# Homotopy invariance of vector bundles by parallel transport: reference needed for my students.

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).