Relationship between Chern and Levi-Civita Connections on Kahler Manifolds

In summary, the statement "On a complex manifold with a hermitian metric the Levi-Civita connection on the real tangent space and the Chern connection on the holomorphic tangent space coincide iff the metric is Kahler" means that there exists an isomorphism between the real and holomorphic tangent spaces that preserves the actions of the complex structures. This is achieved by extending the real connection on the real tangent space to a complex connection on the holomorphic tangent space. This complex connection will be a holomorphic connection if and only if the metric is Kahler.
  • #1
cogito²
98
0
So I'm trying to understand the statement: On a complex manifold with a hermitian metric the Levi-Civita connection on the real tangent space and the Chern connection on the holomorphic tangent space coincide iff the metric is Kahler.

I basically understand the meaning of this statement, but I'm becoming incredibly confused by the details. The isomorphism between the real tangent space and the holomorphic tangent space is basically just (as far as I've understood it):

[tex]\frac{\partial}{\partial x_i} \mapsto \frac{\partial}{\partial z_i} = \frac{1}{2}(\frac{\partial}{\partial x_i} - i\frac{\partial}{\partial y_i})[/tex]

[tex]\frac{\partial}{\partial y_i} \mapsto i\frac{\partial}{\partial z_i} = \frac{1}{2}(\frac{\partial}{\partial x_i} + i\frac{\partial}{\partial y_i})[/tex]

with the holomorphic space viewed as sitting inside the complexified tangent space.

So now what I wonder is the following. What exactly does

[tex]\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j}[/tex]

correspond to in terms of the Chern connection? Is it simply this?

[tex]\nabla_\frac{\partial}{\partial z_i} \frac{\partial}{\partial z_j}[/tex]

What I mean is that

[tex]\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j} = A_{i j}^k \frac{\partial}{\partial x_k} + B_{i j}^k \frac{\partial}{\partial y_k}[/tex]

so does this mean then that the following is true?

[tex]\nabla_\frac{\partial}{\partial z_i} \frac{\partial}{\partial z_j} = (A_{i j}^k + iB_{i j}^k)\frac{\partial}{\partial z_k}[/tex]

If that we're the case, then it would seem like

[tex]\nabla_\frac{\partial}{\partial x_i} f\frac{\partial}{\partial x_j} \neq \nabla_\frac{\partial}{\partial z_i} f\frac{\partial}{\partial z_j}[/tex]

because

[tex]\frac{\partial f}{\partial x_i} \neq \frac{\partial f}{\partial z_i}[/tex]

So I guess I'm just confused exactly what is meant in the books talking about this. I'm having trouble truly understanding the proofs I've found because I can't understand exactly what identifications are being made. If anyone can shed light on my problems or knows of a book that does this pretty explicitly I would be very thankful.
 
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  • #2
I have thought about this some more and have become convinced that the following identification is correct:

Let [tex]X[/tex] be a complex manifold and [tex](TX,J)[/tex] be the real tangent space with canonical complex structure. Then [tex](TX,J)[/tex] is isomorphic to the holomorphic tangent bundle [tex](T_X, i)[/tex] where the isomorphism connects the action of [tex]J[/tex] to that of [tex]i[/tex]. Then say you have a connection [tex]\nabla[/tex] on [tex]TX[/tex] such that [tex]\nabla J = 0[/tex]. Then you can put a real connection on [tex]T_X[/tex] which will preserve the action of [tex]i[/tex]. I.e.:

[tex]\nabla : TX \times TX \to TX[/tex]
[tex]\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j}[/tex]

corresponds to:

[tex]\nabla : TX \times T_X \to T_X[/tex]
[tex]\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial z_j}[/tex]

Then because [tex]\nabla J = 0[/tex] you get

[tex]\nabla_\frac{\partial}{\partial x_i} i\frac{\partial}{\partial z_j} = i\nabla_\frac{\partial}{\partial x_i} \frac{\partial}{\partial z_j}[/tex]

Then after that you can just extend

[tex]\nabla : TX \times T_X \to T_X[/tex]

to

[tex]\nabla : TX \otimes_\mathbb{R} \mathbb{C} \times T_X \to T_X[/tex]

linearly. Then this will be a complex connection. Now you can ask under which conditions the induced complex connection on [tex]T_X[/tex] will end up being a holomorphic connection. I've written up some fairly unilluminating conditions on the Christoffel symbols for this to be the case.

Now if you have a real metric [tex]g[/tex] on [tex]TX[/tex] and a hermitian metric [tex]h[/tex] on [tex]T_X[/tex] (and you'd probably like [tex]h[/tex] to be induced by [tex]g[/tex]), you can ask whether under this process the Levi-Civita connection will induce the Chern connection.

I'm now fairly certain I am understanding this correctly, however if any of you see that I'm still doing something wrong, I'd appreciate the input! Also I'm still interested in seeing any source that does this clearly.
 

Related to Relationship between Chern and Levi-Civita Connections on Kahler Manifolds

1. What is the relationship between the Chern and Levi-Civita connections on Kahler manifolds?

The Chern and Levi-Civita connections are both types of connections used in differential geometry to define a notion of parallel transport on a manifold. On a Kahler manifold, these two connections are related by a specific formula known as the Chern-Levi-Civita equation. This equation expresses the Chern connection in terms of the Levi-Civita connection and the complex structure of the manifold.

2. What is a Kahler manifold?

A Kahler manifold is a type of complex manifold that satisfies a set of geometric conditions, including the existence of a Riemannian metric compatible with the complex structure. This means that the manifold has both a complex and a Riemannian structure, making it a useful object for studying geometric and analytic properties of complex spaces.

3. How are the Chern and Levi-Civita connections related to curvature?

The Chern and Levi-Civita connections are both used to define notions of curvature on a manifold. Specifically, the curvature of the Chern connection is related to the curvature of the Levi-Civita connection by the Chern-Levi-Civita equation. This equation allows for the computation of the Chern curvature in terms of the Levi-Civita curvature and the complex structure of the manifold.

4. What is the significance of the relationship between the Chern and Levi-Civita connections on Kahler manifolds?

The relationship between these two connections is significant because it allows for a deeper understanding of the geometric and analytic properties of Kahler manifolds. By studying the interplay between the Chern and Levi-Civita connections, one can gain insights into the curvature, holomorphic structure, and other important features of these complex spaces.

5. How is the relationship between the Chern and Levi-Civita connections on Kahler manifolds used in mathematical research?

The relationship between these two connections has been extensively studied and used in various areas of mathematics, including complex geometry, algebraic geometry, and mathematical physics. It has applications in understanding the geometry of complex surfaces, studying the moduli space of algebraic varieties, and developing mathematical models in theoretical physics. Additionally, the Chern-Levi-Civita equation has been used to prove important theorems in differential geometry, such as the Kodaira embedding theorem and the Calabi-Yau theorem.

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