*Paula Tretkoff*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and ...
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This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.Less

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface *X* and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.

*Naichung Conan Leung and Shing‐Tung Yau*

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199534920
- eISBN:
- 9780191716010
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534920.003.0015
- Subject:
- Mathematics, Geometry / Topology

Mirror symmetry conjecture says that for any Calabi–Yau (CY) manifold M near the large complex/symplectic structure limit, there is another CY manifold X, called the mirror manifold, such that the ...
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Mirror symmetry conjecture says that for any Calabi–Yau (CY) manifold M near the large complex/symplectic structure limit, there is another CY manifold X, called the mirror manifold, such that the B-model superstring theory on M is equivalent to the A-model superstring theory on X, and vice versa. Mathematically speaking, it roughly says that the complex geometry of M is equivalent to the symplectic geometry of X, and vice versa. It is conjectured that this duality can be realized as a Fourier-type transformation along fibers of special Lagrangian fibrations on M and X, called the SYZ mirror transformation FSY Z. This chapter addresses the following two questions: (i) What is the SYZ transform of the elliptic fibration structure on M? (ii) What is the SYZ transform of the FM transform FFM cx? Section 15.2 reviews the SYZ mirror transformation and show that the mirror manifold to an elliptically fibered CY manifold has a twin Lagrangian fibration structure. Section 15.3 reviews the FM transform in complex geometry in general and also for elliptic manifolds. Section 15.4 first defines the symplectic FM transform between Lagrangian cycles on X and Y, and then defines twin Lagrangian fibrations, giving several examples of them, and studies their basic properties. Section 15.5 shows that the SYZ transformation of the complex FM transform between M and W is the symplectic FM transform between X and Y, which is actually the identity transformation.Less

Mirror symmetry conjecture says that for any Calabi–Yau (CY) manifold *M* near the large complex/symplectic structure limit, there is another CY manifold *X*, called the mirror manifold, such that the B-model superstring theory on *M* is equivalent to the A-model superstring theory on *X*, and vice versa. Mathematically speaking, it roughly says that the complex geometry of *M* is equivalent to the symplectic geometry of *X*, and vice versa. It is conjectured that this duality can be realized as a Fourier-type transformation along fibers of special Lagrangian fibrations on *M* and *X*, called the SYZ mirror transformation F^{SY Z}. This chapter addresses the following two questions: (i) What is the SYZ transform of the elliptic fibration structure on *M*? (ii) What is the SYZ transform of the FM transform F^{FM} _{cx}? Section 15.2 reviews the SYZ mirror transformation and show that the mirror manifold to an elliptically fibered CY manifold has a twin Lagrangian fibration structure. Section 15.3 reviews the FM transform in complex geometry in general and also for elliptic manifolds. Section 15.4 first defines the symplectic FM transform between Lagrangian cycles on *X* and *Y*, and then defines twin Lagrangian fibrations, giving several examples of them, and studies their basic properties. Section 15.5 shows that the SYZ transformation of the complex FM transform between *M* and *W* is the symplectic FM transform between *X* and *Y*, which is actually the identity transformation.

*Johanna Mangahas*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, ...
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This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, you can tell a group by how it acts. That is, a good group action (for example, action by isometries on a metric space) can reveal a lot about the group itself. This theme occupies a central place in geometric group theory. The ping-pong lemma, also dubbed Schottky lemma or Klein's criterion, gives a set of circumstances for identifying whether a group is a free group. The chapter first presents the statement, proof, and first examples using ping-pong before discussing ping-pong with Möbius transformations and hyperbolic geometry. Exercises and research projects are included.Less

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, you can tell a group by how it acts. That is, a good group action (for example, action by isometries on a metric space) can reveal a lot about the group itself. This theme occupies a central place in geometric group theory. The ping-pong lemma, also dubbed Schottky lemma or Klein's criterion, gives a set of circumstances for identifying whether a group is a free group. The chapter first presents the statement, proof, and first examples using ping-pong before discussing ping-pong with Möbius transformations and hyperbolic geometry. Exercises and research projects are included.