TITLE: Homotopy invariance of vector bundles by parallel transport: reference needed for my students.
QUESTION [8 upvotes]: Let $M$ be a smooth manifold and $V \to [0,1] \times M$ be a smooth vector bundle. The homotopy invariance states that the restrictions $V_0$ and $V_1$ to the bottom and top of the cylinder are isomorphic.
One can prove that using parallel transport: pick a connection on $V$. For each $x \in M$, take the curve $c_x:t  \mapsto (t,x)$. Parallel transport defined by the connection along the curves $c_x$ gives an isomorphism $V_0 \cong V_1$.
This is of course an easy exercise, and my question is to find a self-contained reference that is accessible for undergraduates who know the notions of manifolds, vector bundles and integral curves of vector fields on manifolds (I wish to assign this as a seminar talk).

REPLY [2 votes]: You can check pages 33-35 of my course notes. Also you can check section 3.4 of Husemoller's book  Fiber Bundles, Graduate Texts in Math, vol. 20, Springer Verlag, 1994.  In my notes  I prove the result for arbitrary compact spaces, while Husemoller does this for paracompact spaces. In both cases no smoothness is assumed.