# What is an (almost) complex manifold in simple words

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## Summary:

What is a complex manifold in simple words? What, in contrast, are manifolds equipped with some unusual metric (skew-symmetric parts, complex entrances)?
I try to understand (almost) complex manifolds and related stuff. Am I right that the condition for almost complexity simply is that the metric locally can be written in terms of the complex coordinates ##z##, i.e. ##g = g(z_1, .... z_m)## (complex conjugate coordinates must not appear)? These relations may involve complex constants?

If one starts from a space with real coordinates and a real metric it is not sure that the degrees of freedom can be grouped appropriately, but the other way round it is trivial, even globally (complex manifolds)?

What if I start from a space of even number of dimensions (the coordinates being real numbers), and equip it with any crazy metric which can have skew-symmetric parts and/or complex entrances? Do these manifolds have names? Can this lead to self-contradicitons? Are there examples where such structures are used?

Sorry if I may have confused a lot. But I am asking because this entire topic for me is hard to comprehend.

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