I am reading "Morse Theory and Floer Homology" by Audin and Damian to understand the construction and basic properties of Hamiltonian Floer homology. On page 235 of their book, they consider $u:\mathbb R\times S^1\to (W,\omega,J)$, a Floer trajectory (i.e., $\frac{\partial u}{\partial s}+J(u)\frac{\partial u}{\partial t} + \nabla H_t(u)=0$) for a hamiltonian $H\in C^\infty(W\times S^1)$ with only nondegenerate closed orbits, connecting two contractible closed orbits $x$ and $y$.
They then pick a unitary trivialization $Z_1,\ldots,Z_{2n}$ of $u^*TW$ with the property that $\lim_{s\to\pm\infty}\frac{\partial Z_i}{\partial s} = \lim_{s\to\pm\infty}\frac{\partial^2 Z_i}{\partial s^2}=\lim_{s\to\pm\infty}\frac{\partial^2 Z_i}{\partial t\partial s} = 0$ for each $1\le i\le {2n}$ (to make sense of these equalities, we consider $W$ as embedded in a Euclidean space $\mathbb R^{N}$ with $N\gg 2n$). My question is why we can pick a unitary trivialization with these limiting properties.
I can see that a unitary trivialization exists since $c_1(u^*TW)=0$, since $H^2(\mathbb R\times S^1,\mathbb Z) = 0$, and thus, $u^*TW$ is a trivial $U(n)$ vector bundle. But I am not able to produce a trivialization with the additional properties stated above. The book does not give any details regarding this.
One way of constructing such a trivialization is as follows:
The orbits are non-degenerate so isolated. Let $\gamma$ be some orbit. You first trivialize $\gamma^*TW$. You can extend this to a trivialization in a small tubular neighbourhood of $\gamma$. Call these local trivializations $V_1, ..., V_{2n}$ for the orbit at the $+\infty$ limit and $V_1', \dots, V_{2n}'$ at the $-\infty$ limit of $u$. (To be clear, $V_i(t) \in T_{\gamma(t)} W$.)
If you pull these back by $u$, these give local trivializations of $u^*TW$ near $\pm \infty$. They satisfy the required decay estimates by the fact that in this non-degenerate situation, $u$ converges exponentially fast to its asymptotic limits (this is non-trivial!). There is now an obstruction to extending the trivialization over all of $\mathbb{R} \times S^1$, basically given by $\pi_1( U(n))$. $U(n)$ acts on the trivializations, however. If you change the trivialization $V_1', \dots, V_{2n}'$ by an appropriate loop of matrices in $U(n)$, you end up with a trivialization that does extend. (It suffices even to use your favourite $U(1)$ subgroup of $U(n)$.)
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Edit: Audin & Damian assume that $c_1(TW)$ vanishes on all spherical classes (i.e. in the image of $\pi_2(W) \to H_2(W)$). In this case, since $x$ and $y$ are contractible, you can cap them by disks and choose a trivialization that extends across each of the disks. Then, if you have a Floer cylinder between $x$ and $y$, the obstruction to extending the trivialization across the cylinder is given by the pairing of $c_1(TW)$ with the spherical homology class obtained by gluing the cylinder $u$ to the cap of $x$ and to the reverse of the cap of $y$ (or the other way around, it depends on which of $x$ or $y$ is at $-\infty$). By their assumption, this is 0, so the trivialization extends across $u$ with no problem.