In the case of Riemannian manifolds, there are ways to take two manifolds and glue them together to get a new Riemannian manifold. For example, taking connected sums in local regions where the two metrics coincide. In the case of real hyperbolic surfaces, one can cut along geodesics and glue along these.

Complex manifolds are much more rigid objects, so I don't know if it is reasonable to expect gluing constructions to exist. Is anyone aware of such things?

I am most interested in the case of complex 2-manifolds. For example, is there a way to cut along real 3-manifolds inside a pair of complex 2-manifolds and glue to assemble a new complex 2-manifold?

I assume the integrability condition for upgrading almost-complex to complex must play a role somewhere, but I don't understand it well enough to have an intuition about how.

  • 1
    $\begingroup$ Cannot one just check that the Nijenhuis tensor vanishes? See en.wikipedia.org/wiki/…, $\endgroup$ – Igor Belegradek Jan 22 at 3:01
  • 1
    $\begingroup$ It seems as though the answer is “sometimes, and not always.” You might be interested in this question about a particular case, and some of the other questions linked there. $\endgroup$ – Santana Afton Jan 22 at 4:08

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.