### 数学代写|黎曼几何代写Riemannian geometry代考|MATH3342

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## 数学代写|黎曼几何代写Riemannian geometry代考|The Minimal Control and the Length of an Admissible Curve

We start by defining the sub-Riemannian norm for vectors that belong to the distribution of a sub-Riemannian manifold.

Definition 3.8 Let $v \in \mathcal{D}{q}$. We define the sub-Riemannian norm of $v$ as follows: $$|v|:=\min \left{|u|, u \in U{q} \text { s.t. } v=f(q, u)\right} .$$
Notice that since $f$ is linear with respect to $u$, the minimum in $(3.9)$ is always attained at a unique point. Indeed, the condition $f(q, \cdot)=v$ defines an affine subspace of $U_{q}$ (which is nonempty since $v \in \mathcal{D}_{q}$ ) and the minimum in (3.9) is uniquely attained at the orthogonal projection of the origin onto this subspace (see Figure 3.2).

Exercise 3.9 Show that $|\cdot|$ is a norm in $\mathcal{D}{q}$. Moreover prove that it satisfies the parallelogram law, i.e., it is induced by a scalar product $\langle\cdot \mid \cdot\rangle{q}$ on $\mathcal{D}{q}$ that can be recovered by the polarization identity $$\langle v \mid w\rangle{q}=\frac{1}{4}|v+w|^{2}-\frac{1}{4}|v-w|^{2}, \quad v, w \in \mathcal{D}{q} .$$ Exercise $3.10$ Let $u{1}, \ldots, u_{m} \in U_{q}$ be an orthonormal basis for $U_{q}$. Define $v_{i}=f\left(q, u_{i}\right)$. Show that if $f(q, \cdot)$ is injective then $v_{1}, \ldots, v_{m}$ is an orthonormal basis for $\mathcal{D}_{q}$.

An admissible curve $\gamma:[0, T] \rightarrow M$ is Lipschitz, hence differentiable at almost every point. Hence the unique control $t \mapsto u^{*}(t)$ associated with $\gamma$ and realizing the minimum in $(3.9)$ is well defined a.e. on $[0, T]$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Equivalence of Sub-Riemannian Structures

In this section we introduce the notion of the equivalence of sub-Riemannian structures on the same base manifold $M$ and the notion of isometry between sub-Riemannian manifolds.

Definition $3.18$ Let $(\mathbf{U}, f),\left(\mathbf{U}^{\prime}, f^{\prime}\right)$ be two sub-Riemannian structures on a smooth manifold $M$. They are said to be equivalent as distributions if the following conditions hold:

(i) there exist a Euclidean bundle $\mathbf{V}$ and two surjective vector bundle morphisms $p: \mathbf{V} \rightarrow \mathbf{U}$ and $p^{\prime}: \mathbf{V} \rightarrow \mathbf{U}^{\prime}$ such that the following diagram is commutative:

The structures $(\mathbf{U}, f)$ and $\left(\mathbf{U}^{\prime}, f^{\prime}\right)$ are said to be equivalent as sub-Riemannian structures (or simply equivalent) if (i) is satisfied and moreover
(ii) the projections $p, p^{\prime}$ are compatible with the scalar product, i.e., it holds that
\begin{aligned} |u| &=\min {|v|, p(v)=u}, & \forall u \in \mathbf{U}, \ \left|u^{\prime}\right| &=\min \left{|v|, p^{\prime}(v)=u^{\prime}\right}, & \forall u^{\prime} \in \mathbf{U}^{\prime} . \end{aligned}
Remark $3.19$ If $(\mathbf{U}, f)$ and $\left(\mathbf{U}^{\prime}, f^{\prime}\right)$ arc cquivalcnt as sub-Ricmannian structures on $M$ then:
(a) the distributions $\mathcal{D}{q}$ and $\mathcal{D}{q}^{\prime}$ defined by $f$ and $f^{\prime}$ coincide, since $f\left(U_{q}\right)=f^{\prime}\left(U_{q}^{\prime}\right)$ for all $q \in M$;
(b) for each $w \in \mathcal{D}_{q}$ we have $|w|=|w|^{\prime}$, where $|\cdot|$ and $|\cdot|^{\prime}$ are the norms induced by $(\mathbf{U}, f)$ and $\left(\mathbf{U}^{\prime}, f^{\prime}\right)$ respectively.
In particular the lengths of admissible curves for two equivalent subRiemannian structures are the same.

Exercise 3.20 Prove that $(M, \mathbf{U}, f)$ and $\left(M, \mathbf{U}^{\prime}, f^{\prime}\right)$ are equivalent as distributions if and only if the moduli of the horizontal vector fields $\mathcal{D}$ and $\mathcal{D}^{\prime}$ coincide.

## 数学代写|黎曼几何代写Riemannian geometry代考|Sub-Riemannian Distance

In this section we introduce the sub-Riemannian distance and prove the Rashevskii-Chow theorem.

Recall that, thanks to the results of Section 3.1.4, in what follows we can assume that the sub-Riemannian structure on $M$ is free, with generating family $\mathcal{F}=\left{f_{1}, \ldots, f_{m}\right}$. Notice that, by the definition of a sub-Riemannian manifold, $M$ is assumed to be connected and $\mathcal{F}$ is assumed to be bracketgenerating.

Definition 3.30 Let $M$ be a sub-Riemannian manifold and $q_{0}, q_{1} \in M$. The sub-Riemannian distance (or Carnot-Carathéodory distance) between $q_{0}$ and $q_{1}$ is
$d\left(q_{0}, q_{1}\right)=\inf \left{\ell(\gamma) \mid \gamma:[0, T] \rightarrow M\right.$ admissible, $\left.\gamma(0)=q_{0}, \gamma(T)=q_{1}\right} .$
We now state the main result of this section.
Theorem $3.31$ (Rashevskii-Chow) Let $M$ be a sub-Riemannian manifold. Then
(i) $(M, d)$ is a metric space,
(ii) the topology induced by $(M, d)$ is equivalent to the manifold topology.
In particular, $d: M \times M \rightarrow \mathbb{R}$ is continuous.
One of the main consequences of this result is that, thanks to the bracketgenerating condition, for every $q_{0}, q_{1} \in M$ there exists an admissible curve that joins them. Hence $d\left(q_{0}, q_{1}\right)<+\infty$.

In what follows $B(q, r)$ (sometimes denoted also $B_{r}(q)$ ) is the (open) subRiemannian ball of radius $r$ and center $q$ :
$$B(q, r):=\left{q^{\prime} \in M \mid d\left(q, q^{\prime}\right)<r\right} .$$

## 数学代写|黎曼几何代写Riemannian geometry代考|The Minimal Control and the Length of an Admissible Curve

|v|:=\min \left{|u|, u \in U{q} \text { st } v=f(q, u)\right} 。|v|:=\min \left{|u|, u \in U{q} \text { st } v=f(q, u)\right} 。

⟨在∣在⟩q=14|在+在|2−14|在−在|2,在,在∈Dq.锻炼3.10让在1,…,在米∈在q是一个正交基在q. 定义在一世=F(q,在一世). 证明如果F(q,⋅)是内射的在1,…,在米是一个正交基Dq.

## 数学代写|黎曼几何代写Riemannian geometry代考|Equivalence of Sub-Riemannian Structures

(i) 存在欧几里得丛在和两个满射向量丛态射p:在→在和p′:在→在′使得下图是可交换的：

(ii)p,p′与标量积兼容，即它认为

\begin{对齐} |u| &=\min {|v|, p(v)=u}, & \forall u \in \mathbf{U}, \ \left|u^{\prime}\right| &=\min \left{|v|, p^{\prime}(v)=u^{\prime}\right}, & \forall u^{\prime} \in \mathbf{U}^{\主要} 。\end{对齐}\begin{对齐} |u| &=\min {|v|, p(v)=u}, & \forall u \in \mathbf{U}, \ \left|u^{\prime}\right| &=\min \left{|v|, p^{\prime}(v)=u^{\prime}\right}, & \forall u^{\prime} \in \mathbf{U}^{\主要} 。\end{对齐}

(a) 分布Dq和Dq′被定义为F和F′巧合，因为F(在q)=F′(在q′)对所有人q∈米;
(b) 对于每个在∈Dq我们有|在|=|在|′， 在哪里|⋅|和|⋅|′规范是由(在,F)和(在′,F′)分别。

## 数学代写|黎曼几何代写Riemannian geometry代考|Sub-Riemannian Distance

d\left(q_{0}, q_{1}\right)=\inf \left{\ell(\gamma) \mid \gamma:[0, T] \rightarrow M\right.$可接受，$\left .\gamma(0)=q_{0}, \gamma(T)=q_{1}\right} 。d\left(q_{0}, q_{1}\right)=\inf \left{\ell(\gamma) \mid \gamma:[0, T] \rightarrow M\right.$可接受，$\left .\gamma(0)=q_{0}, \gamma(T)=q_{1}\right} 。

（一）(米,d)是一个度量空间，
（ii）由(米,d)等价于流形拓扑。

B(q, r):=\left{q^{\prime} \in M \mid d\left(q, q^{\prime}\right)<r\right} 。

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