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Let $E\rightarrow M$ be a real vector bundle over a real manifold $M$. By complexification of this bundle we mean the complex vector bundle $E_{\mathbb{C}}$ whose fibers are given by complexifying the corresponding fibers in $E$, i.e. $(E_\mathbb {C})_x = E_x\otimes _{\mathbb {R}}\mathbb{C} \space\space \forall x\in M$.

My question: Is the complexification same as the tensor product bundle of $E$ with the trivial bundle $M\times \mathbb{C}$? Intuitively, I think this is true since the fibers seem to be isomorphic, but not sure how to prove they are isomorphic as bundles over $M$. Any help would be appreciated.

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    $\begingroup$ Look at transition functions or at local trivialisations. $\endgroup$ – s.harp Feb 25 '17 at 15:29
  • $\begingroup$ @s.harp: I think it works out... is the answer yes? $\endgroup$ – yojusmath Feb 25 '17 at 17:43

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