### 数学代写|黎曼几何代写Riemannian geometry代考|MATH3968

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

Now we state the global version of the Gauss-Bonnet theorem. In other words we want to generalize $(1.33)$ to the case when $\Gamma$ is a region of $M$ that is not

necessarily homeomorphic to a disk; see for instance Figure 1.4. As we will find, the result depends on the Euler characteristic $\chi(\Gamma)$ of this region.

In what follows, by a triangulation of $M$ we mean a decomposition of $M$ into curvilinear polygons (see Definition $1.31$ ). Notice that every compact surface admits a triangulation. 3

Definition 1.34 Let $M \subset \mathbb{R}^{3}$ be a compact oriented surface with piecewise smooth boundary $\partial M$. Consider a triangulation of $M$. We define the Euler characteristic of $M$ as
$$\chi(M):=n_{2}-n_{1}+n_{0},$$
where $n_{i}$ is the number of $i$-dimensional faces in the triangulation.
The Euler characteristic can be defined for every region $\Gamma$ of $M$ in the same way. Here, by a region $\Gamma$ on a surface $M$ we mean a closed domain of the manifold with piecewise smooth boundary.

## 数学代写|黎曼几何代写Riemannian geometry代考|Consequences of the Gauss–Bonnet Theorems

Definition $1.39$ Let $M, M^{\prime}$ be two surfaces in $\mathbb{R}^{3}$. A smooth map $\phi: \mathbb{R}^{3} \rightarrow$ $\mathbb{R}^{3}$ is called a local isometry between $M$ and $M^{\prime}$ if $\phi(M)=M^{\prime}$ and for every $q \in M$ it satisfies
$$\langle v \mid w\rangle=\left\langle D_{q} \phi(v) \mid D_{q} \phi(w)\right\rangle, \quad \forall v, w \in T_{q} M$$
If, moreover, the map $\phi$ is a bijection then $\phi$ is called a global isometry. Two surfaces $M$ and $M^{\prime}$ are said to be locally isometric (resp. globally isometric) if there exists a local isometry (resp. global isometry) between $M$ and $M^{\prime}$. Notice that the restriction $\phi$ of an isometry of $\mathbb{R}^{3}$ to a surface $M \subset \mathbb{R}^{3}$ always defines a global isometry between $M$ and $M^{\prime}=\phi(M)$.

Formula (1.52) says that a local isometry between two surfaces $M$ and $M^{\prime}$ preserves the angles between tangent vectors and, a fortiori, the lengths of curves and the distances between points.

By Corollary $1.33$, thanks to the fact that the angles and the volumes are preserved by isometries, one obtains that the Gaussian curvature is invariant under local isometries, in the following sense.

Theorem 1.40 (Gauss’ theorema egregium) Let $\phi$ be a local isometry between $M$ and $M^{\prime}$. Then for every $q \in M$ one has $\kappa(q)=\kappa^{\prime}(\phi(q))$, where $\kappa$ (resp. $\kappa^{\prime}$ ) is the Gaussian curvature of $M$ (resp. $\left.M^{\prime}\right)$.

This result says that the Gaussian curvature $\kappa$ depends only on the metric structure on $M$ and not on the specific fact that the surface is embedded in $\mathbb{R}^{3}$ with the induced inner product.

## 数学代写|黎曼几何代写Riemannian geometry代考|The Gauss Map

We end this section with a geometric characterization of the Gaussian curvature of a manifold $M$, using the Gauss map. The Gauss map is a map from the surface $M$ to the unit sphere $S^{2}$ of $\mathbb{R}^{3}$.

Definition 1.44 Let $M$ be an oriented surface. We define the Gauss map associated with $M$ as
$$\mathcal{N}: M \rightarrow S^{2}, \quad q \mapsto v_{q}$$
where $v_{q} \in S^{2} \subset \mathbb{R}^{3}$ denotes the external unit normal vestor to $M$ at $q$.
Let us consider the differential of the Gauss map at the point $q$,
$$D_{q} \mathcal{N}: T_{q} M \rightarrow T_{\mathcal{N}(q)} S^{2}$$

Notice that a tangent vector to the sphere $S^{2}$ at $\mathcal{N}(q)$ is by construction orthogonal to $\mathcal{N}(q)$. Hence it is possible to identify $T_{\mathcal{N}(q)} S^{2}$ with $T_{q} M$ and to think of the differential of the Gauss map $D_{q} \mathcal{N}$ as an endomorphism of $T_{q} M$

Theorem 1.45 Let $M$ be a surface of $\mathbb{R}^{3}$ with Gauss map $\mathcal{N}$ and Gaussian curvature к. Then
$$\kappa(q)=\operatorname{det}\left(D_{q} \mathcal{N}\right),$$
where $D_{q} \mathcal{N}$ is interpreted as an endomorphism of $T_{q} M$.
We start by proving an important property of the Gauss map.
Lemma $1.46$ For every $q \in M$, the differential $D_{q} \mathcal{N}$ of the Gauss map is a symmetric operator, i.e., it satisfies
$$\left\langle D_{q} \mathcal{N}(\xi) \mid \eta\right\rangle=\left\langle\xi \mid D_{q} \mathcal{N}(\eta)\right\rangle, \quad \forall \xi, \eta \in T_{q} M .$$
Proof The statement is local, hence it is not restrictive to assume that $M$ is parametrized by a function $\phi: \mathbb{R}^{2} \rightarrow M$. In this case $T_{q} M=\operatorname{Im} D_{u} \phi$, where $\phi(u)=q$. Let $v, w \in \mathbb{R}^{2}$ such that $\xi=D_{u} \phi(v)$ and $\eta=D_{u} \phi(w)$. Since $\mathcal{N}(q) \in T_{q} M^{\perp}$ we have
$$\langle\mathcal{N}(q) \mid \eta\rangle=\left\langle\mathcal{N}(q) \mid D_{u} \phi(w)\right\rangle=0$$

χ(米):=n2−n1+n0,

## 数学代写|黎曼几何代写Riemannian geometry代考|Consequences of the Gauss–Bonnet Theorems

⟨在∣在⟩=⟨Dqφ(在)∣Dqφ(在)⟩,∀在,在∈吨q米

## 数学代写|黎曼几何代写Riemannian geometry代考|The Gauss Map

ñ:米→小号2,q↦在q

Dqñ:吨q米→吨ñ(q)小号2

ķ(q)=这⁡(Dqñ),

⟨Dqñ(X)∣这⟩=⟨X∣Dqñ(这)⟩,∀X,这∈吨q米.

⟨ñ(q)∣这⟩=⟨ñ(q)∣D在φ(在)⟩=0

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