# Tagged Questions

Surface is a two-dimensional submanifold of three-dimensional Euclidean space.

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### Any tangent vector field on S^2 has a singular point [closed]

Is there an intuitive geometric proof to this?
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### How many parametrisations are needed to cover a sphere?

I have seen that a sphere can be covered with 6 parametrisations, but is it possible to totally cover a sphere with less parametrisations/charts?
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### Parameterization of tubular surface and analysis of its geometry

Consider the tube defined by $$α(s,v) = c(s) + r\big( \cos(v)\,b(s) + \sin(v)\,n(s)\big) , \quad r > 0.$$ Here $c$ is a Frenet curve with curvature $k>0$, torsion $\tau$ and $(t ,n, b)$ ...
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### Finding the equation of the affine tangent plane

I understand the first part but for the second part of the question how do I find the equation of the affine tangent plane at the given point? In the solution I can see that they are working out the ...
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### For surfaces in $\mathbb R^3$, does $ds^2 = du^2 + G(u)^2 dv^2$ implies $|\frac{\partial G(u)}{\partial u}| ≤1$?

Let we have a regular surface in $\mathbb R^3$, parameterized by $\vec r(u,v)$. Suppose that the first fundamental form (metric) of this surface is given by: $ds^2 = du^2 + G(u)^2 dv^2$ (G(u) only ...
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### Counterexample of the fundamental theorem for hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$ in global sense

Are there any example of a closed Riemannian $n$-manifold $(M,g)$ and a symmetric bilinear form $A=h_{ij}dx^i\otimes dx^j\in\Gamma(T^\ast M\otimes T^\ast M)$ satisfying Gauss' equation \begin{eqnarray}...
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### Prove that normal vector to the surface does not depend on parameterization

A given surface can be parameterize in many different ways. How to prove that a change in parameters, given a smooth, invertible map between the two parameter domains, does not change a normal vector ...
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### Axis of a cylinder

Could you explain to me what the axis of a cylinder is? Is it a line that passes through the center?
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### local equation of a divisor, localization and local ring at a point

Let $X$ be a regular, proper surface (integral, separated, of fin. type) over a perfect field $k$ and let $Z\subset X$ be an integral irreducible subscheme. Moreover $\eta$ is the generic point of $X$ ...
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### Show curve is tangent to a surface using gradient

The question is this: 'Show that the sphere $h(x,y,z) = x^2+y^2+z^2-8x-8y-6z+24=0$ is tangent to $f(x,y,z)=x^2+3y^2+2z^2=9$ at the point (2,1,1).' My approach was that grad(f) at P should give a ...
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### Create a smooth surface based on 4 points?

To understand this question, please first understand the question and answer here: Create a formula that creates a curve between two points We are essentially transcending a 2d problem into a 3d ...
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### Fundamental group of complement of image of diagonal embedding from $S_g$ to $S_g \times S_g$?

Let $S_g$ be a surface of genus $g \ge 0$. Let $\Delta \subset S_g \times S_g$ be the image of the diagonal embedding $x \mapsto (x, x)$. Let $X$ be the complement of $\Delta$ in $S_g \times S_g$. My ...
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### Definition of Gauss curvature

In GR class a definition of Gauss curvature was introduced which I cannot understand. It says: In a curved surface in $R^3(x,y,z)$. For any small area (circle?) $\Delta A_1$ of the surface. ...
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### I would like to find the inverse of $X$

Let $X(u,v)=(v−u,u^2−v^2,u+v)$, $(u,v)\in U=\mathbb R^2$ and $S=X(U)$. what is the inverse of $X$ where $X$ is the function which maps a $2$D object into a $3$D object $X^{-1}(x,y,z)$
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### Does there exists any $x$ such that $x\geq AM\geq GM?$ If not,how do I prove it is unbounded?
We know $AM\geq GM$.Or,in words,$AM$ has a minimum value when it is equal to $GM$?But,by any chance is there any way to find $x$ such that it is the upper limit of $AM$? My inspiration for this ...