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February 10th, 2018, 09:57 PM   #1
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Problems on Surfaces

I have seeked in other forums as well, but I get no replies, and I really need some solutions to these kind of problems as the examples in my book are just not good enough to help me further solve problems at the end of the chapter. The problem goes like this:

If we have a unit sphere denoted by $S^2$ , in $\mathbb{R}^3$, and some function $z$ that maps $z:S^2 \rightarrow (0,\infty)$ which is defined to be smooth and has $+$ value. Now given some defined set $\Lambda$:


$$\Lambda=\{{z(x)x: x\in S^2}\}$$


Question 1) If $Z(u_1,u_2):U\rightarrow S^2$ is a parametrisation of $S^2$ (smooth), how can one show that another parametrisation on $\Lambda$ defined by $F:U \rightarrow \Lambda$, $(u_1,u_2) \rightarrow$ $z(Z(u_1,u_2))Z(u_1,u_2)$, then $\Lambda$ is a regular surface?
Question 2) Now, we have coinciding parametrisations of $S^2$ which are given by $Z_i(u_1,u_2):U_i \rightarrow S^2$, and $F_i:U_i\rightarrow \Lambda$ the parametrization of $\Lambda$ caused by $Z_i$. Prove that $F_1^{-1}(F_2)=Z_1^{-1}(Z_2)$ for $i=1,2$.


Question 3)
Is $\Lambda$ and $S^2$ diffeomorhpic? If so give an explicit form of the diffeomorphism.


Question 4) Let us now define some map $\Psi: \mathbb{R}^3 -\{(0,0,0)\} \rightarrow$ $\mathbb{R}^3 - \{(0,0,0)\}$, which is given by:


$$\Psi(x) = \frac{x}{|x|^2}$$


Denote $\Lambda^*=\Psi(\Lambda)$.


a) Is $\Lambda^*$ a regular surface?


b) For some map $\psi:\Lambda \rightarrow \Lambda ^*$which is some restriction on $\Psi$ on $\Lambda$, prove that $\psi$ is a diffeomorphism.


c) For any $c \in \Lambda$, the tangent map $d\psi_c:T_c \Lambda \rightarrow T_{\psi(c)} \Lambda^*$ at $c$ (which is a map from the tangent plane of $\Lambda$ at $c$ to the tangent plane at $\Lambda ^*$ at $\psi(c)$), prove that:


$$d\psi_c(W) =\frac{|c|^2 W-2(c.W)c}{|c|^4}$$ where $c.W$ is the dot product in $\mathbb{R}^3$.
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February 11th, 2018, 01:34 PM   #2
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If you haven't tried it already, try Mathematics Stack Exchange.

https://math.stackexchange.com/
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February 11th, 2018, 05:11 PM   #3
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I also have, no reply yet. Maybe I will try mathoverflow.
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