Let $M$ be a smooth manifold and $TM\otimes \mathbb C$ be its complexified tangent bundle. Is there always a complex manifold $N$ s.t $TM\otimes \mathbb C$ is its tangent bundle? If yes, is $M$ embedded as a smooth submanifold in $N$?

  • $\begingroup$ What does it mean for a bundle on $M$ to be the tangent bundle of a different manifold $N$? $\endgroup$ – Michael Albanese May 13 at 2:15

If the real dimension of $M$ is odd, so is the real dimension of $TM\otimes \mathbb{C}$.

  • $\begingroup$ $TM\otimes\mathbb{C}$ is a vector bundle over $M$ whose fiber at $x$ is $TM_x\otimes\mathbb{C}$. $\endgroup$ – Tsemo Aristide Sep 3 '17 at 13:14
  • $\begingroup$ How is Spenser's comment not right? $\endgroup$ – Ennar Sep 3 '17 at 13:26
  • $\begingroup$ look for a trivialization $(U_i)$, the restriction of $TM\otimes\mathbb{C}$ on $U_i$ is $U_i\times V$ where $V$ is the complexification of the fiber so the real dimension is $dim(M)+2dim(M)$. $\endgroup$ – Tsemo Aristide Sep 3 '17 at 13:29
  • $\begingroup$ Thanks. So what if the real dimension of $M $ is even? And i am now confused. Since if $M$ is a compact Lie group with any dimension then $M$ has a complexification! $\endgroup$ – Ronald Sep 3 '17 at 20:51
  • $\begingroup$ The real dimension of $TM\otimes \mathbb C$ is even. Do you mean the complex dimension? $\endgroup$ – user99914 Sep 3 '17 at 22:05

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