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 $T^{1,0}M\cap T^{0,1}M\not=(0)$?

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 $\vec 0$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.