I am reading the proof of lemma 14.8 page 183 of the Taubes’ book Differential Geometry.
Lemma 14.8 asserts that:Suppose $M$ a compact manifold and $\pi:E \rightarrow M$ a real vector bundle with even dimension fiber. Assume $t \rightarrow j_t$ a smoothly varying 1-parameter family of almost complex structure.Denotes $E^t$ the bundle equips with almost complex structure $j_t$ .Then there exists a bundle isomorphism $E^0$ to $E^t$ for any t.
Following is the proof given by Taubes:Consider $p:[0,1] \times M \rightarrow M$ the projection and the pullback bundle $p^*E$ , for any given $t$, restrict $p^*E$ to $M \times {t}$ as the bundle $E^t$ .Fix a connection $A$ on $E$.Parallel transport by $A$ along the fibers of $p$ defines a complex linear isomorphism.
My questions are:
1)Seems we don't define a connection on $p^*E$, how can we parallel transport along the fiber?
2)If we extend $A$ to a "connection" on $p^*E$, such as the pullback connection , how to prove that this "connection" commutes with the almost complex structure to obtain a real connection?
Thanks you for your answer!
I still don't know how to solve the problems above, but now I have another approach to proof lemma 14.8，here I use Tietze extension theorem to gain the extension of isomorphism instead of using parallel transport.
First, consider the manifold $(0,1) \times M$ and replaces the almost complex structure on $(1-\delta,1)$ and $(0,\delta)$ to be $j_1$ and $j_0$. Then we have the new pullback bundle $p^*E$ which restricts to$ {t}\times M$ is $E^t$.
Second, we have the narual bundle $(0,1)\times E$ maps to $(0,1)\times M$. Now consider the bundle $Hom(p^*E,(0,1) \times E)$, where $(0,1)\times E$ admits the complex structure $j_0$ when restricts to ${t}\times M$ for all $t$.
Now, we have $s:{\frac{\delta}{2}}\times M \rightarrow Hom(p^*E, (0,1) \times E)$ , more explicit, it is identity map from $p^*E|_{\frac{\delta}{2}}$ to ${\frac{\delta}{2}} \times E$. Hence we consider $Hom(p^*E,(0,1) \times E)$ restricts to $[\frac{\delta}{2},1-\frac{\delta}{2}]\times M$ ，thus we can apply tietze extension theorem to extend $s$ to a section on $Hom(p^*E,(0,1) \times E)$ restricts to $[\frac{\delta}{2},1-\frac{\delta}{2}]\times M$, hence a section on $Hom(p^*E,(0,1) \times E)$.Since smooth map near a isomorphic map is isomorphic.We can extend $s$ to the isomorphism in a neighborhood of $\frac{\delta}{2}$ , by the compatness of $[\frac{\delta}{2},1-\frac{\delta}{2}]$ , we can extend it to the isomorphism for the whole $(0,1)$.