Posted on Categories:Differential Manifold, 微分流形, 数学代写

# 数学代考|微分流形代考Differential Manifold代写|MATH544 Transversality

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## 数学代考|微分流形代考Differential Manifold代写|Transversality

In this subsection we want to explain rigorously a phenomenon with which the reader may already be intuitively acquainted. We describe it first in a special case.

Suppose $M$ is a submanifold of dimension 2 in $\mathbb{R}^3$. Then, a simple thought experiment suggests that most horizontal planes will not be tangent to $M$. Equivalently, if we denote by $f$ the restriction to $M$ of the function $(x, y, z) \mapsto z$, then for most real numbers $h$ the level set $f^{-1}(h)=M \cap{z=h}$ does not contain a point where the differential of $f$ is zero, so that most level sets $f^{-1}(h)$ are smooth submanifolds of $M$ of codimension 1 , i.e., smooth curves on $M$.
We can ask a more general question. Given two smooth manifolds $X, Y$, a smooth map $f: X \rightarrow Y$, is it true that for “most” $y \in Y$ the level set $f^{-1}(y)$ is a smooth submanifold of $X$ of codimension $\operatorname{dim} Y$ ? This question has a positive answer, known as Sard’s theorem.
Definition 2.1.14. Suppose that $Y$ is a smooth, connected manifold of dimension $m$.
(a) We say that a subset $S \subset Y$ is negligible if, for any coordinate chart of $Y, \Psi: U \rightarrow \mathbb{R}^m$, the set $\Psi(S \cap U) \subset \mathbb{R}^m$ has Lebesgue measure zero in $\mathbb{R}^m$.
(b) Suppose $F: X \rightarrow Y$ is a smooth map, where $X$ is a smooth manifold. A point $x \in X$ is called a critical point of $F$, if the differential $D_x F: T_x X \rightarrow T_{F(x)} Y$ is not surjective.

We denote by $C r_F$ the set of critical points of $F$, and by $\Delta_F \subset Y$ its image via $F$. We will refer to $\Delta_F$ as the discriminant set of $F$. The points in $\Delta_F$ are called the critical values of $F$.

## 数学代考|微分流形代考Differential Manifold代写|Vector bundles

The tangent bundle $T M$ of a manifold $M$ has some special features which makes it a very particular type of manifold. We list now the special ingredients which enter into this special structure of $T M$ since they will occur in many instances. Set for brevity $E:=T M$, and $F:=\mathbb{R}^m$ $(m=\operatorname{dim} M)$. We denote by $\operatorname{Aut}(F)$ the Lie group $\mathrm{GL}(n, \mathbb{R})$ of linear automorphisms of $F$. Then
(a) $E$ is a smooth manifold, and there exists a surjective submersion $\pi: E \rightarrow M$. For every $U \subset M$ we set $\left.E\right|U:=\pi^{-1}(U)$ (b) From (2.1.4) we deduce that there exists a trivializing cover, i.e., an open cover $\mathcal{U}$ of $M$, and for every $U \in \mathcal{U}$ a diffeomorphism $$\Psi_U:\left.E\right|_U \rightarrow U \times F, \quad v \mapsto\left(p=\pi(v), \Phi_p^U(v)\right)$$ (b1) $\Phi_p$ is a diffeomorphism $E_p \rightarrow F$ for any $p \in U$ (b2) If $U, V \in \mathcal{U}$ are two trivializing neighborhoods with non empty overlap $U \cap V$ then, for any $p \in U \cap V$, the $\operatorname{map} \Phi{V U}(p)=\Phi_p^V \circ\left(\Phi_p^U\right)^{-1}: F \rightarrow F$ is a linear isomorphism, and moreover, the map
$$p \mapsto \Phi_{V U}(p) \in \operatorname{Aut}(F)$$
is smooth.
In our special case, the map $\Phi_{V U}(p)$ is explicitly defined by the matrix (2.1.4)
$$A(p)=\left(\frac{\partial y^j}{\partial x^i}(p)\right)_{1 \leq i, j \leq m}$$

## 数学代考|微分流形代考Differential Manifold代写|Transversality

（b）假设 $F: X \rightarrow Y$ 是 个光滑的地图，其中 $X$ 是 个光滑的流形。一个点 $x \in X$ 称为临界点 $F$ ，如果微分 $D_x F: T_x X \rightarrow T_{F(x)} Y$ 不是主观的。

## 数学代考|微分流形代考Differential Manifold代写|Vector bundles

（一） $E$ 是一个光滑流形，并且存在一个满射浸没 $\pi: E \rightarrow M$. 对于每一个 $U \subset M$ 我们设置 $E \mid U:=\pi^{-1}(U)(\mathrm{b})$ 从(2.1.4)我 们推昌出存在一个平凡鞜盖，即一个开覆盖 $\mathcal{U}$ 的 $M$ ，并且对于每个 $U \in \mathcal{U}$ 微分同胚
$$\Psi_U:\left.E\right|U \rightarrow U \times F, \quad v \mapsto\left(p=\pi(v), \Phi_p^U(v)\right)$$ (b1) $\Phi_p$ 是微分同顺 $E_p \rightarrow F$ 对于任何 $p \in U(\mathrm{~b} 2)$ 如果 $U, V \in \mathcal{U}$ 是两个具有非空重殒的平凡邻域 $U \cap V$ 那么，对于佳何 $p \in U \cap V ，$ 这map $\Phi V U(p)=\Phi_p^V \circ\left(\Phi_p^U\right)^{-1}: F \rightarrow F$ 是 个线性同构，而且，映射 $$p \mapsto \Phi{V U}(p) \in \operatorname{Aut}(F)$$

$$A(p)=\left(\frac{\partial y^j}{\partial x^i}(p)\right)_{1 \leq i, j \leq m}$$

## MATLAB代写

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