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# 数学代写|常微分方程代考Ordinary Differential Equations代写|Associated sphere and projective bund

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Associated sphere and projective bund

Definition 9.2.3 Let $\mathcal{B}$ be a vector bundle over a manifold $\mathcal{M}$.
(1) By the sphere bundle $\mathcal{S B}$ associated with $\mathcal{B}$ we shall mean the quotient of $\mathcal{B} \backslash 0$ modulo the equivalence relation
$$(\wp, \mathbf{v}) \cong\left(\wp^{\prime}, \mathbf{v}^{\prime}\right) \text { meaning: } \wp=\wp^{\prime} \text { and } \exists \lambda>0, \mathbf{v}=\lambda \mathbf{v}^{\prime}$$

(2) By the projective bundle $\mathbb{P} \mathcal{B}$ associated with $\mathcal{B}$ we shall mean the quotient of $\mathcal{B} \backslash 0$ modulo the equivalence relation
$$(\wp, \mathbf{v}) \cong\left(\wp^{\prime}, \mathbf{v}^{\prime}\right) \text { meaning: } \wp=\wp^{\prime} \text { and } \exists \lambda \in \mathbb{K} \backslash{0}, \mathbf{v}=\lambda \mathbf{v}^{\prime} \text {. }$$
The projective bundle $\mathbb{P} \mathcal{B}$ associated with a regular vector bundle $\mathcal{B}$ can be equipped with the structure of a regular fiber bundle, with $\mathbf{G L}(r, \mathbb{K})$ as structure group $(r=\operatorname{rank} \mathcal{B})$. The same is true of the sphere bundle $\mathcal{S B}$ provided $\mathbb{K}=\mathbb{R}$, i.e., provided $\mathcal{B}$ is of class $C^{\infty}$ or $C^\omega$. But when $\mathbb{K}=\mathbb{C}$ and the bundle $\mathcal{B}$ is complexanalytic the sphere bundle $\mathcal{S B}$ is merely real-analytic (it does carry another structure, that of a $C R$ bundle – which we shall not discuss here).

A point of $\mathcal{S} \mathcal{B}{\wp}(\wp \in \mathcal{M})$ can be identified with a ray of $\mathcal{B}{\wp}$, namely the set of points $\lambda \mathbf{v}, \mathbf{v} \in \mathcal{B}{\wp} \backslash{0}, \lambda>0$. A point of $\mathbb{P} \mathcal{B}{\wp}$, can be identified with a “line” in $\mathcal{B}{\wp}$, the set of points $\lambda \mathbf{v}, \mathbf{v} \in \mathcal{B}{\vartheta} \backslash{0}, \lambda \in \mathbb{K}$.

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Tangent bundle

Let $(\mathcal{U}, \varphi)$ be a local chart in a manifold $\mathcal{M}$. Consider a vector field
$$X=\sum_{j=1}^n a_j(x) \frac{\partial}{\partial x_j}$$
in the open set $\varphi(\mathcal{U}) \subset \mathbb{K}^n$. The coefficients of $X$ can be selected in any function class of our choice; for the time being we shall take them to be smooth, although this requirement may be very much weakened (or strengthened) if need be. What is more important is that they should be valued in $\mathbb{K}$, i.e., they should be real in the $C^{\infty}$ and $C^\omega$ cases. We point out, however, that when $\mathbb{K}=\mathbb{C}$, the partial derivative $\frac{\partial}{\partial x_j}$ means the holomorphic derivative: $\frac{\partial}{\partial x_j}=\frac{1}{2}\left(\frac{\partial}{\partial \operatorname{Re} x_j}-\sqrt{-1} \frac{\partial}{\partial \operatorname{Im} x_j}\right)$. In Euclidean space $\mathbb{K}^n$ the vector field $X$ is often identified with the vector $\mathbf{a}=\left(a_1(x), \ldots, a_n(x)\right)$ visualized as originating at the point $x$. This attaches to $x$ a vector space over $\mathbb{K}$, the tangent space $T_x \mathbb{K}^n$ at $x$. We also have the option of regarding $X$ as a first-order differential operator in $\varphi(\mathcal{U})$ without zero-order term, which is what we shall do most of the time.

Consider an arbitrary $C^1$ map $\psi: \varphi(\mathcal{U}) \longrightarrow \mathcal{U}$. The pushforward $\psi_* X$ of the vector field $X$ is a differential operator acting on a function $f \in C^1(\mathcal{U})$ according to the rule
$$\psi_* X f=(X(f \circ \psi)) \circ \stackrel{-1}{\psi}$$
Now suppose $\psi={ }{\varphi}^{-1}$. In the local coordinates $x_j$ of the chart $(\mathcal{U}, \varphi)$ we can write $X f(\wp)=\sum{j=1}^n a_j(x(\wp)) \frac{\partial f}{\partial x_j}(\wp)$. In practice we shall use the same notation (9.3.1) for ${ }_{\varphi}^{-1} X$ as for $X$; in other words, we think of (9.3.1) as a vector field in $\mathcal{U}$, with the understanding that this is only permitted as long as the local coordinates are not changed.

Obviously $X$ defines a linear map $C^1(\mathcal{U}) \longrightarrow C^0(\mathcal{U})$; if we follow it up with the valuation at an arbitrary point $\wp \in \mathcal{U}$ we get the linear functional $C^1(\mathcal{U}) \ni f \longrightarrow$ $X f(p) \in \mathbb{K}$ which we shall denote here by $X_{\wp}$ and interpret as a tangent vector to $\mathcal{M}$ at $\wp$. One often refers to $X_{\wp}$ as the freezing of the vector field $X$ at the point $\wp$. The tangent vectors at $\wp$ form an $n$-dimensional vector space (with the scalars in $\mathbb{K})$ denoted by $T_\rho \mathcal{M}$. This definition is coordinate-free: the choice of coordinates $x_1, \ldots, x_n$ in $\mathcal{U}$ merely provides us with a basis of the vector space $T_\rho \mathcal{M}$, the partial derivatives $\frac{\partial}{\partial x_j}$ evaluated at $\wp$; a change of coordinates produces a linear change of basis in conformity with the chain-rule of differentiation, i.e., a linear automorphism of $T_\rho \mathcal{M}$. It ensues directly from this and from (M1)-(M2)-(M3) that the disjoint union $T \mathcal{M}=\bigcup_{\rho \in \mathcal{M}} T_\rho \mathcal{M}$ carries a natural regular vector bundle structure over $\mathcal{M}$; this is the tangent bundle of $\mathcal{M}$. Of course, $\operatorname{rank} T \mathcal{M}=n$.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Associated sphere and projective bund

9.2.3设$\mathcal{B}$为流形$\mathcal{M}$上的向量束。
(1)与$\mathcal{B}$相关联的球束$\mathcal{S B}$是指$\mathcal{B} \backslash 0$模等价关系的商
$$(\wp, \mathbf{v}) \cong\left(\wp^{\prime}, \mathbf{v}^{\prime}\right) \text { meaning: } \wp=\wp^{\prime} \text { and } \exists \lambda>0, \mathbf{v}=\lambda \mathbf{v}^{\prime}$$

(2)与$\mathcal{B}$相关联的射影束$\mathbb{P} \mathcal{B}$是指$\mathcal{B} \backslash 0$模等价关系的商
$$(\wp, \mathbf{v}) \cong\left(\wp^{\prime}, \mathbf{v}^{\prime}\right) \text { meaning: } \wp=\wp^{\prime} \text { and } \exists \lambda \in \mathbb{K} \backslash{0}, \mathbf{v}=\lambda \mathbf{v}^{\prime} \text {. }$$

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