Riemann surface黎曼曲面的研究起源于Riemann， 复分析当然是18世纪和19世纪的潮流， 而Riemman师从伟大的数学家Guass， Guass对几何也深有研究，比如发现了 Guass 绝妙定理 *Theorema Egregium*

维基百科上对于这个定理是这样描述的：

## Guass绝妙定理

The theorem is”remarkable” because the beginning definition of Gaussian curvature makes immediate use of standing of the surface in distance. So it’s quite surprising that the result does not depend on its embedding regardless of all bending and twisting deformations undergone.

In modern mathematical language, the theorem can be stated as follows:

Gauss introduced the theorem in this manner (translated from Latin):

Gauss’s Theorema Egregium (Latin for”Remarkable Theorem”) is a major result of differential geometry (demonstrated by Carl Friedrich Gauss in 1827) that worries the curvature of surfaces. The theorem is that Gaussian curvature can be determined entirely by measuring distances, angles and their rates on a surface, with no reference to the particular way the surface has been embedded from the neighboring 3-dimensional Euclidean distance. To put it differently, the Gaussian curvature of a surface does not change if a person pops the surface without stretching it. Hence the Gaussian curvature is an intrinsic invariant of a surface.

The Gaussian curvature of a surface is invariant under local isometry.

Thus the formulation of the preceding article leads itself into the remarkable Theorem. In case a curved surface has been developed upon any other surface whatever, the measure of curvature in every point stays unchanged.

## Chern–Gauss–Bonnet theorem

除此之外，Guass还发现了Chern–Gauss–Bonnet theorem的原型

Riemann-Roch and Atiyah-Singer are additional generalizations of this Gauss-Bonnet theorem.

It is an extremely non-trivial generalization of this timeless Gauss–Bonnet theorem (for 2-dimensional manifolds / surfaces) to higher even-dimensional Riemannian manifolds. In 1943, Carl B. Allendoerfer and André Weil proved a special case for extrinsic manifolds. In a classic paper published in 1944, Shiing-Shen Chern demonstrated the theorem in complete generality linking worldwide topology with neighborhood geometry. [4]

In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem[1][2][3] after Shiing-Shen Chern, Carl Friedrich Gauss, along with Pierre Ossian Bonnet) says the Euler-Poincaré characteristic (a topological invariant called the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to this integral of a particular polynomial (that the Euler class) of its curvature form (an analytical invariant).

One useful form of the **Chern theorem** is that

$$

\chi(M)=\int_{M} e(\Omega)

$$

where $\chi(M)$ denotes the Euler characteristic of $M$. The Euler class is defined as

$$

e(\Omega)=\frac{1}{(2 \pi)^{n}} \operatorname{Pf}(\Omega)

$$

where we have the Pfaffian $\mathrm{Pf}(\Omega)$. Here $M$ is a compact orientable 2 n-dimensional Riemannian manifold without boundary, and $\Omega$ is the associated curvature form of the Levi-Civita connection. In fact, the statement holds with $\Omega$ the curvature form of any metric connection on the tangent bundle, as

as for other vector bundles over $M$.

also an invariant polynomial. However, Chern’s theorem in general is that for any closed $C^{\infty}$ orientable n-dimensional $M^{[5]}$ $\chi(M)=(e(T M),[M])$

where the above pairing () denotes the cap product with the Euler class of the tangent bundle TM.

## Riemman曲面的提出

而作为RH的提出者， Riemann在复分析上的造诣自然是他的老师Guass所不能及的

Known as the Riemann Zeta function. The Riemann hypothesis asserts that intriguing solutions of this equation

Some figures have the special property that they can’t be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc.. Such amounts are called prime numbers, plus they play an important part, both in pure mathematics and its programs. The distribution of such prime numbers one of all natural numbers doesn’t follow any regular pattern. However, the German mathematician G.F.B. Riemann (1826 – 1866) observed that the frequency of prime numbers is quite closely related to the behaviour of an elaborate function

It was assessed for the initial 10,000,000,000,000 solutions. A proof it is accurate for each and every intriguing solution would shed light on lots of the mysteries surrounding the distribution of prime numbers.

ζ(s) = 1 + 1/2s + 1/3s + 1/4s + …

Lie on a specific perpendicular straight line.

当然数论不是我们今天的主题，我们主要聊一聊Riemann surface， Riemann surface的动机和Riemann Geometry类似，他都是为了提出并且研究某种几何，比如微分流形是带微分结构的拓扑流形，黎曼流形是带黎曼度量的拓扑流形，而黎曼面是带复结构的一维复流形，因为他是一维的所以他某种意义上很简单，比如说我们有所谓的归一化定理

Many classical signs of the uniformization theorem rely on building a real-valued harmonic function on the simply connected Riemann surface, possibly with a singularity in one or two points and frequently corresponding to a kind of Green’s function. Four methods of constructing the harmonic function are widely employed: the Perron method; the Schwarz alternating method; Dirichlet’s principle; and Weyl’s method of orthogonal projection. These include the Beltrami equation out of Teichmüller theory and an equivalent formulation in terms of harmonic maps; Liouville’s formula , already studied by Poincaré; and Ricci flow in addition to other nonlinear flows.

The uniformization theorem is a generalization of the Riemann mapping theorem from appropriate simply connected open subsets of this airplane to arbitrary simply connected Riemann surfaces. The uniformization theorem has an equivalent statement concerning closed Riemannian 2-manifolds: each such manifold includes a conformally equivalent Riemannian metric with constant curvature.

In mathematics, the uniformization theorem claims that each just connected Riemann surface is conformally equivalent to among three Riemann surfaces: the open unit disc , the complicated plane, or the Riemann sphere. In particular it indicates that each Riemann surface admits a Riemannian metric of continuous curvature. For compact Riemann surfaces, people that have worldwide cover the unit disc are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian basic group; people with universal cover the complex plane are the Riemann surfaces of genus 1, especially the complex tori or elliptic curves with basic group Z2; and those with universal cover that the Riemann sphere are those of genus zero, namely the Riemann world itself, with trivial fundamental group.

On an oriented 2 -manifold, a Riemannian metric induces a complex structure using the passage to isothermal coordinates. If the Riemannian metric is given locally as $d s^{2}=E d x^{2}+2 F d x d y+G d y^{2}$

then in the complex coordinate $z=x+i y$, it takes the form

$$

d s^{2}=\lambda|d z+\mu d \bar{z}|^{2}

$$

where

$$

\lambda=\frac{1}{4}\left(E+G+2 \sqrt{E G-F^{2}}\right), \quad \mu=(E-G+2 i F) / 4 \lambda

$$

so that $\lambda$ and $\mu$ are smooth with $\lambda>0$ and $|\mu|<1$. In isothermal coordinates $(u, \eta)$ the metric should take the form $d s^{2}=\rho\left(d u^{2}+d v^{2}\right)$ with $\rho>0$ smooth. The complex coordinate $w=u+\mathrm{i} v$ satisfies

$$

\rho|d w|^{2}=\rho\left|w_{z}\right|^{2}\left|d z+\frac{w_{\bar{z}}}{w_{z}} d \bar{z}\right|^{2}

$$

so that the coordinates $(u, \eta)$ will be isothermal locally provided the Beltrami equation $\frac{\partial w}{\partial \bar{z}}=\mu \frac{\partial w}{\partial z}$

has a locally diffeomorphic solution, i.e. a solution with non-vanishing Jacobian.

## Riemann–Roch theorem

Riemann surface上面另外一个重要的定理就是Riemann–Roch theorem

The Riemann–Roch theorem is an important theorem in math , specifically in complex analysis and algebraic geometry, for the computation of the size of the space of meromorphic functions with prescribed zeros and enabled sticks . It relates the complex analysis of a connected compact Riemann surface using the surface’s purely topological genus gram , in a way that can be carried over into purely algebraic settings.

Originally established as Riemann’s inequality by Riemann (1857), the theorem reached its definitive form for Riemann surfaces following work of Riemann’s short lived student Gustav Roch (1865). It was later generalized to algebraic curves, to higher-dimensional varieties and outside.

The Riemann-Roch theorem for a compact Riemann surface of genus $g$ with canonical divisor $K$ states $\ell(D)-\ell(K-D)=\operatorname{deg}(D)-g+1$

Typically, the number $\ell(D)$ is the one of interest, while $\ell(K-D)$ is thought of as a correction term (also called index of speciality $[2][3]$ ) so the theorem may be roughly paraphrased by saying

dimension – correction $=$ degree $-$ genus $+1 .$

Because it is the dimension of a vector space, the correction term $\ell(K-D)$ is always non-negative, so that

$\ell(D) \geq \operatorname{deg}(D)-g+1$

This is called Riemann’s inequality. Roch’s part of the statement is the description of the possible difference between the sides of the inequality. On a general Riemann surface of genus $g$, $K$ has degree $2 g-2$, independently of the meromorphic form chosen to represent the divisor. This follows from putting $D=K$ in the theorem. In particular, as long as $D$ has degree at least $2 g-1$, the correction term is 0 , so that $\ell(D)=\operatorname{deg}(D)-g+1$

The theorem will now be illustrated for surfaces of low genus. There are also a number other closely related theorems: an equivalent formulation of this theorem using line bundles and a generalization of the theorem to algebraic curves.

Riemann-Roch定理后来被发展为代数几何中的lefschetz fixed point theorem，然后被Deligne用来证明Weil conjecture，当然这是另外一个故事了。

##### math319 Riemann surface

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黎曼曲面代写，数学代写Riemann surface请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。

Fourier analysis代写

**微分几何代写**

## 离散数学代写

Partial Differential Equations代写可以参考一份偏微分方程midterm答案解析

# Available Sections

Open

# Riemann Surfaces

Spring 2021 – MATH 234-01

### Description

Riemann surfaces, conformal maps, harmonic forms, holomorphic forms, the Reimann-Roch theorem, the theory of moduli.

### Capacity and Available Seats

**Available Seats**2**Enrollment Capacity**20**Enrolled**18

### Enrollment Requirements

Enrollment is restricted to graduate students.

### Class Details

**Career**Graduate**Grading**Student Option**Class Number**62992**Type**Lecture**Credits**5

### Meeting Information

**Days & Times**MW 3:20 PM – 4:55 PM**Room**Remote Instruction**Instructor**Sanders, B.**Meeting Dates**3/29/2021 – 6/4/2021

Riemann Surfaces (B-KUL-G0B05A)

MATH 890 – Riemann Surfaces

WISL514 2020 – OSIRIS

### 2021 Complex Analysis III

Academic unit or majorUndergraduate major in MathematicsInstructor(s)Fujikawa Ege Course component(s)Lecture Day/Period(Room No.)-Group-Course numberMTH.C331Credits2Academic year2021Offered quarter4QSyllabus updated2021/3/19Lecture notes updated-Language usedJapaneseAccess Index

### Course description and aims

The goal of this course is to outline the new epoch of classical complex analysis.At the beginning, we will introduce the hyperbolic geometry in the upper half plane. After discussing the normal family, we will show Riemann’s mapping theorem which has many applications in the complex analysis. We will explain Riemann surfaces. The theory of Riemann surfaces provides a new foundation for complex analysis on a higher level. As in elementary complex analysis, the subject matter is analytic functions. But the notion of an analytic function will have now a broader meaning as we show.

### Student learning outcomes

By the end of this course, students will be able to:

1) understand the hyperbolic geometry.

2) obtain the notion of normal family and its applications.

3) know Riemann’s mapping theorem and its applications.

4) understand Riemann surfaces.

### Keywords

Normal family, Riemann’s mapping theorem, Riemann surface.

### Competencies that will be developed

✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |

### Class flow

Standard lecture course.

### Course schedule/Required learning

Course schedule | Required learning | |
---|---|---|

Class 1 | Harmonic functions and their properties | Details will be provided during each class session. |

Class 2 | Dirichlet problem | Details will be provided during each class session. |

Class 3 | The linear fractional transformations : classification and properties | Details will be provided during each class session. |

Class 4 | Biholomorphic functions | Details will be provided during each class session. |

Class 5 | Normal family | Details will be provided during each class session. |

Class 6 | Montel’s theorem and its applications | Details will be provided during each class session. |

Class 7 | Riemann’s mapping theorem | Details will be provided during each class session. |

Class 8 | The hyperbolic plane and the Poincare disk | Details will be provided during each class session. |

Class 9 | Analytic continuation | Details will be provided during each class session. |

Class 10 | Modular function | Details will be provided during each class session. |

Class 11 | Covering and lifts of maps | Details will be provided during each class session. |

Class 12 | The definition of Riemann surfaces and a construction | Details will be provided during each class session. |

Class 13 | Functions on Riemann surfaces, degree and genus | Details will be provided during each class session. |

Class 14 | Differential form | Details will be provided during each class session. |

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.

They should do so by referring to textbooks and other course material.

### Textbook(s)

None.

### Reference books, course materials, etc.

J. Gilman, I. Kra and R. Rodriguez: Complex Analysis (Springer, GTM 245).

Junjiro Noguchi, Introduction to complex analysis, Shokabo

### Assessment criteria and methods

Final exam

### Related courses

- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed [MTH.C301 ： Complex Analysis I] and [MTH.C302 ： Complex Analysis II].

### Other

None in particular.