For questions about surfaces.

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0
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1answer
18 views

what is the definition of “two parallel copies of a surface S”

As indicated in the title, suppose $S$ is a surface with genus $g$, then what is the definition of "two parallel copies of S"?
3
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1answer
131 views

Confusing Analysis proof

I have a question about a proof of the Beltrami-Enneper theorem: In the following $\nu$ is the surface-normal and $e_1,e_2,e_3$ the Frenet 3-frame. It states: Every asymptotic curve $c: I \rightarrow ...
2
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0answers
29 views

Parameterization of surface of revolution with constant mean curvature

Let $x(u,v) = (g(u), h(u) \cos v, h(u) \sin v)$ be a parameterization of a surface of revolution $M$, arising from rotating the regular curve $\alpha(u) = (g(u),h(u),0)$ around the $x$-axis with ...
1
vote
0answers
15 views

Is the standard embedding of the torus the tight embedding?

Definition of tight: A mapping of a surface into $\mathbb{R}^3$ is called tight if its image, equipped with the induced metric, has minimal total absolute curvature. (A definition of this kind is ...
1
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1answer
39 views

On a flat surface, can a holonomy can be nontrivial around certain curves

On a flat surface, can a holonomy can be nontrivial around certain curves? How is this possible?
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0answers
20 views

Compact hypersurface in $\mathbb{R}^n$

Let $S$ be an $(n-1)$ dimensional hypersurface in $\mathbb{R}^n$. If we say that $S$ is compact, does this necessarily mean that $S$ has no boundary? Eg. $S$ can be a sphere but not a sphere cut in ...
1
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1answer
31 views

Mutlivariable Calculus: Surface Area

This was a question a students had asked me earlier today regarding surface area. Find the surface area of the hemisphere $x^2+y^2+z^2 = 4$ bounded below by $z=1$. I decided to approach ...
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1answer
39 views

obtaining a surface equation by rotation

Let $C$ be a curve on the plane $(xoz)$ given by equation $g(x,z)=0$. How to find the equation in cylindric coordinates of the surface obtain rotating $C$ around zz axis?
2
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1answer
46 views

Blow-ups followed by contractions

Let $S$ be a minimal, non-singular complex projective surface. $\widehat S$ is the surface obtained by $r$ blow-ups of $S$ at the points $x_1,\ldots,x_r\in S$. Clearly $\widehat S$ contains exactly ...
9
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1answer
98 views

A “trivial” implication I don't understand.

I'm reading the article "Belyi's theorem for complex surfaces - Gabino Gonzalez Diez" and there are few lines of a certain proof that I don't understand (the author claims that all is trivial): ...
3
votes
1answer
36 views

Surfaces on which not every pair of points is connected by a geodesic

Let $S$ be a surface in $\mathbb{R}^3$. I believe that, if $S$ is smooth, bounded, and closed, then, for every pair of points $x,y \in S$, there is at least one geodesic $\gamma$ connecting $x$ to ...
0
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0answers
32 views

Number of base points of a linear system on a surface

I use Hartshorne's notations: Let $S$ be a non-singular complex surface and moreover let $D$ be a (Weil) divisor of $S$. Now consider a linear system $\delta\subset |D|$; we say that a point $p\in S$ ...
1
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1answer
48 views

Proof of the existence of Lefschetz Pencils.

Let $S$ be a smooth complex projective surface. A Lefschetz pencil over $S$ is a rational map (which is not a morphism) $f:S--\rightarrow\mathbb P^1_{\mathbb C}$ with the following property: All but ...
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0answers
30 views

3D equation of a cone-like shape

Imagine there are two parallel planes (base plane and plane1) in the following image: There is one point on the base plane and there are several points on the plane1. The positions of these points ...
1
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0answers
16 views

Inverse function theorem and parametric surfaces as graphs

Let $\psi$ be a regular surface at the point $(u_{0}, v_{0})$ ($\psi \in C^{1}, T_{u} \times T_{v} \neq 0$ at $u_{0}, v_{0}$). Use the implicit function theorem to show the image of $\psi$ near ...
0
votes
3answers
51 views

surface area of the solid (column side)

I made a problem But I'm stuck in solving .. :-( the problem is following. Find the surface area of the solid that lies under the paraboloid $z =x^2 + y^2$, above the $xy$-plane, ...
1
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0answers
16 views

Torsion of an asymptotic curve with nonzero curvature

I wish to solve the following problem using the matrix of the shape operator $S_{P}$. Suppose $K= \text{det} S_{P} <0$, and $C$ is an asymptotic curve with curvature $\kappa (P)$ nonzero. I want to ...
0
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1answer
36 views

minimum number of points on the surface of a 3D ellipsoid to define it uniquely

An ellipsoid in 3 D is described by 9 independent parameters: 3 for the coordinates of its centre + 6 independent components of a symmetric 3 x 3 matrix. What is the minimum number of points on the ...
1
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2answers
52 views

Parameterizing a surface

The question I was asked goes like this: The part of the hyperboloid $5x^2 − 5y^2 − z^2 = 5$ that lies in front of the yz-plane. Let x, y, and z be in terms of u and/or v. Find a parametric ...
1
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0answers
21 views

Integrate Gaussian over elliptic area

I have got a two-dimensional gaussian distribution, where $\sigma_x = \sigma_y$ and $\mu_x = \mu_y = 0$. $ f(x,y) = \frac{1}{2\pi \sigma^2} e ^{-\left( \frac{x^2 + y^2}{2\sigma^2} \right)}$ I would ...
0
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1answer
21 views

Determine the best cup of coffee to have faster cooling possible

Assume that a cup of coffee is a cylinder. The coffee machine at my workplace always produces the same amount of coffee, so the volume is constant. The coffee is always really hot, so I'm looking (out ...
0
votes
1answer
29 views

Surface integrals and surface areas of arbitrary parameter domains

I'm having trouble evaluating this surface integral. This would be very simple to solve if the parameter domain of the variables u and u was a square region. However, that isn't the case here. I've ...
0
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0answers
34 views

Integrals Over Paths and Surfaces proof

Let S be a sphere of radius r and p be a point inside or outside the sphere (but not on S). Show that $\iint\limits_S \frac{1}{\mid x-p \mid}\ dS=4\pi r $ if r is is inside S and $=4\pi r^2/d$, if p ...
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2answers
79 views

Intersection and Curvature of Surfaces

a) Describe the intersection $(C)$ of sphere $x^2 + y^2 + z^2 = 1$ and the elliptic cylinder $x^2 + 2z^2 = 1$, and find out the total arc-length of this intersection. b) Determine the points on the ...
0
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0answers
39 views

What is the definition of boundary-parallel Dehn twist?

I have not been able to find a working definition for the term: "boundary-parallel Dehn twist ". I know what a boundary-parallel surface is, and what a parallel surface is, but I have not been able to ...
1
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0answers
31 views

Embedding Klein Bottle in $\mathbb{R}^4$ using a figure 8 loop.

I'm trying to show that we can embed the Klein bottle in $\mathbb{R}^4$. I've previously shown that a figure 8 curve can be embedded in $\mathbb{R}^3$ by a bump function that pushes away the ...
8
votes
2answers
602 views

Is the Euler characteristc defined wrong? If not, why not?

Ever since learning that $$\chi(S_0\# S_1) = \chi(S_0)+\chi(S_1)-2$$ (where $\chi$ denote the Euler characteristic), I've wondered whether $\chi$ isn't "defined wrong." If we let $\chi' = 2-\chi,$ ...
1
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2answers
72 views

Maps between Riemann surfaces are open and continuous

I'm having some trouble with a couple of concepts in Riemman surfaces that I would really appreciate some help clarifying! Firstly, is it true that a holomorphic map between two Riemann surfaces $f:R ...
1
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0answers
21 views

Function for a sphere

I believe that there is something fundamentally wrong with my understanding of functions but I can't pin point what it is, so I would greatly appreciate any guidance. Consider a unit sphere, ...
2
votes
1answer
51 views

Surface has Euler characteristic 2 iff equal to sphere

Let $\Sigma$ be a connected (not necessarily compact) surface with or without boundary. Is it true that $\Sigma$ is homeomorphic to the sphere if it has euler characteristic $\chi(\Sigma)\geq 2$? I ...
2
votes
1answer
91 views

Differential geometry: Conformal map

Let $f:\mathbb{R}_{>0} \times (0,2\pi) \rightarrow \mathbb{R}^3$ $$f(t,\xi) := (r(t) \cos( \xi) , r(t) \sin(\xi),z(t))$$ be a surface of revolution, where we assume that $r>0$ and ...
1
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0answers
24 views

Understanding how connected sum of smooth surfaces is a surface

I have two smooth surfaces $M_1$ and $M_2$ I''m trying to understand how the connected sum $M_1 \mathop{\#} M_2$ is a smooth surface. I will write my understanding of the proof and then explain where ...
-1
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0answers
16 views

Find the exact surface area by rotating curve around x axis

Can someone explain step by step how I would solve this problem, I tried and came up with $1/800(17620\cdot\sqrt{401} + 79 \sinh^{-1}(20))$ plugging this into web assign tells me it is incorrect. ...
0
votes
1answer
23 views

Understanding why Euler's Formula applies to planar graphs

I'm trying to prove that given a planar graph (by that I mean a graph where every pair of points is joined without crossings) $V-E+F = 2$. I can prove this by induction directly on the edges except ...
2
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2answers
44 views

Reference request for equality of torsion of H1 and H2

I have heard that for a surface $X$ (algebraic? smooth? compact?) the torsion part of $H_1(X,\mathbb{Z})$ is the same as that of $H_2(X,\mathbb{Z})$. Please could you give me a correct statement? I ...
0
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0answers
14 views

Cutting a surface at critical levels produces cylinders

Let $F$ a closed surface with isolated critical points and a homeomorphism $g: F \rightarrow F$ that maps critical levels to critical levels. Let us cut the surface $F$ by the critical levels. Do we ...
0
votes
3answers
28 views

Find the intersection of two surfaces

I have been looking into this question : we have two surfaces : $$\big\{(x,y,z)\in \mathbb{R}^3 \mid\;\; S_1\colon\;\; x+z=1 ,\;\; S_2\colon\;\; x^2+y^2=1 \big\}$$ we need to draw or describe the ...
4
votes
2answers
59 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
2
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0answers
44 views

Is there a surface S$\subset R^3$ whose Gaussian curvature is -1 at each point S?

Is there a surface $S\subset \Bbb R^3$ whose Gaussian curvature is $-1$ at each point $S$? At first I think this does not make a sense. But googling and googling.. I found a 'final exam problem' ...
0
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0answers
40 views

Lattice representation of the Klein bottle

I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems. To ...
3
votes
1answer
48 views

Surface of an onion-shaped church tower

I am wondering how to calculate surface of the church tower in the picture, for painting purposes. Especially, I am interested in the two 'onion-shaped' parts. I am thinking, that it is not really ...
4
votes
1answer
25 views

Volumes and surfaces of revolution?

Please can someone explain to me why we use $dx$ in a volume of revolution i.e. $$\pi \int{f(x)^2 dx}$$ but $ds$ (an elementary bit of arc) in a surface of revolution i.e. $$2\pi \int{f(x)ds}$$ does ...
4
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1answer
108 views

Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
4
votes
0answers
87 views

Number of fibrations over a curve.

Fix a non-singular complex projective curve $C$. I would like to know how many non-singular complex projective surfaces $S$ have the following properties (up to isomorphism): There is a fibration ...
3
votes
3answers
340 views

Surface area of a sphere limits

If I am finding the surface area of a sphere in spherical coordinates my intergral would be like this: $$\int^{\pi}_0 \int^{2\pi}_0 R^2 \sin (\theta) d\phi d\theta =4\pi R^2$$ But if I do the ...
1
vote
2answers
33 views

Analytically isomorphic fibers.

Suppose that $S$ is a non-singular complex projective surface with a fibration $f$ over $\mathbb P^1(\mathbb C)$. Suppose also that: There are only finitely many points $y_1,\ldots,y_n\in\mathbb ...
0
votes
2answers
64 views

Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured ...
1
vote
1answer
40 views

a problem with Stokes' theorem(curl)

If L is the circle which you get from the intersection between the sphere $$ x^2+y^2+z^2=1, y=x\sqrt(3) $$ and $$ I= \int_L (y-z)dx+(z-x)dy+(x-y)dz $$ so |I| equals to? but i dont understand how the ...
0
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0answers
38 views

Roulette of a parabola - Delaunay-Surface

I've problems to understand an equation I've found in various books and papers. Maybe someone could help me and explain it a little bit more precisely. I colored the equation in yellow (the picture ...
4
votes
0answers
25 views

Given the equation for a surface, how to find enclosed volume?

Suppose we give an equation of the form $f(x_1,x_2,..., x_n)=C$, with $f$ a smooth function, and assume this is such that defines a closed surface in $\mathbb{R}^{n+1}$. Assume also that the equation ...