Kahler Manifolds: Understanding Mutual Compatibility

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In summary, the three structures are mutually compatible, and the unitary group is presented as a way of describing the compatibility condition.
  • #1
WWGD
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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.
 
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  • #2
Hi WWGD,

The compatibility beween the three structures means that

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

where [itex]\mathbf{u}[/itex] and [itex]\mathbf{v}[/itex] are tangent vectors, [,] is the symplectic structure, (,) is the riemannian structure and [itex]i[/itex] 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 <,>:

[tex]<\mathbf{u},\mathbf{v}> = (\mathbf{u},\mathbf{v}) + i [\mathbf{u},\mathbf{v}].[/tex]
 
Last edited:
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
Never mind, I think I got it, thanks.
 

Related to Kahler Manifolds: Understanding Mutual Compatibility

1. What is the definition of a Kahler manifold?

A Kahler manifold is a type of complex manifold that has both a Riemannian metric and a complex structure. This means that it is a smooth manifold with a specific type of geometric structure that allows for complex analysis to be applied.

2. What is the significance of mutual compatibility in Kahler manifolds?

Mutual compatibility refers to the relationship between the Riemannian metric and the complex structure in a Kahler manifold. In order for a manifold to be considered Kahler, these two structures must be compatible with each other, meaning they can be used together to define a unique geometric structure.

3. How is the complex structure defined in a Kahler manifold?

The complex structure in a Kahler manifold is defined by a set of complex-valued functions that satisfy certain equations known as the Cauchy-Riemann equations. These equations ensure that the complex structure is compatible with the Riemannian metric and allows for complex analysis to be applied.

4. What is the relationship between Kahler manifolds and symplectic manifolds?

Kahler manifolds are a special type of complex manifold that also have a symplectic structure. In fact, any Kahler manifold can be seen as a symplectic manifold with additional geometric structure. This relationship allows for connections to be made between complex and symplectic geometry.

5. What are some applications of Kahler manifolds?

Kahler manifolds have many applications in mathematics and physics, particularly in areas such as symplectic geometry, complex analysis, and algebraic geometry. They are also used in string theory, which is a theoretical framework for understanding the fundamental structure of the universe.

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