# Tagged Questions

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### tubular neighborhoods - problem

Consider $\alpha:[0,L]\to \mathbb R^3$ a smooth regular simple closed curve, arc-length parametrized. Denote its trace by $\gamma$. Let $\epsilon >0$ be a real number and $N_\epsilon (\alpha)$ the ...
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### An immersive map is locally left invertible

Question: Suppose that $m < n$, that $U$ is an open set in $\mathbb R^m$ and that $f : U \rightarrow \mathbb R^n$ is a $C^1$ function that has rank $m$ everywhere in $U$. Show that for every ...
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### Meridians of surfaces of revolutions

First off, I know there is another question asking the same thing, but that one was concerning where to start, whereas for this one, I'm almost complete, but I can't get something at the end to ...
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### Parallel Curve of Regular Plane Curves

Let $\gamma$ be a regular plane curve and let $\lambda$ be a constant. The parallel curve $\gamma^\lambda$ of $\gamma$ is defined as $$\gamma^\lambda(t)=\gamma(t)+(\lambda)n_s(t),$$ where $n_s$ is the ...
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### Given the normal vector n(s), determines the curvature k(s) and the torsion

Given the normal vector n(s) of a curve $\alpha$, with non zero torsion everywhere, determines the curvature k(s) and the torsion $\tau$(s) of $\alpha$. I am first trying to show the following which ...
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### Line containing a vector equation

In Do Carmo 1.5.1d, it asks: Show that, for the parametrized curve $\alpha(s) = (a$ cos $\frac{s}{c}, a$ sin $\frac{s}{c}, b\frac{s}{c})$, the lines containing the normal $n(s)$ passing through ...
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### Equation of a plane from cross product

I'm working from Do Carmo, and I ran into another snag. More specifically, 1.4.5: Given points $p_1, p_2, p_3 \in \mathbb{R}^3$, show that the following expression gives the equation for the ...
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### Finding limit of a parametrized curve

Related to this question: Parametrized curve tangent to a line I'm working on Do Carmo 1.3.5c, which is: Given a parametrized curve $\alpha(t) = (\frac{3at}{1+t^3},\frac{3at^2}{1+t^3})$ and the ...
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### Hadamard space: property of the Busemann function

I have a question about a property of Busemann functions on Hadamard spaces. Let $X$ be a complete CAT($0$) space. If $r:[0, \infty) \to X$ is a geodesic ray, and $x\in X$ the Busemann function is ...
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### Four circles & a square in a circle

Radius of the big triangle is $2$. ABCD is a square. What is the difference between $T_{1}$ and $(M_{1}+M_{2})$. I have solved it already though I don't know if my answer is right or wrong. My ...
I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...