Are there many almost complex structures on a (complex) manifold? I guess one can have many almost complex structures on a manifold, can someone give me an example? How about when the manifold is complex? is the almost complex structure induced by the complex structure the only one?
 A: Suppose $J :TM \to TM$ is an almost complex structure on $M$, and let $\varphi : TM \to TM$ be a bundle automorphism. Set $J_{\varphi} = \varphi\circ J\circ \varphi^{-1}$, then 
\begin{align*}
J_{\varphi}\circ J_{\varphi} &= \varphi\circ J\circ \varphi^{-1}\circ\varphi\circ J\circ \varphi^{-1}\\
&= \varphi\circ J\circ J\circ \varphi^{-1}\\
&= \varphi\circ (-\operatorname{id}_{TM})\circ\varphi^{-1}\\
&= -\varphi\circ\operatorname{id}_{TM}\circ\varphi^{-1}\\
&= -\operatorname{id}_{TM}
\end{align*}
where we have used the fact that $\varphi$ is linear on fibres in the penultimate equality. 
Therefore, if $M$ admits an almost complex structure $J$, every bundle automorphism $\varphi$ gives rise to another almost complex structure $J_{\varphi}$, but not every almost complex structure on $M$ arises in this way; for example, $-J$. 
Note, the previous paragraph is nothing more than a global version of the following linear algebra statement: if $J$ is a matrix which squares to $-I_n$, then any matrix which is similar to $J$ also squares to $-I_n$, but not every such matrix is similar to $J$; for example, $-J$.
A: One important example is the 1-dimensional one: For Riemann surfaces, all almost complex structures are integrable (see e.g. Theorem 11.1.6 of these notes). On the other hand, surfaces generally admit many different complex structures, and these are parametrized by Teichmüller space.
