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Questions tagged [surfaces]

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

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Klein bottle with constant curvature is flat

A torus (equipped with a Riemannian or Lorentzian metric) which has constant curvature must be flat because of Gauss-Bonnet theorem. Is it true that a Klein bottle (equipped with a Riemannian or ...
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+50

Completion of the local ring at a point on arithmetic surfaces.

Let $K$ be a number field and consider a arithmetic surface $X\to B=\operatorname{Spec} O_K$, i.e. $X$ is integral, regular, flat, proper over $O_K$ and it has dimension $2$. Now pick a closed point $...
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Theorem of Pappus

Given a surface of revolution $S$ which can be parametrized by the map $$ \mathbf x(u,v) = (f(v)\cos u,f(v)\sin u,g(v)), $$ over the open set $U =\{(u,v) \in \mathbb R^2 \mid 0 < u < 2\pi, a <...
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General form of quadric surfaces

The general form of quadric surfaces is $$Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0$$ I want to classify all of the possibilities including degenerate cases with the help of ...
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1answer
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Definition of closed surface/manifold

This question might appear silly, I was reading on wikipedia that a closed surface (or manifold in general) is a surface without a boundary, I'd like to elaborate a bit on such definition. Assuming ...
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Enneper surface is an injective inmmersed surface

Show that the map $\varphi:\mathbb{R}^2\to \mathbb{R}^3$ defined by, $$\varphi(u,v)=\left(u-\frac{u^3}{3}+uv^2,v-\frac{v^3}{3}+vu^2,u^2-v^2\right),$$ is an injective immersed surface. The problem is ...
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Implicit Function Theorem on Surfaces.

(a) State the Implicit Function Theorem in the most general way that you know (b) Let $\Sigma$ the set of $2 \times 2$ matrices with determinant zero. Show that if $0 \neq M \in \Sigma$, then ...
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Surface covered by hyperbolic plane admits a hyperbolic metric

Let $S$ be a surface. Is it true that if $S$ is covered by the hyperbolic plane (or a subset thereof) then it admits a Riemannian metric of constant negative curvature? How does the metric (or ...
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Chapter 2.4 Exercise 1 in Do Carmo (Tangent Plane)

Show that the equation of the tangent plane at $p=(x_0,y_0,z_0)$ of a regular surface is given by $f(x,y,z)=0$ where $0$ is a regular value of $f$, is $$f_x(p)(x-x_0)+f_y(p)(y-y_0)+f_z(p)(z-z_0)$$
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Connected components of Enneper surface

Let $\varphi:\mathbb{R}^2\to\mathbb{R}^3$ be the inmersed surface (Enneper's surface) given by, $$\varphi(u,v)=\left(u-\frac{u^3}{3}+uv^2,v-\frac{v^3}{3}+vu^2,u^2-v^2\right).$$ Prove that a connected ...
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Connection between coordinates of surfaces

Let $$dx^2+dy^2+dz^2 = E(u,v)du^2+2F(u,v)dudv+G(u,v)dv^2$$ and $$dx^2+dy^2+dz^2 = \widetilde{E}d\widetilde{u}^2+2\widetilde{F}d\widetilde{u}d\widetilde{v}+\widetilde{G}d\widetilde{v}^2$$ Then ...
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Finding a vector function for the curve of intersection of two surfaces

I wanted to find a vector function $\mathbb{r}(t)$ for the curve of intersection between $z=\sqrt{x^2+y^2}$ and $z = y+5$. I understand that one way would be to let $x=t$ and then we can set $ \sqrt{...
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1answer
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Infinite solids similar to Gabriel's Horn

Do any solids of revolution exist with properties similar to Gabriel's Horn (i.e. a geometric solid with finite volume but infinite surface area)? Please restrict your answers to functions not in the ...
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Least Energy Path, Contour Following, around Hills toward Goal

I have a matrix of elevation values which could be said represents $h(x,y)$. I can obtain contours using this function that are like sides of hills, and I have a starting point and an end point. How ...
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2answers
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Killing cohomology of surfaces in finite covers

Let $S$ be a closed orientable surface and $R$ a commutative ring. Given a nonzero element $\alpha \in H^1(S;R)$ is there a finite cover $p : \tilde{S} \to S$ such that $p^*(\alpha) = 0 \in H^1(\...
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2answers
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A Suitable Function for Terrain, Mountain Modeling

On Google Maps and various other mapping programs, one can see contour lines that correspond to elevation. Sometimes these contour lines are concentric corresponding to a mountain. My question is ...
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Topological space formed by the identification of a unit square

I know that the first identification "rolls up" the unit square to form a cylinder with open ends but I'm unsure on how the next two "close up" the ends. Does it become a cylinder or more of a cone ...
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Show that the space $X$ is not a surface

I would like to show that the following space is not a surface: $X$ is made as an identification space of the unit square $Q=\{(x,y)\mid 0\leq x,y\leq1\}$ with the identifications: $(0,y)\sim(1,y)$ ...
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+300

Computing singularities of a surface

Let $Y$ be the abelian variety $\mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i] $. Let $X$ be the quotient of $Y$ by action of the group generated by the map $\eta(x,y)=(ix,iy)$. This group ...
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Proof about trace

Let f: $\mathbb{R}$ $\rightarrow$ $\mathbb{R}^2$ be a smooth parameterized curve. Defined by f(u) = ($t^3$,$t^2$). Prove that the trace of f (image of f) is $y^3=x^2$. Well, the image of f is $\{$ a $...
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What is the class of implicit surfaces for which the distance to an external point is returned by its equation?

Let's define an implicit surface as the set of all points for which a function f : R3 -> R equals 0. A sphere around origin, for example, can be defined as ...
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1answer
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Finding parametrization of the curve of intersection

Given 2 equations $z = x^2 - y^2$ and $z = x^2 +xy -1$, find a parametrization of the curve of the intersection of the surfaces. By equating them together, I get $y^2 +xy -1 =0$. Letting $x=t$, I ...
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2answers
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Retract of noncompact surface to its boundary?

Suppose $M$ is a connected, noncompact 2-manifold, and its boundary $\partial M$ is a circle. What's the simplest way to show there is a retraction $r: M\rightarrow \partial M$? Here are some ...
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1answer
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Gaussian curvature of surface

Consider the set $ \Sigma = \{ (x, y, z) \in \mathbb R ^3 : xyz=1 \} > $. Show that $ \Sigma$ is a surface and evaluate the Gaussian curvature at a general point $(x,y,z) \in \Sigma $ I've ...
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Name of the Quartic Surface $z=(x^2-a^2)(y^2-a^2)$

Does the surface defined by the following equation have an specific name? $$ f(x,y)=(x^2-a^2)(y^2-a^2)$$ I've searched a lot and found that $z=x^2y^2$ is called Crossed Trough. However I didn't find ...
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Classifying a surface using the Euler characteristic

I want to prove that if a surface is divisible in hexagons, in a way that every vertex is adjacent to three or more hexagons, then the surface is not a sphere nor the projective plane. I know that ...
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Shape operator of pseudo-spheres

In the following, I will refer by O'Neil to Barrett O'Neil's book Semi-Riemannian Geometry With Applications to Relativiy. Consider the $M=\{ x \in \mathbb{R}^n_s | f(x)=r^2\}$ with $f(x)=g_s(x,x)$ ...
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Distance a from point in R3 to a surface defined by a parametric curve and a radius function?

I'm interested in studying the class surfaces defined by: Take an arbitrary parametric curve f : {0..1} -> ℝ3. Pick an arbitrary radius function ...
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What is the type of this surface? (square with two bridges)

Take a square (without border) and build two bridges on it. You can go under the bridge or across the bridge (as my attempt at drawing it poorly attempts to describe). Since you can freely move the ...
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1answer
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Calculate $\int_{\Gamma}\omega$ when $\omega = x^2yzdx+xy^2zdy+xyz^2dz$ and $\Gamma$ is the intersection of surfaces $x=1$ and $y^2+z^2=1$

Calculate $\int_{\Gamma}\omega$, when: $\omega = x^2yzdx+xy^2zdy+xyz^2dz$ and $\Gamma$ is the intersection of the surfaces: $x=1$, $y^2+z^2=1$. Can you help me with that? At first, I thought of ...
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Help evaluating this surface integral, how to evaluate $dS$ in this?

Evaluating this surface integral $\int_{S}(a^2x^2+b^2y^2+c^2z^2)^{1/2}dS$ over the ellipsoid $S:ax^2+by^2+cz^2=1$ I am well aware of evaluating surface integral over any given sphere $x^2+y^2+z^2=a^...
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Prove the injectivity and differentiability of $x(u,v) = (u \cos(v), u \sin(v), \frac{u^2}{2}).$

Prove the injectivity and differentiability of $$x(u,v) = (u \cos(v), u \sin(v), \frac{u^2}{2})$$ where $u \in (0,\infty),$ $v \in (0,2\pi).$ What I really want to prove is that (U,x) is a simple ...
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1answer
42 views

Is valid the Brioschi Formula of Gaussian curvature in $R^n$?

Is the Brioschi formula, for Gaussian Curvature of a Surface, valid only in codimension 1, or is valid for a surface in $R^n$ with $n>3$?
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How to Find the area of the portion of the sphere $ x^2 + y^2 + z^2 = 1$ between the two parallel planes .

Find the area of the portion of the sphere $ x^2 + y^2 + z^2 = 1$ between the two parallel planes $ z = a$ and $z = b$ where $-1 < a < b < 1$ are parameters. How to solve this question using ...
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Differential Geometry - Local Isometries

Consider the following surfaces in $ \mathbb R ^3 $: $$ \Sigma _ 1 = \{ (x_1, x_2, x_3) \in \mathbb R ^3 : x_3 = x_1x_2 \} \\ \Sigma_2 = \{ (x_1, x_2, x_3) \in \mathbb R ^3 : x_3 = \dfrac{x_1^2-...
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Can we prove that two circles lie on a sphere?

There is a problem in my book In the question the circles are given in the form $S+kP=0$ where $S$ is plane and $P$ is a plane and $k$ is real number. The question asks us to prove that the circles ...
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Differential of inverse function to a tubular neighborhood

Suppose $S$ is a (regular) compact differentiable surface embedded in $\mathbb{R}^3$ so that tubular neighborhoods exist. Consider the diffeomorphism to one of them: $$F:S\times(-\epsilon,\epsilon)\...
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4answers
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Tangent plane to $x^2+y^2+z^2=50$ at $(3,4,5)$

Prove that the tangent plane to $x^2+y^2+z^2=50$ at $(3,4,5)$ is $3x+4y+5z=50$ My workings are shown below but get the answer wrong completly, have I made a simple mistake is my method flawed. So if ...
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Do these two definitions of hypersurfaces coincide?

According to Wikipedia and some other sources, a hypersurface is a manifold of dimension $n-1$ embedded in an ambient space of dimension $n$ which is simple to understand. Now here https://homepages....
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4answers
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Find the equation of the tangent plane to $xy+yz+zx=11$ when $x=1$ and $y=2$

Find the equation of the tangent plane to $xy+yz+zx=11$ when $x=1$ and $y=2$ giving the answer in the form $f(x,y,z)=k$, where $k$ is a constant and $k\in \Bbb{Z}$. So I know that the tangent plane ...
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0answers
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Newman's “proof” that surface groups are LERF?

In trying to find alternatives to Scott's paper, I came across Tretkoff Covering Spaces, Subgroup Separability, and the Generalized M. Hall Property, which references this paper of Newman to show ...
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Find the coordinates and nature of the stationary point of $z=x^3+y^3-6xy$

Find the coordinates and nature of the stationary point of $$z=x^3+y^3-6xy$$ So I have found all the partial derivatives but I'm not sure how to then find the stationary point. All I know is $\frac{\...
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0answers
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Geodesics with inverse accelerations in a submanifold

Let $S \subseteq \mathbb{R}^n$ be a smooth embedded submanifold. Consider $S$ as a Riemannian manifold, with the induced metric which comes from the standard Euclidean metric on $\mathbb{R}^n$. Is ...
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Partial derivatives of surface curvature relative to a tangent plane

If we take the tangent plane (say, $\mathsf{T}$) to a convex surface $\mathcal{S}$ associated with a given surface normal vector $\mathtt{n}$, I assume the surface will have a fixed rate of change ...
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Calculate surface normal and area for a non-planar quadrilateral

Given the four coordinates of the vertices, what is the best possible approximation to calculate surface area and outward normal for a quad? I currently join the midpoints of the sides, thus ...
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2answers
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Curves on surfaces lifting to embedded curves in finite covers

Let $S$ be a orientable closed surface with genus $g \geq 1$ and let $\gamma \subset S$ be an immersed curve. Does there exist a finite cover of $S$ where $\gamma$ lifts to a curve that is homotopic ...
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0answers
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Why is $i(\gamma,\mathcal{F}^u)+i(\gamma,\mathcal{F}^s)\geq\varepsilon$?

This is still about measured foliations... (I have asked a few questions about them.) The setting is as the following. Let $f:M\rightarrow M$ be a pseudo-Anosov map and $\mathcal{F}^s$ and $\mathcal{...
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2answers
53 views

Two Variable Function with Specific Properties (Challenging)

I am looking for a two variable function (a surface), $g(x,b)$, with the following properties: $$g:\ [0,\pi] \times (0,\pi) \to[0,\pi] $$ $$g(0,b)=0$$ $$g(\pi,b)=\pi$$ $$g(b,b)=\frac{\pi}{2}$$ $$g\...
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1answer
50 views

Calculate the area of $z=\frac{x^2}{2}+\frac{y^2}{2}\;$ that is enclosed by $x^2+\frac{y^2}{4}=1$

The exercise is the text in the title. I'm studying surface integrals. To start I thougt to make a change to cartesian coordinates with $z$ as a function of $x$ and $y$, that is, $z=\frac{x^2+y^2}{2}$ ...
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2answers
49 views

Geometric presentation of fundamental group of a surface

Let $S = S_g$ be a closed surface. An author of a paper writes: We say $\langle a_1, b_1, \cdots, a_{2g}, b_{2g} \ | \ R \rangle$ is a geometric presentation of the fundamental group $\pi_1(S)$ ...