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).



Homotopy invariance of vector bundles by parallel transport: reference needed for my students.
Greub, Halperin, Vanstone, Connections, curvature and cohomology, Volume II, ch VII.7.18, Theorem I.


