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Miller, R. (2020). Stochastic Methods in Differential Geometry. Springer.

Johnson, L. (2021). Applications of Differential Geometry in Engineering. Cambridge University Press.

Chen, H. (2022). Advanced Topics in Differential Geometry: Curvature and Topology. MIT Press.

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 $\mathbb{C}^2$.

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