which page has this proposition in the second edition.). 

For (3), I don't understand the　detailed meaning. If it means "whether there is a piecewise smooth homotopy between the piecewise smooth loops that are homotopic to each other?", then your question probably included in the following two questions.


*

*The piecewise-smooth homotopy

*piecewise smooth homotopy
I haven't completed the proof, but I think it's probably yes.
Reference:
(ref.1)John M. Lee; "Introduction to Smooth Manifolds (Graduate Texts in Mathematics, 218)" Springer (2002/9/23)
P.S:
I'm not very good at English, so I'm sorry if I have some impolite or unclear expressions. I welcome any corrections and English review. (You can edit my question and description to improve them)
