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数学代写|黎曼几何代写Riemannian geometry代考|Negative Curvature: The Hyperbolic Plane

The negative constant curvature model is the hyperbolic plane $H_{r}^{2}$ obtained as the surface of $\mathbb{R}^{3}$, endowed with the hyperbolic metric, defined as the zero level set of the function
$$a(x, y, z)=x^{2}+y^{2}-z^{2}+r^{2} .$$
Indeed, this surface is a two-fold hyperboloid, so we can restrict our attention to the set of points $H_{r}^{2}=a^{-1}(0) \cap{z>0}$.

In analogy with the positive constant curvature model (which is the set of points in $\mathbb{R}^{3}$ whose Euclidean norm is constant) the negative constant curvature model can be seen as the set of points whose hyperbolic norm is constant in $\mathbb{R}^{3}$. In other words,
$$H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{2}\right} \cap{z>0} .$$
The hyperbolic Gauss map associated with this surface can be easily computed, since it is explicitly given by
$$\mathcal{N}: H_{r}^{2} \rightarrow H^{2}, \quad \mathcal{N}(q)=\frac{1}{r} \nabla_{q} a .$$
Exercise 1.63 Prove that the Gaussian curvature of $H_{r}^{2}$ is $\kappa=-1 / r^{2}$ at every point $q \in H_{r}^{2}$.

Wé can now discuss the structure of geodesics and curves with constant geodesic curvature on the hyperbolic space. We start with a result that can be proved in an analogous way to Proposition 1.60. The proof is left to the reader.
Proposition 1.64 Iet $\gamma:[0, T] \rightarrow H_{r}^{2}$ he a curve with unit speed and constant geodesic curvature equal to $c \in \mathbb{R}$. For every vector $w \in \mathbb{R}^{3}$, the function $\alpha(t)=\langle\dot{\gamma}(t) \mid w\rangle_{h}$ is a solution of the differential equation
$$\ddot{\alpha}(t)+\left(c^{2}-\frac{1}{r^{2}}\right) \alpha(t)=0 .$$

数学代写|黎曼几何代写Riemannian geometry代考|Tangent Vectors and Vector Fields

Let $M$ be a smooth $n$-dimensional manifold and let $\gamma_{1}, \gamma_{2}: I \rightarrow M$ be two smooth curves based at $q=\gamma_{1}(0)=\gamma_{2}(0) \in M$. We say that $\gamma_{1}$ and $\gamma_{2}$ are equivalent if they have the same first-order Taylor polynomial in some (or, equivalently, in every) coordinate chart. This defines an equivalence relation on the space of smooth curves based at $q$.

Definition 2.1 Let $M$ be a smooth $n$-dimensional manifold and let $\gamma: I \rightarrow$ $M$ be a smooth curve such that $\gamma(0)=q \in M$. Its tangent vector at $q=\gamma(0)$, denoted by
$$\left.\frac{d}{d t}\right|_{t=0} \gamma(t) \quad \text { or } \quad \dot{\gamma}(0)$$
is the equivalence class in the space of all smooth curves in $M$ such that $\gamma(0)=$ $q$ (with respect to the equivalence relation defined above).

It is easy to check, using the chain rule, that this definition is well posed (i.e., it does not depend on the representative curve).

Definition 2.2 Let $M$ be a smooth $n$-dimensional manifold. The tangent space to $M$ at a point $q \in M$ is the set
$$T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \gamma(0)=q\right} .$$ It is a standard fact that $T{q} M$ has a natural structure of an $n$-dimensional vector space, where $n=\operatorname{dim} M$.

Definition 2.3 A smooth vector field on a smooth manifold $M$ is a smooth map
$$X: q \mapsto X(q) \in T_{q} M$$
that associates with every point $q$ in $M$ a tangent vector at $q$. We denote by $\operatorname{Vec}(M)$ the set of smooth vector fields on $M$.

黎曼几何代考

数学代写|黎曼几何代写Riemannian geometry代考|Negative Curvature: The Hyperbolic Plane

$$a(x, y, z)=x^{2}+y^{2}-z^{2}+r^{2} .$$

$\backslash 1 \mathrm{eft}$ 的分隔符缺失或无法识别

$$\mathcal{N}: H_{r}^{2} \rightarrow H^{2}, \quad \mathcal{N}(q)=\frac{1}{r} \nabla_{q} a .$$

$$\ddot{\alpha}(t)+\left(c^{2}-\frac{1}{r^{2}}\right) \alpha(t)=0 .$$

数学代写|黎曼几何代写Riemannian geometry代考|Tangent Vectors and Vector Fields

$$\left.\frac{d}{d t}\right|{t=0} \gamma(t) \quad \text { or } \quad \dot{\gamma}(0)$$ 是所有平滑曲线空间中的等价类 $M$ 这样 $\gamma(0)=q$ (关于上面定义的等价关系)。 使用链式法则很容易检查这个定义是否恰当（即，它不依赖于代表曲线）。 定义 $2.2$ 让 $M$ 做一个光滑的 $n$ 维流形。的切线空间 $M$ 在某一点 $q \in M$ 是集合 $\backslash 1$ eft 的分隔符缺失或无法识别 个标准的事实是 $T q M$ 有一个自然的结构 $n$ 维向量空间，其中 $n=\operatorname{dim} M$. 定义 $2.3$ 光滑流形上的光滑矢量场 $M$ 是一张光滑的地图 $$X: q \mapsto X(q) \in T{q} M$$

有限元方法代写

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MATLAB代写

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