Welcome to MathOverflow

Question

0

1

Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the covariant derivative. What is the computation of $\nabla_{Z}Y$ (i.e. what is $\nabla_{\frac{\partial}{\partial z^{\beta}}}\frac{\partial}{\partial z^{\alpha}}$, i think its $0$ but why ?)?

edited Jun 12 2011 at 13:22

asked Jun 12 2011 at 11:11

gregor
23●3

1

You don't have a Levi-Civita covariant derivative unless you also impose a Riemannian metric. It does not come for free on a complex manifold. Your question has nothing to do with complex manifolds at all. The answer is in absolutely any text on Riemannian geometry. – Spiro Karigiannis Jun 12 2011 at 11:51

1

The again perhaps the OP does not mean the Levi-Civita connection, but only a complex connection: one relative to which the complex structure is parallel. – José Figueroa-O'Farrill Jun 12 2011 at 12:09

ok sorry the without levi civita. but does it mean that: for some complex connection one has $\nabla_{\frac{\partial}{\partial z^{\beta}}}\frac{\partial}{\partial z^{alpha}} = 0$ ? du you know some literature where i can find this ? – gregor Jun 12 2011 at 13:22

No. If $z^{\alpha}$ and $z^{\beta}$ are local holomorphic coordinates on the manifold, as I assume you mean, then you can choose a connection such that in that single coordinate chart, this condition holds, but you cannot always ensure this holds in every single coordinate chart. Because if it did, this would be a "flat" connection, and not all complex manifolds admit flat connections. – Spiro Karigiannis Jun 12 2011 at 13:32

1

Any of the three following references should give you enough background to work this out, though the answer to your question may not be spelled out in detail: [1] "Complex differential geometry" by F. Zheng, [2] "Lectures on Kahler geometry" by A. Moroianu, [3] "Complex analytic and differential geometry" by J-P. Demailly (tinyurl.com/2dcnycm). – Gunnar Magnusson Jun 12 2011 at 15:41

show 2 more comments

covariant derivative complex manifold

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.