Kahler Manifolds: Understanding Mutual Compatibility

[WWGD]

Hi, everyone:
I am doing some reading on the Frolicher Spec Seq. and I am trying to
understand better the Kahler mflds. Specifically:

What is meant by the fact that the complex structure, symplectic structure
and Riemannian structure (from being a C^oo mfld.) are "mutually compatible"?

I read this in both Griffiths and Harris and in the Wiki page.

I realized I chose a handle that does not have a good abbreviation. Choices
are:

What,
What Would,
What Would Gauss,
Do,

None of them too good.

Anyway, chronic burnout is getting to me. Thanks for any answer or suggestions
for refs.

 

[OrderOfThings]

Hi WWGD,

The compatibility beween the three structures means that

$$[\mathbf{u},\mathbf{v}] = (\mathbf{u},i\mathbf{v}),$$

where $\mathbf{u}$ and $\mathbf{v}$ are tangent vectors, [,] is the symplectic structure, (,) is the riemannian structure and $i$ is the complex structure. So, given two of the structures, the third structure is defined (if it can be defined at all) by this relation.

This also means that the riemannian and the symplectic structures are the real and imaginary parts of a hermitian structure <,>:

$$<\mathbf{u},\mathbf{v}> = (\mathbf{u},\mathbf{v}) + i [\mathbf{u},\mathbf{v}].$$

 

[WWGD]

Thanks, O.O.Things, that was helpful.

Any chance you could help me with the aspect of the "integrability condition" of the

Kahler mfld?. (from http://en.wikipedia.org/wiki/Kahler_manifold ).

Thanks.

 

[OrderOfThings]

Well, the integrability condition is that the imaginary part of the hermitian metric must be closed, which is required for it to be a symplectic structure.

 

[WWGD]

Thanks again, O.O.T. I hope it is not too much to ask for a comment on the same
entry, on the statement that the compatibility between all three structures is
equivalent to the presentation of the unitary group as:

U(n)=O(2n)/\Gl(n,C)/\Sp(2n)


as in the link above. Sorry, I don't see the relation between this presentation
of U(n) and the compatibility condition. Brother:Can you spare a paradigm?

Thanks.

 

[WWGD]

Never mind, I think I got it, thanks.