微分几何代写|Differential Geometry代考|高质保分代写微分几何作业

Problem 1. Consider the compact differentiable manifold
$$
S^2:=\left{\left(x_1, x_2, x_3\right): x_1^2+x_2^2+x_3^2=1\right}
$$
An element $\eta \in S^2$ can be written as
$$
\eta=(\cos (\phi) \sin (\theta), \sin (\phi) \sin (\theta), \cos (\theta))
$$
where $\phi \in[0,2 \pi)$ and $\theta \in[0, \pi]$. The stereographic projection is a map
$$
\text { ПI : } S^2 \backslash{(0,0,-1)} \rightarrow \mathbb{R}^2
$$
given by
$$
x_1(\theta, \phi)=\frac{2 \sin (\theta) \cos (\phi)}{1+\cos (\theta)}, \quad x_2(\theta, \phi)=\frac{2 \sin (\theta) \sin (\phi)}{1+\cos (\theta)}
$$
(i) Let $\theta=0$ and $\phi$ arbitrary. Find $x_1, x_2$. Give a geometric interpretation.
(ii) Find the inverse of the map, i.e., find
$$
\Pi^{-1}: \mathbb{R}^2 \rightarrow S^2 \backslash{(0,0,-1)}
$$

Problem 7. The $n$-dimensional complex projective space $\mathbb{C P}^n$ is the set of all complex lines on $\mathbb{C}^{n+1}$ passing through the origin. Let $f$ be the map that takes nonzero vectors in $\mathbb{C}^2$ to vectors in $\mathbb{R}^3$ by
$$
f\left(z_1, z_2\right)=\left(\frac{z_1 \bar{z}_2+\bar{z}_1 z_2}{z_1 \bar{z}_1+\bar{z}_2 z_2}, \frac{z_1 \bar{z}_2-\bar{z}_1 z_2}{i\left(z_1 \bar{z}_1+\bar{z}_2 z_2\right)}, \frac{z_1 \bar{z}_1-\bar{z}_2 z_2}{z_1 \bar{z}_1+\bar{z}_2 z_2}\right)
$$
The map $f$ defines a bijection between $\mathbb{C} \mathbf{P}^1$ and the unit sphere in $\mathbb{R}^3$. Consider the normalized vectors in $\mathbb{C}^2$
$$
\left(\begin{array}{l}
1 \
0
\end{array}\right), \quad\left(\begin{array}{l}
0 \
1
\end{array}\right), \quad \frac{1}{\sqrt{2}}\left(\begin{array}{l}
1 \
1
\end{array}\right), \quad \frac{1}{\sqrt{2}}\left(\begin{array}{c}
1 \
-1
\end{array}\right), \quad \frac{1}{\sqrt{2}}\left(\begin{array}{c}
i \
-i
\end{array}\right) .
$$
Apply $f$ to these vectors in $\mathbb{C}^2$.