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I'm interested in the first Chern class of the cotangent bundle. I concretely work on the sphere $S^2$, but the reasoning below seems to work for any manifold.

I take the symplectic point of view considering the standard symplectic form $\omega$ on $T^*S^2$. We may also take a Riemannian metric on $T^*S^2$ in order to get an almost complex structure so that $TT^*S^2$ is a complex vector bundle. My advisor pointed out that now one can choose a splitting $TT^*S^2\cong H\oplus V$ (via a connection) in the horizontal and vertical bundles. These are then identified with $\pi^*(TS^2)$ and $\pi^*(T^*S^2)$ so that from the fact that $c_1(E^*)=-c_1(E)$ and $c_1(E\oplus F)=c_1(E)+ c_1(F)$ one concludes that $c_1(TT^*S^2)=0$. Here $\pi: T^*S^2 \to S^2$.

I would like to understand the details of this. In particular, I feel that we have to make sure that $TT^*S^2$ and $\pi^*(TS^2)\oplus\pi^*(T^*S^2)$ are isomorphic as complex vector bundles as $c_1$ depends on the complex structure. But I can't see what the complex structure on $\pi^*(TS^2)\oplus\pi^*(T^*S^2)$ should be.

Furthermore, I'm not sure about $c_1(E\oplus F)=c_1(E)+ c_1(F)$. Wiki says that the chern class of the direct sum is the cup product of the chern classes so I guess it should say $c_1(E\oplus F)=c_1(E)c_1(F)$, but many authors use addition.

Bernard
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bas
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    A note on your last paragraph: in general, one has $c(E \oplus F) = c(E)c(F)$ where $c$ denotes total Chern class (multiplication is cup product). Thus, picking out the portion of this equation in degree 2, we have $c_1(E \oplus F) = c_1(E) + c_1(F)$. The equation $c_1(E \oplus F) = c_1(E)c_1(F)$ doesn't quite make sense because the left-hand side is in $H^2$ and the right-hand side is in $H^4$. – kamills Nov 23 '21 at 16:23
  • Perfect, so that solves the second of my questions! – bas Nov 23 '21 at 16:24
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    Can you specify how you define your almost complex structure on $TT^*S^2$ ? – Jonas Nov 23 '21 at 19:57
  • I define an almost complex structure on $T^S^2$ so that $TT^S^2$ is a complex vector bundle. The almost complex structure is defined as a the very top here, that is: for $u\in TT^M$, $J(u)$ is the unique $v$ so that $g(v,u)=\omega(v,u)$. Here $g$ is the Riemannian metric on $T^S^2$. – bas Nov 24 '21 at 05:44

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