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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As you probably already know, a (smooth) manifold is a space locally homeomorphic to open subsets of ##\mathbb{R}^n## with smooth transition functions (and Hausdorff, second countable).

A complex manifold is defined the same way, except that it is locally homeomorphic to open subsets of ##\mathbb{C}^n## and the transition functions are holomorphic (complex differentiable). Every complex manifold can be viewed as a (real) smooth manifold, but not vice-versa even for even dimensions, since being holomorphic is a much stronger condition than being smooth.

One important property of complex manifolds is that their tangent spaces are naturally vector spaces over ##\mathbb{C}##. Suppose we want to replicate this feature in real manifolds, meaning that we want a way to view the tangent spaces of a real manifold as complex vector spaces. To view a real vector space ##V## as complex vector space, we need to specify how to scale a vector by ##i##; that is, we need to specify a linear map ##J:V\to V## such that ##J^2=-1##. Then ##V## would be a vector space with the scalar multiplication law ##(a+bi)v=av+bJv.## For (necessarily even-dimensional) manifolds, an almost complex structure is an endomorphism of the tangent bundle that squares to ##-1.## This makes each tangent space into a vector space over ##\mathbb{C}.## All complex manifolds are of course also almost-complex, but the converse is not true. Not every almost complex structure comes from a complex structure. The Nijenhuis tensor associated to an almost complex structure has to vanish for this to happen, and it is a difficult theorem that this is actually sufficient.
 
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