### 数学代写|黎曼几何代写Riemannian geometry代考|MATH6205

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## 数学代写|黎曼几何代写Riemannian geometry代考|Geodesics and Optimality

Let $M \subset \mathbb{R}^{3}$ be a surface and $\gamma:[0, T] \rightarrow M$ be a smooth curve in $M$. The length of $\gamma$ is defined as
$$\ell(\gamma):=\int_{0}^{T}|\dot{\gamma}(t)| d t,$$
where $|v|=\sqrt{\langle v \mid v\rangle}$ denotes the norm of a vector $v$ in $\mathbb{R}^{3}$.
Notice that the definition of length in (1.1) is invariant under reparametrizations of the curve. Indeed, let $\varphi:\left[0, T^{\prime}\right] \rightarrow[0, T]$ be a smooth monotonic function. Define $\gamma_{\varphi}:\left[0, T^{\prime}\right] \rightarrow M$ by $\gamma_{\varphi}:=\gamma \circ \varphi$. Using the change of variables $t=\varphi(s)$, one gets
$$\ell\left(\gamma_{\varphi}\right)=\int_{0}^{T^{\prime}}\left|\dot{\gamma}{\varphi}(s)\right| d s=\int{0}^{T^{\prime}}|\dot{\gamma}(\varphi(s))||\dot{\varphi}(s)| d s=\int_{0}^{T}|\dot{\gamma}(t)| d t=\ell(\gamma) .$$
The definition of length can be extended to piecewise-smooth curves on $M$ by adding the length of every smooth piece of $\gamma$.

When the curve $\gamma$ is parametrized in such a way that $|\dot{\gamma}(t)| \equiv c$ for some $c>0$ we say that $\gamma$ has constant speed. If moreover $c=1$, we say that $\gamma$ is parametrized by arclength (or arclength parametrized).

The distance between two points $p, q \in M$ is the infimum of the lengths of curves that join $p$ to $q$ :
$d(p, q)=\inf {\ell(\gamma) \mid \gamma:[0, T] \rightarrow M$ piecewise-smooth, $\gamma(0)=p, \gamma(T)=q}$
Now we focus on length-minimizers, i.e., piecewise-smooth curves $\gamma:[0, T]$ $\rightarrow M$ realizing the distance between their endpoints, i.e., satisfying $\ell(\gamma)=$ $d(\gamma(0), \gamma(T))$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Existence and Minimizing Properties of Geodesics

As a direct consequence of Proposition $1.8$ one obtains the following existence and uniqueness theorem for geodesics.

Corollary 1.10 Let $q \in M$ and $v \in T_{q} M$. There exists a unique geodesic $\gamma:[0, \varepsilon] \rightarrow M$, for $\varepsilon>0$ small enough, such that $\gamma(0)=q$ and $\dot{\gamma}(0)=v$.
Proof By Proposition 1.8, geodesics satisfy a second-order ordinary differential equation (ODE), hence they are smooth curves characterized by their initial position and velocity.

To end this section we show that small pieces of geodesics are always global minimizers.

Theorem 1.11 Let $\gamma:[0, T] \rightarrow M$ be a geodesic. For every $\tau \in[0, T[$ there exists $\varepsilon>0$ such that
(i) $\left.\gamma\right|{[\tau, \tau+\varepsilon]}$ is a minimizer, i.e., $d(\gamma(\tau), \gamma(\tau+\varepsilon))=\ell\left(\left.\gamma\right|{[\tau, \tau+\varepsilon]}\right)$,
(ii) $\left.\gamma\right|_{[\tau, \tau+\varepsilon]}$ is the unique minimizer joining $\gamma(\tau)$ and $\gamma(\tau+\varepsilon)$ in the class of piecewise-smooth curves, up to reparametrization.

Proof Without loss of generality let us assume that $\tau=0$ and that $\gamma$ is arclength parametrized. Consider an arclength parametrized curve $\alpha$ on $M$, such that $\alpha(0)=\gamma(0)$ and $\dot{\alpha}(0) \perp \dot{\gamma}(0)$, and denote by $(t, s) \mapsto x_{s}(t)$ a smooth variation of geodesics such that $x_{0}(t)=\gamma(t)$ and (see also Figure 1.1)
$$x_{s}(0)=\alpha(s), \quad \dot{x}{s}(0) \perp \frac{\partial}{\partial s} \alpha(s) .$$ The map $\psi:(t, s) \mapsto x{s}(t)$ is smooth and is a local diffeomorphism near $(0,0)$. Indeed, we can compute the partial derivatives
$$\left.\frac{\partial \psi}{\partial t}\right|{t=s=0}=\left.\frac{\partial}{\partial t}\right|{t=0} x_{0}(t)=\dot{\gamma}(0),\left.\quad \frac{\partial \psi}{\partial s}\right|{t=s=0}=\left.\frac{\partial}{\partial s}\right|{s=0} x_{s}(0)=\dot{\alpha}(0),$$

and they are linearly independent. Thus $\psi$ maps a neighborhood $U$ of $(0,0)$ to a neighbōthood $W$ of $\gamma(0)$. We now considèr a function $\phi$ and a vecctor field $X$ defined on $W$ by
$$\phi: x_{s}(t) \mapsto t, \quad X: x_{s}(t) \mapsto \dot{x}_{s}(t)$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Absolutely Continuous Curves

Notice that formula (1.1) defines the length of a curve even if $\gamma$ is only absolutely continuous, if one interprets the integral in the Lebesgue sense (recall that absolutely continuous curves are differentiable almost everywhere).

The proof of Theorem $1.11$, and in particular estimates (1.20) and (1.21), can be extended to the class of absolutely continuous curves. This proves that small pieces of geodesics are also minimizers in the larger class of absolutely continuous curves on $M$. As a byproduct, we have the following corollary.
Corollary $1.13$ Any length-minimizer (in the class of absolutely continuous curves) is a geodesic, and hence smooth.

In this section we want to introduce the notion of parallel transport on a surface (along a curve), which allows us to define its main geometric invariant: the Gaussian curvature.

Definition 1.14 Let $\gamma:[0, T] \rightarrow M$ be a smooth curve. A smooth curve of tangent vectors $\xi(t) \in T_{\gamma(t)} M$ is said to be parallel if $\dot{\xi}(t) \perp T_{\gamma(t)} M$.

This notion generalizes the notion of parallelism of vectors on the plane, where it is possible to canonically identify every tangent space to $M=\mathbb{R}^{2}$ with $\mathbb{R}^{2}$ itself. ${ }^{2}$ In this case a smooth curve of tangent vectors $\xi(t) \in T_{\gamma(t)} M$ is parallel if and only if $\dot{\xi}(t)=0$.

When $M$ is the zero level of a smooth function $a: \mathbb{R}^{3} \rightarrow \mathbb{R}$, as in (1.14), we have the following description:

Proposition $1.15$ A smooth curve of tangent vectors $\xi(t)$ defined along $\gamma:[0, T] \rightarrow M$ is parallel if and only if it satisfies
$$\dot{\xi}(t)=-\frac{\dot{\gamma}(t)^{T}\left(\nabla_{\gamma(t)}^{2} a\right) \xi(t)}{\left|\nabla_{\gamma(t)} a\right|^{2}} \nabla_{\gamma(t)} a, \quad \forall t \in[0, T] .$$
Proof As in Remark 1.7, $\xi(t) \in T_{\gamma(t)} M$ implies that $\left\langle\nabla_{\gamma(t)} a, \xi(t)\right\rangle=0$. Moreover, by assumption, $\dot{\xi}(t)=\alpha(t) \nabla_{\gamma(t)} a$ for some smooth function $\alpha$. With computations analogous to those in the proof of Proposition $1.8$ we get that
$$\dot{\gamma}(t)^{T}\left(\nabla_{\gamma(t)}^{2} a\right) \xi(t)+\alpha(t)\left|\nabla_{\gamma(t)} a\right|^{2}=0,$$
from which the statement follows.

## 数学代写|黎曼几何代写Riemannian geometry代考|Geodesics and Optimality

ℓ(C):=∫0吨|C˙(吨)|d吨,

$$\ell\left(\gamma_{\varphi}\right)=\int_{0}^{T^{\prime}}\left|\dot{\gamma} {\varphi}(s) \对| ds=\int {0}^{T^{\prime}}|\dot{\gamma}(\varphi(s))||\dot{\varphi}(s)| ds=\int_{0}^{T}|\dot{\gamma}(t)| dt=\ell(\gamma) 。$$

d(p,q)=信息ℓ(C)∣C:[0,吨]→米$p一世和C和在一世s和−s米○○吨H,$C(0)=p,C(吨)=q

## 数学代写|黎曼几何代写Riemannian geometry代考|Existence and Minimizing Properties of Geodesics

(i)C|[τ,τ+e]是一个极小值，即d(C(τ),C(τ+e))=ℓ(C|[τ,τ+e]),
(ii)C|[τ,τ+e]是唯一的最小化器加入C(τ)和C(τ+e)在分段平滑曲线的类中，直到重新参数化。

Xs(0)=一个(s),X˙s(0)⊥∂∂s一个(s).地图ψ:(吨,s)↦Xs(吨)是光滑的并且是附近的局部微分同胚(0,0). 事实上，我们可以计算偏导数

∂ψ∂吨|吨=s=0=∂∂吨|吨=0X0(吨)=C˙(0),∂ψ∂s|吨=s=0=∂∂s|s=0Xs(0)=一个˙(0),

φ:Xs(吨)↦吨,X:Xs(吨)↦X˙s(吨)

## 数学代写|黎曼几何代写Riemannian geometry代考|Absolutely Continuous Curves

X˙(吨)=−C˙(吨)吨(∇C(吨)2一个)X(吨)|∇C(吨)一个|2∇C(吨)一个,∀吨∈[0,吨].

C˙(吨)吨(∇C(吨)2一个)X(吨)+一个(吨)|∇C(吨)一个|2=0,

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