数学代写|黎曼几何代写Riemannian geometry代考|MATH4068

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|黎曼几何代写Riemannian geometry代考|Parallel Transport and the Levi-Civita Connection

Definition $1.19$ An orientation of a surface $M$ is a smooth map $v: M \rightarrow \mathbb{R}^{3}$, defined globally on $M$, such that $v(q) \perp T_{q} M$ and $|v(q)| \equiv 1$ for every $q \in M$. Notice that if $v$ is an orientation of $M$, then $-v$ also defines an orientation of $M$.

A surface $M$ is oriented if it is given (when this exists) an orientation. On an oriented surface $M$, an orthonormal frame $\left{e_{1}, e_{2}\right}$ of $T_{q} M$ is said to be positively oriented (resp. negatively oriented) if $e_{1} \wedge e_{2}=k v(q)$ with $k>0$ (resp. $k<0$ ).
In the following we assume that $M$ is an oriented surface.
Definition 1.20 The spherical bundle $S M$ on $M$ is the disjoint union of all unit tangent vectors to $M$ :
$$S M=\bigsqcup_{q \in M} S_{q} M, \quad S_{q} M=\left{v \in T_{q} M,|v|=1\right}$$
The spherical bundle $S M$ can be endowed with the structure of a smooth manifold of dimension 3 , and more precisely of a fiber bundle with base manifold $M$, typical fiber $S^{1}$ and canonical projection
$$\pi: S M \rightarrow M, \quad \pi(v)=q \quad \text { if } v \in T_{q} M .$$

Remark $1.21$ Fix a positively oriented local orthonormal frame $\left{e_{1}(q), e_{2}(q)\right}$ on $M$. Since every vector in the fiber $S_{q} M$ has norm 1, we can write every $v \in S_{q} M$ as $v=(\operatorname{sos} \theta) e_{1}(q)+(\sin \theta) e_{2}(q) \operatorname{for} \theta \in S^{1}$.

The choice of such an orthonormal frame then induces coordinates $(q, \theta)$ on $S M$. Notice that the choice of a different positively oriented local orthonormal frame $\left{e_{1}^{\prime}(q), e_{2}^{\prime}(q)\right}$ induces coordinates $\left(q^{\prime}, \theta^{\prime}\right)$ on $S M$, where $q^{\prime}=q$ and $\theta^{\prime}=\theta+\phi(q)$ for $\phi \in C^{\infty}(M)$

The orientation of $M$ permits us, once a unit tangent vector is given, to define a canonical orthonormal frame.

数学代写|黎曼几何代写Riemannian geometry代考|Gauss–Bonnet Theorems

In this section we will prove both the local and the global version of the GaussBonnet theorem. A strong consequence of these results is the celebrated Gauss’ theorema egregium, which says that the Gaussian curvature of a surface is independent of its embedding in $\mathbb{R}^{3}$.

Definition 1.29 Let $\gamma:[0, T] \rightarrow M$ be a smooth curve parametrized by arclength. The geodesic curvature of $\gamma$ is defined as
$$\rho_{\gamma}(t)=\omega_{\gamma(t)}(\ddot{\gamma}(t)) .$$
Notice that if $\gamma$ is a geodesic, then $\rho_{\gamma}(t)=0$ for every $t \in[0, T]$. The geodesic curvature measures how far a curve is from being a geodesic.

Remark $1.30$ The geodesic curvature changes sign if we move along the curve in the opposite direction. Moreover, if $M=\mathbb{R}^{2}$, it coincides with the usual notion of the curvature of a planar curve.

数学代写|黎曼几何代写Riemannian geometry代考|Gauss–Bonnet Theorem: Local Version

A regular polygon in $\mathbb{R}^{2}$ is a polygon that is equiangular and equilateral. We include disks among regular polygons (as a limiting case, when the number of edges is infinite).

Definition 1.31 A curvilinear polygon $\Gamma$ on an oriented surface $M$ is the image of a regular polygon in $\mathbb{R}^{2}$ under a diffeomorphism. We assume that $\partial \Gamma$ is oriented consistently with the orientation of $M$.

Notice that a curvilinear polygon is always homeomorphic to a disk, and the case when $\partial \Gamma$ is smooth (and $\Gamma$ is diffeomorphic to the disk) is included in the definition.

In what follows, given a curvilinear polygon $\Gamma$ on an oriented surface $M$ (see Figure 1.2), we denote by

• $\gamma_{i}: I_{i} \rightarrow M$, for $i=1, \ldots, m$, the smooth curves parametrized by arc length, with orientation consistent with $\partial \Gamma$, such that $\partial \Gamma=\cup_{i=1}^{m} \gamma_{i}\left(I_{i}\right)$,
• $\alpha_{i}$, for $i=1, \ldots, m$, the external angles at the points where $\partial \Gamma$ is not $C^{1}$.
Theorem $1.32$ (Gauss-Bonnet, local version) Let $\Gamma$ be a curvilinear polygon on an oriented surface $M$. Then we have
$$\int_{\Gamma} \kappa d V+\sum_{i=1}^{m} \int_{I_{i}} \rho_{\gamma_{i}}(t) d t+\sum_{i=1}^{m} \alpha_{i}=2 \pi$$
Proof (a) The case where $\partial \Gamma$ is smooth. In this case $\Gamma$ is the image of the unit (closed) ball $B_{1}$, centered at the origin of $\mathbb{R}^{2}$, under a diffeomorphism
$$F: B_{1}>M, \quad \Gamma=F\left(B_{1}\right)$$
In what follows we denote by $\gamma: I \rightarrow M$ the curve such that $\gamma(I)=\partial \Gamma$. We consider on $B_{1}$ the vector field $V(x)=x_{1} \partial_{x 2}-x_{2} \partial_{x 1}$ which has an isolated zero at the origin and whose flow is a rotation around zero. Denote by $X:=F_{*} V$ the induced vector field on $M$ with critical point $q_{0}=F(0)$.
For $\varepsilon>0$ small enough, we define (see Figure 1.3)
$$\Gamma_{\varepsilon}:=\Gamma \backslash F\left(B_{\varepsilon}\right) \quad \text { and } \quad A_{\varepsilon}:=\partial F\left(B_{\varepsilon}\right)$$

数学代写|黎曼几何代写Riemannian geometry代考|Parallel Transport and the Levi-Civita Connection

S M=\bigsqcup_{q \in M} S_{q} M, \quad S_{q} M=\left{v \in T_{q} M,|v|=1\right}S M=\bigsqcup_{q \in M} S_{q} M, \quad S_{q} M=\left{v \in T_{q} M,|v|=1\right}

数学代写|黎曼几何代写Riemannian geometry代考|Gauss–Bonnet Theorems

ρC(吨)=ωC(吨)(C¨(吨)).

数学代写|黎曼几何代写Riemannian geometry代考|Gauss–Bonnet Theorem: Local Version

• C一世:我一世→米， 为了一世=1,…,米，由弧长参数化的平滑曲线，方向与∂Γ, 这样∂Γ=∪一世=1米C一世(我一世),
• 一个一世， 为了一世=1,…,米, 点的外角∂Γ不是C1.
定理1.32（Gauss-Bonnet，本地版本）让Γ是有向曲面上的曲线多边形米. 然后我们有
∫Γķd在+∑一世=1米∫我一世ρC一世(吨)d吨+∑一世=1米一个一世=2圆周率
证明 (a) 情况∂Γ是光滑的。在这种情况下Γ是单位（封闭）球的形象乙1, 以原点为中心R2, 在微分同胚下
F:乙1>米,Γ=F(乙1)
下面我们用C:我→米曲线使得C(我)=∂Γ. 我们考虑乙1向量场在(X)=X1∂X2−X2∂X1它在原点有一个孤立的零，其流动是围绕零旋转。表示为X:=F∗在上的诱导矢量场米有临界点q0=F(0).
为了e>0足够小，我们定义（见图 1.3）
Γe:=Γ∖F(乙e) 和 一个e:=∂F(乙e)

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