For questions about surfaces.

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6
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0answers
126 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
6
votes
0answers
126 views

Paper cylinder inside out

My question is related with paper folding: Given a cylinder of paper it is possible to turn it inside out using folding along lines. This is a Martin Gardner recreational puzzle. Secondly, ...
6
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0answers
150 views

Normal subgroups of the fundamental group of a non-orientable surface.

Let $N^2_g$ be a non-orientable closed genus $g\geq 2$ surface. Is there a way to explicitly list the normal subgroups of $\pi_1(N^2_g)$ in terms of generators and relations? I am interested in ...
5
votes
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58 views

Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...
4
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0answers
28 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 ...
4
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0answers
42 views

Property that defines Quadric Surface

The book < Geometry and the Imagination > (written by David Hilbert) introduces a property of a Quadric Surface without a proof. Property : The cone consisting of all the tangents from a ...
4
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0answers
99 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
4
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54 views

What does the metric matrix G tell us here

Let $\phi:U \rightarrow S \subseteq \mathbb{R}^3$ be a chart from $U \subseteq \mathbb{R}^2$ to a surface $S$. $G = g_{ij}$ be the metric matrix such that $ g_{ij} = \frac{\partial \phi}{\partial ...
4
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105 views

Computing a contraction of an exceptional divisor.

For a few days, I have been working on the following problem, from Qing Liu's book: Let $\mathcal{O_K}$ be a discrete valuation ring with uniformizing parameter t and residue characteristic $\neq ...
4
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378 views

The Birman–Hilden Theorem and the Nielsen–Thurston classification

So this post is half question/half reference request, as I'm sure it's the kind of thing people would have thought about before (and indeed the question might even be trivial), but I've been unable to ...
4
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0answers
77 views

Space of solutions to $f(x+y) = f(x) + f(y)$ when $f$ is convex

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as $\Omega_f(x) \triangleq ...
4
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212 views

Morse theory and homology of an algebraic surface (example)

Let $T_n$ denote the $n$-th Chebyshev polynomial and define $f_n(x,y,z)\!:=\!T_n(x)\!+\!T_n(y)\!+\!T_n(z)$ and $$Z_n:=\mathcal{Z}(f_n) \subseteq \mathbb{R}^3,$$ the Bachoff-Chmutov surface, where in ...
4
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0answers
208 views

Understanding surface area of a revolution/length of curve

I don't quite understand why the formula to find the surface area of a revolution is what it is: $$A = 2\pi \int_a^b x\ \sqrt{1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2}\ \text{d} x.$$ I ...
4
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0answers
208 views

Hirzebruch surfaces

How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Is it true that $F_{2}$ is the total space of a bundle with fibre SO(3) over $\mathbb{R}_{+}$?
3
votes
0answers
126 views

Squeezed cylinder parametrization

A Cylinder is such a common surface. But is there a parametrization for an isometrically $ R^2 $ bent cylinder whose major and minor dimensions are along x, y axes? I used an approximation to ...
3
votes
0answers
83 views

If there exists a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?

Let $S(0)$ and $S(t)$ be a hypersurface in $\mathbb{R}^n$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Suppose we have the Laplace-Beltrami operator $\Delta_{S(\cdot)}$. Let $u:S(t) \to ...
3
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65 views

homomorphism of fundamental groups induced by a map of two surfaces

I am trying to find another proof of the following theorem Theorem Let $X$ and $Y$ be two compact surfaces of genus greater than $2$. Then every homomorphism $π_1(X,x_0)→π_1(Y,y_0)$ is induced by a ...
3
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0answers
51 views

Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside a cylinder.

(a) Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside the region $\frac{x^2}{M^2}+\frac{y^2}{N^2}=1$ (b) Find the surface integral of $f=|xy|$ over ...
3
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89 views

Working with projection of areas?

I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting ...
3
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0answers
149 views

Calculating the Solid Angle

$\textbf{Problem:}$ Consider we have a class $C^1$ parameterization $\psi:[a_1,b_1]\times[a_2,b_2]\rightarrow\mathbb{R}^3-\{0\}$ for the surface $S$. Also, consider that $S$ is such that the map ...
3
votes
0answers
107 views

Does the dual of a vector bundle with ample determinant have global sections

Let $E$ be a locally free sheaf on a smooth projective variety $X$ over $\mathbb C$. Suppose that $\det E$ is an ample line bundle on $X$. Is $$H^0(X,E^\vee) =0?$$ In fact, if $E$ of rank $1$, it is ...
3
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0answers
52 views

What are the easiest surfaces of general type

The "easiest type" of curves of general type are those of genus two. In this case $\chi(X,\mathcal O_X) = -1$ and $\deg K = 2$, where $K$ is the canonical sheaf. I'm a bit lost when it comes to ...
3
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0answers
70 views

Do we really need to use the Jordan-Schönflies Theorem to prove that every surface can be triangulated?

I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds ...
3
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0answers
420 views

Ambient Isotopy

From Hirsch's Differential Topology, p. 180. The first of the isotopy extension theorems says; Let $A\subset M$ be a compact submanifold and $F:V\times I \rightarrow S^{3}$ an isotopy of $A$. If ...
2
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0answers
27 views

Cubic surface as P$^2$ blowing up 6 points

This question is from the chapter 1 of Reid's note: Chapters on algebraic surfaces Suppose that L:(x=y=0), M:(z=t=0), and L$_5$:(y=t=0) lie on a nonsingular cubic surface X in P$^3$, define a ...
2
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0answers
34 views

An algorithm to find isometry between surfaces in $\mathbb{R}^3$?

Given two surfaces in $R^3$, i would like to find isometry between these two. Usually, in class, we did some examples, like bending the plane into a cylinder, or cone, and they were not hard, quite ...
2
votes
0answers
11 views

Area of a region on the surface of a prolate spheroid

Is there a general expression for the area of a region bounded by 3 great ellipses on the surface of a prolate spheroid (where a great ellipse is the intersection of the spheroid with a plane passing ...
2
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0answers
33 views

Does every differentiable ruled surfaces possess a global ruled parametrization?

According to my notes, a differentiable ruled surface of $\mathbb R ^3$ is a 2-dimensional $C^k$ submanifold of $\mathbb R ^3$ that can be described as a union of straight lines. I'm working on some ...
2
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0answers
19 views

Calculating the equation of a multivariable surface of revolution

I'm stucked with a surface equation problem so I would be very thankful if someone could help me with it. What the excercise says: Find the equation of the revolution surface that is spanned when ...
2
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0answers
52 views

Criterion for orientability: Derivative of transition map

The definition I have been given for a smooth abstract surface, $S$, to be orientable is that given a continuous family of maps $f_t: D \to S$ that embed the closed unit disk into $S$ with $f_0(D) = ...
2
votes
0answers
50 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' ...
2
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0answers
73 views

Finding the leftmost, rightmost, top, and bottom, points, on a surface, of a sphere.

So I'm making a 3D game, and the player is inside a glass sphere. I'm projecting a bunch of points onto the sphere, and I need to find the leftmost, rightmost, topmost, and bottommost points, so I can ...
2
votes
0answers
31 views

common surface between two equation

what is common surface between: $(x+5)^2+z^2=y$ and $z^2+y^2=25$ ? I have found that at the XY plane the common surface is hiperbola, but it cannot be right because at the paraboloid there aren't any ...
2
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0answers
41 views

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the ...
2
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0answers
99 views

How do you integrate surface area in spherical coordinates?

A single-valued function of spherical coordinates $r(\theta,\phi)$ (where $(\theta,\phi)\in[0,\pi]\otimes[0,2\pi]$) naturally defines a surface in 3D space. How does one calculate the surface area of ...
2
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0answers
30 views

diagrams of twist spun torus knots

Kindly can you explain to me how to obtain the double twist spun of torus knots from tangle diagram of the given torus knot. I found the method here ...
2
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0answers
46 views

What kind of surface is this?

I'm not a math guru, but just fascinated by it, so sorry if my questions are only curiosity and not high level. In some contemporary art website I have found this image: In the right side there is ...
2
votes
0answers
36 views

Difference between diffeomorphisms fixing a point or a whole neighborhood.

Let $S_g$ be a closed orientable surface of genus $g$ and $S_{g}^1$ a closed orientable surface with one boundary component. Let $p$ be in $S_g$ and let's note $\mathrm{Diff}_+(S_g,p)$ the set of ...
2
votes
0answers
26 views

Show that the (Veronese-like) surface is given by zero locus of the following polynomials

Consider the following set in $\mathbb R^6$: $$ S= \bigl \{(x_1^2,x_2^2,x_3^2,x_1x_2,x_2x_3,x_3x_1) \mid (x_1,x_2,x_3) \in \mathbb R^3, \; x_1^2+x_2^2+x_3^2 = 1 \bigr\}. $$ If we denote by ...
2
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0answers
72 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
2
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0answers
112 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
2
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0answers
39 views

Seifert surface and crossing number

i am sitting here with the problem of Seifert Surfaces. I know from a theorem that every knot does have a Seifert surface. We can also make a so called disc-and-band surface $F$ by gluing $v$ discs ...
2
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0answers
330 views

Surfaces are homeomorphic iff are diffeomorphic.

I have read this statement in several places: "Two surfaces are homeomorphic iff are diffeomorphic". I think the nontrivial implication follows in this manner: First, we triangulate the surface and ...
2
votes
0answers
158 views

calculating the area on the surface os a sphere created by intersection of two spherical caps!

Consider a spherical object composed of two compartments (A and B, not necessarily hemispheres) sitting at the interface which is characterized by a plane separating 1 and 2. For this case, ...
2
votes
0answers
112 views

Do K3-surfaces have Weierstrass equations

I've been wondering a bit about K3-surfaces and their analogy to elliptic curves. I've just started so this might be a very silly question. Do all K3-surfaces have a Weierstrass equation (up to ...
2
votes
0answers
57 views

ruling out non Pseudo-Anosov automorphisms

We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus ...
2
votes
0answers
117 views

How to determine whether or not a specified set is a smooth surface?

I know that a given set $M$ is a smooth surface of dimension $k$ in $\mathbb{R}^n$ iff there's a map $r:U\rightarrow\mathbb{R}^n, U\subset \mathbb{R}^k$ is open such that $\forall a\in U, ...
1
vote
0answers
35 views

What are some applications of parametrization of curves and surfaces?

I know that we can find all elements of a quadratic field with norm 1 by rational parametrization of conics, it can be used to show that some Diophantine equations are not so easy to solve, and that ...
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0answers
44 views

Second fundamental form

Is it correct to say that .. the second fundamental form of surface theory determines the Euler characteristic and the genus of the surface ? If not how is it determined?
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20 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 ...