## Calabi-Yau Manifold

### Contents

Introduction
Calabi-Yau Manifold for Dummies
Topology,
Chern Class,
Curvature,
Riemann Surface and Complex Manifolds,
Kahler Manifold,
Calabi's Conjecture,
Calabi-Yau Manifold
Theory of Superstring, and M Theory
Extra Dimensions and Supersymmetry,
Compactification,
Types of Superstring Theory, and M Theory,
Modulus, Flux, Brane, and Vacuum Energy
The Connections
End of the Universe
Testing the Superstring Theory
Apparent and Real Problems

### Introduction

This is a summary on "The Shape of Inner Space" about the Calabi-Yau Manifold by Shing-Tung Yau and Steve Nadis published in 2010. It is for novice who find it difficult to digest the material, and for those who do not want to read the whole book - something like a glorified "Book Review". Manifold can be in one, two, three, ... dimensional space with the special property that it looks flat (Euclidean) in small scale but becomes non-Euclidean (curved) globally as shown as Figure 01. Such property implies smoothness. However, a manifold is characterized as smooth even if it has some points with singularity, e.g., a shape point. The dimension of complex manifold involves real and imaginary axes. Since complex number always shows up in pair such as (x,iy), complex manifold always has even dimension. Riemannian geometry is a generalization of the Euclidean geometry to include manifolds with curvature. It prescribes mathematical formulas to calculate values of angle, length of curves, surface area, and volume. All the information are contained within the metric tensors gij as demonstrated by a very simple example in Eqs.(2)-(6).

#### Figure 01 2-Dimensional Manifold [view large image]

There is a special kind of Riemann surface, which possesses the property that a projection from one surface onto another preserves the angles between things even though the distance may be distorted. This angle-preserving characteristic is known as conformal mapping, which can simplify
calculation involving Riemann surfaces. Riemann surface can be orientable, meaning that the measurement of direction remains consistent, or otherwise such as the Mobius strip where directions are reversed by going through a loop back to the original spot.  Eugenio Calabi (Figure 02) was born Jewish Italian in 1923. He is American mathematician and professor emeritus at the University of Pennsylvania since 1964, specializing in differential geometry, partial differential equations and their applications. In 1953 he raised the Calabi's conjecture, which proposed that certain specific geometric structures are allowed under some topological condition (briefly, topology concerns with the overall shape, while geometry are often related to the exact shape and curvature of some object).
Shing-Tung Yau ( , Figure 03) was born Chinese in 1949. He has been a professor of mathematics at Harvard since 1987 and is the current department chair. In 1976, he proves the existence of the geometric structure as surmised by the Calabi's conjecture. Such structure is now called Calabi-Yau manifold. Its special properties are indispensable for compactification in Superstring Theory.

#### Figure 03 Shing-Tung Yau Yau's perception of the conjecture is somewhat different but equivalent. It asks if there could be gravity even when the space contains no matter - in analogy to the Schwarzschild's solution in vacuum but more general.

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