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