Difference between an orbifold and a Calabi-Yau manifold?

In summary, an orbifold is a type of manifold that allows for certain symmetries and singularities, while a Calabi-Yau manifold is a special type of manifold with certain geometric properties. The exact number of Calabi-Yau manifolds that exist in a cubic meter of space is unknown, as it depends on various factors and is still an active area of research. Wikipedia and Mathworld are good resources for more information, but may not have all the latest developments in the field.
  • #1
meteor
940
0
Hi, here are a pair of questions that I can't find the answer:
What's the difference between an orbifold and a Calabi-Yau manifold?
How many Calabi-Yau manifolds there exist for cubic meter of space?
 
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  • #3
Looks like Wikipedia and Mathworld are slightly out of their depthe too, at least on orbifolds.
 
  • #4
Problem is that both sites rely on volunteers to add new material. If no one has added anyhting, you won't find it.
 

1. What is the difference between an orbifold and a Calabi-Yau manifold?

An orbifold is a special type of manifold that has singular points, while a Calabi-Yau manifold is a smooth, non-singular manifold. In other words, an orbifold has points where the geometry is not well-defined, while a Calabi-Yau manifold is smooth and well-behaved throughout.

2. How are orbifolds and Calabi-Yau manifolds used in string theory?

Calabi-Yau manifolds are used in string theory to describe the extra dimensions beyond the three spatial dimensions we experience in our everyday lives. Orbifolds are used to simplify the mathematics of Calabi-Yau manifolds and make them easier to work with.

3. Can an orbifold be transformed into a Calabi-Yau manifold?

Yes, in some cases an orbifold can be transformed into a Calabi-Yau manifold through a process called "orbifold resolution." This involves smoothing out the singular points of the orbifold to create a smooth, non-singular manifold.

4. What are the geometric differences between an orbifold and a Calabi-Yau manifold?

An orbifold has singular points with special symmetries, while a Calabi-Yau manifold is smooth and has no singular points. Additionally, the curvature of an orbifold is not constant, while a Calabi-Yau manifold has constant curvature.

5. Are all Calabi-Yau manifolds also orbifolds?

No, not all Calabi-Yau manifolds are orbifolds. Orbifolds are a special type of manifold with singular points, while Calabi-Yau manifolds can be either singular or non-singular. However, orbifolds are often used to simplify the study of Calabi-Yau manifolds in string theory.

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