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Let $M$ be a smooth manifold and consider the complexified tangent bundle $TM\otimes\mathbb C$. Then there exists a complex-structure $J:TM\otimes\mathbb C\rightarrow TM\otimes\mathbb C$ with $J^2=-I$. Let $T^{1,0}M $ be the eigenspace of $i$ and $T^{0,1}M$ be the eigenspace of $-i$ (i.e $T^{0,1}M=\overline {T^{1,0}M}$).Therefore, $TM\otimes \mathbb C=T^{1,0}M+ T^{0,1}M$.

  • If $T^{1,0}M\cap T^{0,1}M=(0)$, then is trure that $M$ is an almost complex manifold?
  • What is an example where $T^{1,0}M\cap T^{0,1}M\not=(0)$?
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No and No.

  • You are basically saying that there is a complex vector bundle $TM\otimes \mathbb C$ over $M$. It says nothing about $M$. Indeed, $M$ need not be even dimensional.

  • Eigenspaces with different eigenvalues can intersect only at $\vec 0$.

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