In Chapter 4.2 of Schwarz's book on Morse homology there is a brief discussion of Morse theory on the total space of a smooth vector bundle $E \to M$. In particular, one can take the Morse function $f\_E : E \to \mathbb{R}$ given by
\begin{equation\*}
f\_E(v\_m)=f(m) + q(v\_m) \ ,
\end{equation\*}  

where $f: M \to \mathbb{R}$ is a Morse function on the (closed) base and $q$ is the quadratic form associated to a Riemannian metric on $E$. Then one has $f\_E|\_M = f$ and there is a bijection
\begin{equation\*}
\mbox{Crit}\_\*(f\_E) \cong \mbox{Crit}\_\*(f) \ .
\end{equation\*}
Given that the singular homology of $E$ is just that of $M$, I would assume there is some way of relating the moduli space of gradient flow lines of $f\_E$ on $E$ with those of $f$ on $M$. However, this is not explained in Schwarz's book (there is a different objective) and I have not seen this done anywhere else. In particular, is it possible to get an identification at the chain level or only at the level of Morse homology? 



How does one identify flow lines on a vector bundle with those on the base in Morse theory?
The function $q$ strictly decreases along the solutions of the gradient flow outside of the zero section. Hence any orbit that starts outside the zero section will not converge to a critical point in backwards time, and does not show up in some moduli space of orbits connecting critical points. The orbits on the zero section correspond bijectively to orbits on $M$ if you choose a metric which respects a horizontal-vertical splitting of $TE$ in the obvious way.


If one chooses orientations of all the unstable manifolds in $M$, this naturally also gives you orientations for all the unstable manifolds of the critical points in $E$, hence even the orientations of the moduli spaces on $M$ and $E$ agree. 


Much more interesting is to not take the function you give, but the function


$$
f^2\_E(v\_m)=f(m)-q(v\_m)
$$


The critical points on $E$ correspond to the ones on $M$ shifted by the dimension of $E$, as there are this many more unstable directions. The moduli spaces on $M$ and $E$ again correspond bijectively to each other, but the orientations might not agree. However, if $E$ is an oriented vector bundle, we can orient the unstable manifolds of the critical points on $E$ taking into account this orientation on the fibers. Doing this correctly will show that the moduli spaces are isomorphic as oriented manifolds. This will give an isomorphism


$$
HM\_\*(M)\cong HM\_{\*+\dim E}(E,f\_E^2)
$$


which is the Morse theoretic Thom isomorphism. This can be found for example in the appendix of this paper <http://arxiv.org/abs/0810.1995> by Abbondandolo and Schwarz


To see that the orientation assumptions are necessary, you can compute this explicitly for the Mobius strip. 


