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?
[Math] Are there many almost complex structures on a (complex) manifold
almost-complexcomplex-geometry
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The point is that $V = \mathbb{R}^3 = \operatorname{Im}\mathbb{H}$ and $V = \mathbb{R}^7 = \operatorname{Im}\mathbb{O}$ inherit a cross-product $V \times V \to V$ from quaternion and octonion multiplication. It is given by $u \times v = \operatorname{Im}(uv) = \frac{1}{2}(uv - vu)$.
Given an oriented hypersurface $\Sigma \subset V$ with corresponding Gauß map $\nu \colon \Sigma \to S^n$ ($n \in 2,6$) sending a point $x \in \Sigma$ to the outer unit normal $\nu(x) \perp T_x\Sigma$ one obtains an almost complex structure by setting $$J_x(u) = \nu(x) \times u\qquad\text{for }u \in T_x\Sigma.$$
One economic way to see that this is an almost complex structure: The cross-product is bilinear and antisymmetric. It is related to the standard scalar product via $$ \langle u \times v, w\rangle = \langle u, v \times w\rangle $$ (which shows that $u\times v$ is orthogonal to both $u$ and $v$) and the Graßmann identity $u \times (v \times w) = \langle u,w\rangle v - \langle u,v\rangle w$ (only valid in dimension $3$) has the variant $$ (u \times v) \times w + u \times (v\times w) = 2\langle u,w\rangle v-\langle v,w\rangle u - \langle v,u\rangle w $$ valid in $3$ and $7$ dimensions (which shows that $u \times (u \times v) = -v$ for $u \perp v$ and $|u| = 1$).
Therefore $J_x(u) = \nu(x) \times u$ indeed defines an almost complex structure $J_x \colon T_x\Sigma \to T_x\Sigma$.
Specializing this to $\Sigma = S^2$ embedded as unit sphere in $\operatorname{Im}\mathbb{H}$ you can “see” (or calculate) that $J_x$ acts in exactly the same way as multiplication by $i$ on $\mathbb{CP}^1$.
The classic example of an almost complex manifold that is not a complex manifold is the six-sphere $S^6$. Consider $S^6$ in $\mathbb{R}^7 = \mathrm{im}\,\mathbb{O}$ as the set of unit norm imaginary octonions. The almost complex structure on $S^6$ is defined by $J_p v = p \times v$, where $p\in S^6$ and $v\in T_p S^6$ and $\times$ stands for the vector product on $\mathbb{R}^7$. This almost complex structure cannot be induced by a complex atlas on $S^6$ because the Nijenhuis tensor $N_J$ doesn't vanish (cf. the Nirenberg Newlander theorem).
A result of Borel-Serre states that the only spheres endowed with an almost complex structure are $S^2$ and $S^6$. (The one on $S^2$ is a complex structure.) Up to now, it is not known whether there exists a complex structure on $S^6$.
As a reference I mention the Wikipedia page on almost complex manifolds.
Best Answer
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, $-J$.