# Tagged Questions

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### Find isometry group of the Poincaré disc

I cannot show that any isometry of the Poincaré disc $\mathbb{H}^2$ is given by $e^{i\theta} \frac{z + c}{cz + 1}$ where $\theta$ and $c$ are constants. I know that they are indeed isometries, however ...
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### Prove that $g^{-1}(0)$ is a $n$-dimensional manifold.

Let $A\subset \mathbb R ^n$ be open and let $g:A\to \mathbb R ^p$ be a differentiable function such that $g'(x)$ has rank $p$ whenever $g(x)=0$. Then $g^{-1}(0)$ is an $(n-p)$-dimensional manifold. ...
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### If $\gamma$ is spherical, the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Please help me doing this question. In fact, there is its solution as I posted below. But I don't understand the answer. Please explain this more clearly. Thank you for helping. ...
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### How to obtain $y$

The question was written with dark-blue pen. And I tried to solve this question. I obtained $x$ as it is below. But I cannot obtain $y$ Please show me how to do this. By the way, $\gamma (t)$ ...
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### how to find the signed normal

$$\gamma (t)= (R\cos (t/R), R\sin (t/R))$$ $$\dot {\gamma (t)}=(-\sin (t/R), \cos (t/R))$$ $$n_s= (-\cos (t/R), -\sin (t/R))$$ where $n_s$ is the signed normal. the instructor has found the $n_s$. ...
how can I compute the curvative of an ellipse given by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ do i need to take $x=acos(t)$ and $y=bsin(t)$? please show me a way how to solve this? thank you for ...