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

也就是Uniformization theorem

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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