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## 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 vector 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 $\kappa$. 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.

## 数学代写|黎曼几何代写Riemannian geometry代考|Surfaces in R3 with the Minkowski Inner Product

The theory and the results obtained in this chapter can be adapted to the case when $M \subset \mathbb{R}^{3}$ is a surface in Minkowski 3-space, i.e., $\mathbb{R}^{3}$ endowed with the hyperbolic (or pseudo-Euclidean) inner product
$$\left\langle q_{1}, q_{2}\right\rangle_{h}=x_{1} x_{2}+y_{1} y_{2}-z_{1} z_{2}$$

Here $q_{i}=\left(x_{i}, y_{i}, z_{i}\right)$ for $i=1,2$ are two points in $\mathbb{R}^{3}$. When $\langle q, q\rangle_{h} \geq 0$, we denote by $|q|_{h}=\langle q, q\rangle_{h}^{1 / 2}$ the length of the vector induced by the inner product (1.65).

For the metric structure to be well defined on $M$, we require that the restriction of the inner product (1.65) to the tangent space to $M$ is positive definite at every point. Indeed, under this assumption, the inner product (1.65) can be used to define the length of a tangent vector to the surface (which is nonnegative). Thus one can introduce the length of a (piecewise) smooth curve on $M$ and its distance by the same formulas as in Section 1.1. These surfaces are also called space-like surfaces in Minkowski space.

The structure of the inner product imposes some conditions on the structure of space-like surfaces, as the following exercise shows.

Exercise 1.50 Let $M$ be a space-like surface in $\mathbb{R}^{3}$ endowed with the inner product (1.65).
(i) Show that if $v \in T_{q} M$ is a nonzero vector that is orthogonal to $T_{q} M$ then $\langle v, v\rangle_{h}<0$.
(ii) Prove that if $M$ is compact then $\partial M \neq \emptyset$.
(iii) Show that the restriction to $M$ of the projection $\pi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ defined by $\pi(x, y, z)=(x, y)$ is a local diffeomorphism.
(iv) Show that $M$ is locally a graph, i.e., for every point in $M$ there exists $U \subset \mathbb{R}^{3}$ such that $M \cap U={(x, y, z) \mid z=f(x, y)}$, for a suitable smooth function $f$.

# 黎曼几何代考

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

$$\mathcal{N}: M \rightarrow S^{2}, \quad q \mapsto v_{q}$$

$$D_{q} \mathcal{N}: T_{q} M \rightarrow T_{\mathcal{N}(q)} S^{2}$$

$$\kappa(q)=\operatorname{det}\left(D_{q} \mathcal{N}\right),$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Surfaces in R3 with the Minkowski Inner Product

$$\left\langle q_{1}, q_{2}\right\rangle_{h}=x_{1} x_{2}+y_{1} y_{2}-z_{1} z_{2}$$

(i) 证明如果 $v \in T_{q} M$ 是一个非零向量，与 $T_{q} M$ 然后 $\langle v, v\rangle_{h}<0$.
(ii) 证明如果 $M$ 那么䋈凑 $\partial M \neq \emptyset$.
(iii) 表明限制 $M$ 投影的 $\pi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ 被定义为 $\pi(x, y, z)=(x, y)$ 是同部铂分同脴。
(iv) 证明 $M$ 是同部的图，即，对于 $M$ 那里存在 $U \subset \mathbb{R}^{3}$ 这样 $M \cap U=(x, y, z) \mid z=f(x, y)$, 对于一个合适的平滑函数 $f$.

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