# 数学代写|自旋几何代写spin geometry代考|MATH3349

## 数学代写|自旋几何代写spin geometry代考|Difierential Operators

This section presents the basic notion of a linear elliptic differential operator over a manifold $X$. We begin by fixing notation. For an $n$-tuple of nonnegative integers $\alpha=\left(\alpha_1, \ldots, \alpha_n\right)$, we set $|\alpha|=\sum_k \alpha_k$, and for each $\xi \in \mathbb{R}^n$ we set $\xi^\alpha=\xi_1^{\alpha_1} \xi_2^{\alpha_2} \cdots \xi_n^{\alpha_n}$. In local coordinates $\left(x_1, \ldots, x_n\right)$ on $X$ we define the differentiation operators $D^\alpha$ by $i^{|\alpha|} D^\alpha \equiv \partial^{|\alpha|} \cdot / \partial x^\alpha \equiv \partial^{|\alpha|} / \partial x_1^{a_1} \partial x_2^{\alpha_2} \cdots \partial x_n^{\alpha_n}$. Recall that for a smooth vector bundle $E$ on $X$, the symbol $\Gamma(E)$ denotes the space of smooth (i.e., $C^{\infty}$ ) cross-sections of $E$.

Definirion 1.1. A differential operator of order $m$ on $X$ is a linear map $P: \Gamma(E) \rightarrow \Gamma(F)$, where $E$ and $F$ are smooth complex vector bundles over $X$, with the following property. Each point of $X$ has a neighborhood $U$ with local coordinates $\left(x_1, \ldots, x_n\right)$ and local trivializations: $\left.E\right|U \stackrel{\approx}{\rightarrow} U \times \mathbb{C}^D$ and $\left.F\right|_U \stackrel{\approx}{\rightarrow} U \times \mathbb{C}^q$, in which $P$ can be written in the form: $$P=\sum{|\alpha| \leqq m} A^\alpha(x) \frac{\partial^{|\alpha|}}{\partial x^\alpha}$$
where each $A^\alpha(x)$ is a $q \times p$-matrix of smooth complex-valued functions and where $A^\alpha \neq 0$ for some $\alpha$ with $|\alpha|=m$.

A real differential operator of order $\boldsymbol{m}$ is defined similarly with $\mathbb{C}$ replaced by $\mathbb{R}$.

Observe that if we make a change of the local trivializations of $\left.E\right|U$ and $\left.F\right|_U$ by smooth maps $g_E: U \rightarrow G L_p(\mathbb{C})$ and $g_F: U \rightarrow G L_q(\mathbb{C})$ respectively, then in these new trivializations $P$ has the form \begin{aligned} P &=g_F\left(\sum{|\alpha| \leq m} A^\alpha \frac{\partial^{|\alpha|}}{\partial x^\alpha}\right) g_E^{-1} \ &=\sum_{|\alpha| \leq m} A^\alpha \frac{\partial^{|\alpha|}}{\partial x^\alpha} \end{aligned}
where the $A_{\sim}^\alpha$ ‘s are again $p \times q$-matrices of smooth functions of $x$ and where
$$A^\alpha=g_F A^\alpha g_E^{-1} \quad \text { for }|\alpha|=m .$$

## 数学代写|自旋几何代写spin geometry代考|Sobolev Spaces and Sobolev Theorems

Let $E$ be a hermitian vector bundle with connection $\nabla$ on a compact riemannian manifold $X$. Given $u \in \Gamma(E)$ we have $\nabla u \in \Gamma\left(T^* X \otimes E\right.$ ), and using the tensor product connection on $T^* X \otimes E$, we have $\nabla \nabla u \in$ $\Gamma\left(T^* X \otimes T^* X \otimes E\right)$. This process continues, and for any $k$ we can define the norm
$$|u|_k^2 \equiv \sum_{j=0}^k \int_x \underbrace{|\nabla \nabla \cdots \nabla|^2,}_{j \text { times }}$$
called the basic Sobolev $\boldsymbol{k}$-norm on $\Gamma(E)$. An easy exercise shows the equivalence class of this norm to be independent of the choice of metrics and connection. The completion of $\Gamma(E)$ in this norm is the Sobolev space $L_k^2(E)$. It is straightforward to verify the following:

Proposition 2.1. A differential operator $P: \Gamma(E) \rightarrow \Gamma(F)$ of order $m$ extends to a bounded linear map $P: L_k^2(E) \rightarrow L_{k-m}^2(F)$ for all $k \geqq m$.

Ultimately we shall see that if $P$ is elliptic then these extensions have finite dimensional kernel and “cokernel” which consist of smooth sections and are independent of $k$.

Our aim at present is to establish some analytical tools. This is best done using Fourier transform methods. To this end we select a good system of trivializations of our bundle $E$. To start we choose a finite covering of $X$ by closed coordinate balls $y_\beta: U_\beta \rightarrow \bar{B}^n=\left{y \in \mathbb{R}^n:|y| \leqq 1\right}, \beta=$ $1, \ldots, N$. Over each ball $U_\beta$, we choose a smooth trivialization of $E$
$$\left.E\right|{U\beta} \stackrel{\because}{\longrightarrow} U_\beta \times \mathbb{C}^p$$
which possesses a smooth extension to an open neighborhood of $U_a$. We further assume that the open balls of radius $1 / \sqrt{2}$ cover $X$, i.e., $X=$ $\bigcup_{\alpha=1}^N B_\theta$ where $\boldsymbol{B}\theta \equiv\left{p \in U\beta:\left|y_\theta(p)\right|^2<\frac{1}{2}\right}$.

## 数学代写|自旋几何代写spin geometry代考|Difierential Operators

$$A^\alpha=g_F A^\alpha g_E^{-1} \quad \text { for }|\alpha|=m$$

## 数学代写|自旋几何代写spin geometry代考|Sobolev Spaces and Sobolev Theorems

$$|u|k^2 \equiv \sum{j=0}^k \int_x \underbrace{|\nabla \nabla \cdots \nabla|^2}{j \text { times }},$$ 称为基本的 Sobolev $\boldsymbol{k}$-规范 $\Gamma(E)$. 一个简单的练习表明这个范数的等价类独立于度量和连接的选择。的 完成 $\Gamma(E)$ 在这个范数中是 Sobolev 空间 $L_k^2(E)$. 验证以下内容很简单: 提案 2.1。微分算子 $P: \Gamma(E) \rightarrow \Gamma(F)$ 秩序 $m$ 扩展到有界线性映射 $P: L_k^2(E) \rightarrow L{k-m}^2(F)$ 对所有人 $k \geqq m$.

$$E \mid U \beta \stackrel{\because}{\longrightarrow} U_\beta \times \mathbb{C}^p$$

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