For questions about surfaces.

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6
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0answers
115 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
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124 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
141 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 ...
4
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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 ...
4
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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 ...
4
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0answers
40 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
73 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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48 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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99 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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369 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
76 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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135 views

Union of two self-intersecting planes is not a surface

I need to show that the union of xy-plane and xz-plane, i.e. the set $S:=\lbrace (x,y,z)\in\mathbb{R}^3 : z=0 \mbox{ or } y=0\rbrace$, is not a surface. Here is my claim, $\textbf{Claim :}$ Suppose ...
4
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208 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
206 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
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81 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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0answers
61 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
50 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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86 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
142 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
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0answers
106 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
50 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
391 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
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 ...
2
votes
0answers
45 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
59 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
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0answers
30 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
35 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
83 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
29 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
43 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
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0answers
34 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
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0answers
25 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
votes
0answers
69 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
103 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
36 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
313 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
148 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
108 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
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0answers
42 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
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0answers
114 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
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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
vote
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 ...
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 ...
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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 ...
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0answers
32 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 ...
1
vote
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, ...
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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
vote
0answers
18 views

Principle Lines of curvature

In the text by Manfredo P. Do Carmo entitled Differential Geometry of Curves and Surfaces, an analysis of the principle directions is made near a non umbilic point on pp 160-161. I have followed his ...