如果你也在 怎样代写黎曼几何Riemannian geometry这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。
黎曼几何是研究黎曼流形的微分几何学分支，黎曼流形是具有黎曼公制的光滑流形，即在每一点的切线空间上有一个内积，从一点到另一点平滑变化。
statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。
我们提供的黎曼几何Riemannian geometry及其相关学科的代写，服务范围广, 其中包括但不限于:
- Statistical Inference 统计推断
- Statistical Computing 统计计算
- Advanced Probability Theory 高等概率论
- Advanced Mathematical Statistics 高等数理统计学
- (Generalized) Linear Models 广义线性模型
- Statistical Machine Learning 统计机器学习
- Longitudinal Data Analysis 纵向数据分析
- Foundations of Data Science 数据科学基础
数学代写|黎曼几何代写Riemannian geometry代考|Weighted Rotationally Symmetric Spaces
As already alluded to in Section 17, the works [GreW, Gri1, O’N], provide a complete description of the rotationally symmetric model spaces $M_{w}^{m}$, that we will use to establish our comparison theorems. We remark that these spaces constitute a huge set of comparison spaces that we will now apply in the same way as we used the constant curvature spaces $\mathbb{K}^{n}(b)$ in Section 3. The spaces $\mathbb{K}^{n}(b)$ are but particular examples of rotationally symmetric model spaces. We shall need the formal definition of warped products as follows:
Definition $23.1$ (see [GreW, Ch. 2], [Gri1, Sect. 3], [Pe, Ch. 3]). A $w$-model space is a smooth warped product $\left(M_{w}^{m}, g_{w}\right):=B^{1} \times_{w} F^{m-1}$ with base $B^{1}:=[0, \Lambda[\subset \mathbb{R}$ (where $0<\Lambda \leq \infty$ ), fiber $F^{m-1}:=\mathbb{S}{1}^{m-1}$ (the unit $(m-1)$-sphere with standard metric), and warping function $w:[0, \Lambda[\rightarrow[0, \infty[$ such that $w(r)>0$ for all $r>0$, whereas $w(0)=0, w^{\prime}(0)=1$, and $w^{(k)}(0)=0$ for all even derivation orders $k$. The point $o{w}:=\pi^{-1}(0)$, where $\pi$ denotes the projection onto $B^{1}$, is called the center point of the model space. If $\Lambda=\infty$, then $o_{w}$ is a pole of the manifold (recall that a pole of a complete Riemannian manifold $M$ is a point $o \in M$ such that the exponential map $\exp {o}: T{o} M \rightarrow M$ is a diffeomorphism).
Example 23.2. The simply connected space forms $\mathbb{K}^{m}(b)$ of constant sectional curvature $b$ can be constructed as $w$-models with any given point as center point using the warping functions
$$
w_{b}(r):= \begin{cases}\frac{1}{\sqrt{b}} \sin (\sqrt{b} r) & \text { if } b>0 \ r & \text { if } b=0 \ \frac{1}{\sqrt{-b}} \sinh (\sqrt{-b} r) & \text { if } b<0\end{cases} $$ Note that, for $b>0$, the function $w_{b}(r)$ admits a smooth extension to $r=\pi / \sqrt{b}$. For $b \leq 0$ any center point is a pole.
In a model space the sectional curvatures for 2-planes containing the radial direction from the center point are determined by the radial function
$$
-\frac{w^{\prime \prime}(r)}{w(r)}
$$
where $r=$ dist $_{M_{w}}\left(o_{w}, \cdot\right)$ is the distance from the center $o_{w} \in M_{w}$. Moreover, the mean curvature of the metric sphere of radius $r$ from the center $o_{w}$ is
$$
\frac{w^{\prime}(r)}{w(r)}=\frac{d}{d r} \ln (w(r))
$$
数学代写|黎曼几何代写Riemannian geometry代考|Weighted Curvatures
Let us consider a complete non-compact weighted manifold $\left(N^{n}, g, e^{h}\right)$. Among the several generalizations of the Ricci curvature tensor (which we use to control the Laplacian), the most extensively used are the Bakry-Emery Ricci tensors:
- The finite or $q$-Bakry-Emery Ricci tensor, $q>0$ :
$$
\operatorname{Ric}_{h}^{q}=\text { Ric }-\text { Hess } h-\frac{1}{q} d h \otimes d h .
$$ - The infinite or $\infty$-Bakry-Emery Ricci tensor:
$$
\operatorname{Ric}_{h}=\mathrm{Ric} \quad \mathrm{Hess} h .
$$
Observe that
$$
\mathrm{Ric}{h}=\mathrm{Ric}{h}^{q}+\frac{1}{q} d h \otimes d h,
$$
so lower bounds on $\mathrm{Ric}{h}^{q}$ implies lower bounds on $\mathrm{Ric}{h}$. Here Ric denotes the Ricci tensor in $(N, g)$.
The weighted radial sectional curvatures, (which agree, up to a constant, with the weighted sectional curvatures introduced in [W]), are defined as follows:
Definition 24.1. Consider the radial plane $\sigma_{p} \subseteq T_{p} N$ spanned by linearly independent unit vectors $X$ and $Y$, i.e., $\sigma_{p}=\operatorname{span}{X, Y}$. We define the $\infty$-weighted sectional curvature of $\sigma_{p}$ in a weighted Riemannian manifold $\left(N^{n}, g, e^{h}\right)$ as
$$
\operatorname{Sec}{h}^{\infty}\left(\sigma{p}\right)=K_{N}\left(\sigma_{p}\right)-\frac{1}{n-1}(\operatorname{Hess} h){p}(Y, Y) \text {. } $$ The $q$-weighted sectional curvature of the radial plane $\sigma{p}=\operatorname{span}{X, Y}$ is defined by:
$$
\operatorname{Sec}{h}^{q}\left(\sigma{p}\right)=K_{N}\left(\sigma_{p}\right)-\frac{1}{n-1}(\operatorname{Hess} h){p}(X, Y)-\frac{1}{(n-1) q}(d h \otimes d h)(Y, Y) $$ In the paper [HPR2] an analysis of the intrinsic distance function to a pole in the weighted manifold $M^{m}$ is carried out. Analogously, as in Sections 3 and 8 , the estimates of the Hessian and the weighted Laplacian of the distance from the pole lead to capacity comparison results and parabolicity criteria for weighted manifolds under lower bounds on the Ricci curvatures $\mathrm{Ric}{h}$ and $\mathrm{Ric}_{h}^{q}$.
We also found in [HPR2] bounds for the weighted isoperimetric quotients and volumes of metric balls, along the lines of Section 8 , but now from an intrinsic point of view. We must remark that the works [Q, L, Mo2, WW, PRRS] contain previous comparison results that involve weighted volumes and quotients of weighted volumes (although not weighted isoperimetric quotients) when Ric ${ }_{\infty}^{h}$ is bounded from below.
数学代写|黎曼几何代写Riemannian geometry代考|Analysis of Restricted Distance Functions
In [HPR1] and [HPR2] the restricted distance to a submanifold immersed in an ambient weighted manifold, is analyzed in the same vein as in Section 3. Let $P^{m}$ be an $m$-dimensional submanifold with $\partial P=\emptyset$ properly immersed in a weighted manifold $\left(N^{n}, g, \epsilon^{h}\right)$ with a pole $o \in N$. We consider in $P$ the induced Riemannian metric. We use the notation $\nabla^{P} u$ and $\Delta^{P} u$ for the gradient and Laplacian in $P$ of a function $u \in C^{2}(P)$.
The restriction to $P$ of the weight $e^{h}$ in $N$ produces a structure of weighted manifold in $P$. From (21.2) the associated $h$-Laplacian $\Delta_{h}^{P}$ has the expression
$$
\Delta_{h}^{P} u=\Delta^{P} u+\left\langle\nabla^{P} h, \nabla^{P} u\right\rangle,
$$ for any $u \in C^{2}(P)$. We say that the submanifold $P$ is $h$-parabolic when $P$ is weighted parabolic considered as a weighted manifold. Otherwise we say that $P$ is $h$-hyperbolic. By Theorem $22.1$ the $h$-parabolicity of $P$ is equivalent to $\operatorname{Cap}{h}^{P}(D)=$ 0 for some precompact open set $D \subseteq P$, where $\operatorname{Cap}{h}^{P}$ denotes the $h$-capacity relative to $P$. Clearly a compact submanifold $P$ is $h$-parabolic.
Now we present another necessary ingredient to establish these results: the weighted mean curvature of submanifolds. In the case of two-sided hypersurfaces this was first introduced by Gromov [G], see also [Ba, Ch. 3].
Definition 25.1. The weighted mean curvature vector or h-mean curvature vector of $P$ is the vector field normal to $P$ given by
$$
\bar{H}{h}^{P}:=m \bar{H}^{P}-(\nabla h)^{\prime}, $$ where $(\nabla h)^{\perp}$ is the normal projection of $\nabla h$ with respect to $P$ and $\bar{H}^{P}$ is the mean curvature vector of $P$. This is defined as $m \bar{H}^{P}:=-\sum{i=1}^{n-m}\left(\operatorname{div}^{P} N_{i}\right) N_{i}$, where $\operatorname{div}^{P}$ stands for the divergence relative to $P$ and $\left{N_{1}, \ldots, N_{n-m}\right}$ is any local orthonormal basis of vector fields normal to $P$.
We say that $P$ has constant $h$-mean curvature if $\left|\bar{H}{h}^{P}\right|$ is constant on $P$. If $\bar{H}{h}^{P}=0$, then $P$ is called $h$-minimal. More generally, $P$ has bounded h-mean curvature if $\left|\bar{H}{h}^{P}\right| \leq c$ on $P$ for some constant $c>0$. For later use we note that the equality $$ \left\langle m \bar{H}^{P}, \nabla r\right\rangle+\left\langle\nabla^{P} h, \nabla^{P} r\right\rangle=\left\langle\bar{H}{h}^{P}, \nabla r\right\rangle+\langle\nabla h, \nabla r\rangle
$$
holds on $P-{o}$. This easily follows from the definition of $\bar{H}_{h}^{P}$ and the fact that $\nabla h-(\nabla h)^{\perp}=\nabla^{P} h$.
As in Proposition 3.13, some inequalities for the weighted Laplacian of submanifolds have been established under bounds (which are not necessarily constants) on the radial sectional curvatures of the ambient manifold. In the particular case of a weighted $(w, f)$-model space it can be checked that all the estimates in the next statement become equalities.
黎曼几何代考
数学代写|黎曼几何代写Riemannian geometry代考|Weighted Rotationally Symmetric Spaces
正如第 17 节中已经提到的，作品 [GreW, Gri1, O’N] 提供了旋转对称模型空间的完整描述米在米，我们将使用它来建立我们的比较定理。我们注意到这些空间构成了一组巨大的比较空间，我们现在将以与使用恒定曲率空间相同的方式应用它们ķn(b)在第 3 节中。空间ķn(b)只是旋转对称模型空间的特殊例子。我们需要对翘曲产品的正式定义如下：
定义23.1（参见 [GreW, Ch. 2]、[Gri1, Sect. 3]、[Pe, Ch. 3]）。一个在-模型空间是一个光滑的翘曲产品(米在米,G在):=乙1×在F米−1带底座乙1:=[0,Λ[⊂R（在哪里0<Λ≤∞）， 纤维F米−1:=小号1米−1（那个单位(米−1)-具有标准公制的球体）和翘曲函数在:[0,Λ[→[0,∞[这样在(r)>0对所有人r>0， 然而在(0)=0,在′(0)=1， 和在(ķ)(0)=0对于所有偶数派生订单ķ. 重点○在:=圆周率−1(0)， 在哪里圆周率表示投影到乙1，称为模型空间的中心点。如果Λ=∞， 然后○在是流形的一个极点（回想一下完整黎曼流形的一个极点米是一个点○∈米使得指数映射经验○:吨○米→米是微分同胚）。
例 23.2。简单连通的空间形式ķ米(b)等截面曲率b可以构造为在- 使用扭曲函数将任何给定点作为中心点的模型
在b(r):={1b罪(br) 如果 b>0 r 如果 b=0 1−b出生(−br) 如果 b<0请注意，对于b>0， 功能在b(r)承认平滑扩展至r=圆周率/b. 为了b≤0任何中心点都是极点。
在模型空间中，包含从中心点开始的径向方向的 2 平面的截面曲率由径向函数确定
−在′′(r)在(r)
在哪里r=距离米在(○在,⋅)是到中心的距离○在∈米在. 此外，公制半径球的平均曲率r从中心○在是
在′(r)在(r)=ddrln(在(r))
数学代写|黎曼几何代写Riemannian geometry代考|Weighted Curvatures
让我们考虑一个完整的非紧加权流形(ñn,G,和H). 在 Ricci 曲率张量（我们用来控制拉普拉斯算子）的几个推广中，最广泛使用的是 Bakry-Emery Ricci 张量：
- 有限的或q-Bakry-Emery Ricci 张量，q>0 :
里克Hq= 里克 − 赫斯 H−1qdH⊗dH. - 无限或∞-Bakry-Emery Ricci 张量：
里克H=里克赫斯H.
请注意
里克H=里克Hq+1qdH⊗dH,
所以下界里克Hq意味着下界里克H. 这里 Ric 表示 Ricci 张量(ñ,G).
加权径向截面曲率（与 [W] 中引入的加权截面曲率一致，直到一个常数）定义如下：
定义 24.1。考虑径向平面σp⊆吨pñ由线性独立的单位向量跨越X和是， IE，σp=跨度X,是. 我们定义∞-加权截面曲率σp在加权黎曼流形中(ñn,G,和H)作为
秒H∞(σp)=ķñ(σp)−1n−1(赫斯H)p(是,是). 这q- 径向平面的加权截面曲率σp=跨度X,是定义为：
秒Hq(σp)=ķñ(σp)−1n−1(赫斯H)p(X,是)−1(n−1)q(dH⊗dH)(是,是)在论文[HPR2]中对加权流形中一个极点的内在距离函数的分析米米完成了。类似地，如在第 3 节和第 8 节中，Hessian 和从极点距离的加权拉普拉斯算子的估计导致容量比较结果和加权流形在 Ricci 曲率下限下的抛物线标准里克H和里克Hq.
我们还在 [HPR2] 中发现了公制球的加权等周商和体积的界限，沿着第 8 节的路线，但现在从内在的角度来看。我们必须指出，作品 [Q, L, Mo2, WW, PRRS] 包含以前的比较结果，其中涉及加权体积和加权体积商（尽管不是加权等周商），当 Ric∞H是从下方限定的。
数学代写|黎曼几何代写Riemannian geometry代考|Analysis of Restricted Distance Functions
在 [HPR1] 和 [HPR2] 中，与浸入环境加权流形中的子流形的限制距离按照与第 3 节相同的方式进行分析。让磷米豆米维子流形∂磷=∅正确浸入加权歧管中(ñn,G,εH)用一根杆子○∈ñ. 我们考虑在磷诱导黎曼度量。我们使用符号∇磷在和Δ磷在对于梯度和拉普拉斯算子磷函数的在∈C2(磷).
限制为磷重量的和H在ñ产生一个加权流形结构磷. 从（21.2）相关的H-拉普拉斯算子ΔH磷有表达式
ΔH磷在=Δ磷在+⟨∇磷H,∇磷在⟩,对于任何在∈C2(磷). 我们说子流形磷是H-抛物线当磷是加权抛物线，被视为加权流形。否则我们说磷是H-双曲线。按定理22.1这H- 抛物线磷相当于帽H磷(D)=0 对于一些预紧开集D⊆磷， 在哪里帽H磷表示H-容量相对于磷. 显然是一个紧凑的子流形磷是H-抛物线。
现在我们提出另一个必要的成分来确定这些结果：子流形的加权平均曲率。在双面超曲面的情况下，这首先由 Gromov [G] 引入，另见 [Ba, Ch. 3]。
定义 25.1。的加权平均曲率向量或 h 平均曲率向量磷是垂直于的向量场磷由
H¯H磷:=米H¯磷−(∇H)′,在哪里(∇H)⊥是的正常投影∇H关于磷和H¯磷是平均曲率向量磷. 这被定义为米H¯磷:=−∑一世=1n−米(div磷ñ一世)ñ一世， 在哪里div磷代表相对于的散度磷和\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别是向量场的任何局部正交基，垂直于磷.
我们说磷有常数H- 平均曲率如果|H¯H磷|是恒定的磷. 如果H¯H磷=0， 然后磷叫做H-最小。更普遍，磷有界 h 平均曲率如果|H¯H磷|≤C上磷对于一些常数C>0. 为了以后的使用，我们注意到等式⟨米H¯磷,∇r⟩+⟨∇磷H,∇磷r⟩=⟨H¯H磷,∇r⟩+⟨∇H,∇r⟩
坚持磷−○. 这很容易从定义中得出H¯H磷并且事实上∇H−(∇H)⊥=∇磷H.
与命题 3.13 一样，子流形的加权拉普拉斯算子的一些不等式已在环境流形的径向截面曲率的界限（不一定是常数）下建立。在加权的特定情况下(在,F)-模型空间可以检查下一条语句中的所有估计值是否相等。
