Diff Geom in Complex Spaces: Hermitian, Anti-Symmetric & Affine Connections

In summary, the conversation is about finding resources for understanding differential geometry for 2-dimensional complex spaces, specifically in regards to Hermitian vs anti-symmetric metric tensors, dealing with complex conjugates, and defining affine connections. The person is also interested in how general relativity would need to be modified for complex-valued vectors. A suggestion is made to try math.stackexchange.com for further assistance.
  • #1
thehangedman
69
2
Are there any good papers or books that go over our current understanding of differential geometry for 2-dimensional complex spaces? Hermitian vs anti-symmetric metric tensors, dealing with complex conjugates, and defining affine connections?

Yes, I've already hit up Google, so I was hoping an expect on this forum might know a good book or paper to start, perhaps someone who already knows this topic rather well?

I am also interested in how the current formalism of general relativity would need to be modified if it were to deal with complex-valued vectors rather than just real-valued vectors.

Thanks!
 
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  • #2
You might want to try math.stackexchange.com.
 

Related to Diff Geom in Complex Spaces: Hermitian, Anti-Symmetric & Affine Connections

1. What is the difference between a Hermitian connection and an anti-symmetric connection?

A Hermitian connection is a type of connection on a complex manifold that preserves the Hermitian inner product structure. It is compatible with the complex structure and preserves the Hermitian metric. On the other hand, an anti-symmetric connection is a connection that satisfies the anti-symmetry condition, meaning that the torsion tensor of the connection is skew-symmetric. This means that the connection preserves the complex structure, but does not necessarily preserve the Hermitian metric.

2. How do affine connections relate to complex spaces?

An affine connection is a generalization of the notion of a connection on a vector bundle. It is defined on a manifold and assigns to each tangent space a linear connection, which in turn gives a way to differentiate vector fields. In the case of complex spaces, affine connections are used to study the geometry of complex manifolds and their associated vector bundles. These connections are compatible with the complex structure and provide a way to differentiate complex-valued functions and vector fields.

3. What is the role of Hermitian connections in complex geometry?

Hermitian connections play a crucial role in complex geometry as they are used to define the curvature of a complex manifold. This curvature is a measure of how much the manifold deviates from being flat, and it encodes important information about the geometry of the manifold. Hermitian connections also allow us to define holomorphic vector bundles and study their properties, such as holomorphic sections and holomorphic curvature.

4. How are Hermitian connections and anti-symmetric connections related?

Hermitian connections and anti-symmetric connections are related in that they both preserve the complex structure of a manifold. In fact, a Hermitian connection can be decomposed into its anti-symmetric and symmetric parts. The symmetric part of a Hermitian connection is called the Levi-Civita connection and plays a key role in Riemannian geometry. The anti-symmetric part is related to the torsion tensor and encodes information about how the Hermitian metric is preserved under parallel transport.

5. What are some applications of complex geometry in physics?

Complex geometry has many applications in physics, particularly in the study of quantum mechanics and string theory. In quantum mechanics, complex geometry is used to study the properties of particles and their interactions, while in string theory, it is used to understand the behavior of strings in higher-dimensional spaces. Complex geometry is also used in the study of supersymmetric field theories and their related symmetries. Additionally, complex geometry plays a crucial role in the study of mirror symmetry, a phenomenon that relates different Calabi-Yau manifolds and has implications for the study of black holes and the structure of the universe.

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