Let $\varphi : U \rightarrow X$ be a holomorphic mapping of some open set $U\subseteq\mathbb{C}$ into a complex $n-$dimensional manifold $X$. If we know that this mapping is diffeomorphic onto its immage does it follow that it is also biholomorphic onto its image ?


1 Answer 1


Yes. Since the map is a diffeomorphism onto its image it means that the induced map on tangent spaces (thought of as $C^{\infty}$ manifolds) is an injection. But the tangent space as a complex manifold is the same space with an added complex structure, so it is still an injection. The (holomorphic) implicit function theorem then implies that the map is biholomorphic onlto its image.

  • 2
    $\begingroup$ It is even sufficient that $\varphi$ be bijective and holomorphic for it to be biholomorphic. $\endgroup$ Jun 11, 2011 at 9:12
  • 7
    $\begingroup$ @Olivier: This is true only if we assume that the image is a submanifold. Consider for instance $t\mapsto (t^2,t^3)$ from $\mathbb C$ to $\mathbb C^2$. $\endgroup$ Jun 11, 2011 at 11:58

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.