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Questions tagged [surfaces]

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

2
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2answers
73 views

Let $X$ be a K3 surface, show that $H_1(X,\mathbb Z)=0$

Let $X$ be a(n algebraic) K3 surface, i.e., $X$ is a smooth algebraic surface with trivial canonical bundle and $H^1(X,\mathcal{O})=0$. This assumption directly implies that $H^1(X,\mathbb C)=0$, so $...
0
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1answer
29 views

calc help: rate of change w/ respect to t of the surface area

As a spherical balloon is being inflated, its radius (in centimeters) after $t$ minutes is given by $r(t)=3\left[(t+8)^{1/3}\right]$ where $0<t<10$. What is the rate of change with respect to $t$...
2
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0answers
46 views

About the covariant derivative - is this correct?

Let $S \subset \mathbb{R}^3$ be a regular surface, the image of a parametrization $X: U \subset \mathbb{R}^2 \to S$ and $\alpha:I \to W \subset S$ be a curve in $S$. We can write $\alpha(t) = X(u(t), ...
1
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0answers
20 views

Is there a solvable subgroup with finite index and finite type in the mapping class group of a surface?

I want to find a subgroup $H$ of the (orientation preserving) mapping class group $G=MCG(g,n)$ of a surface with genus $g$ and $n$ boundary components that satisfies the following properties: $[G:H]&...
0
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1answer
49 views

Compact surface with constant strictly positive curvature is a sphere

I'm following Cartan's Differential forms. I'm trying to do exercise 8 on page 161. The chapter is about moving frames and differential forms in surface theory. Consider the frame of Ex. 2 (...
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1answer
31 views

Can someone check if my proof is correct?

I was working a tutorial and it had this proof listed below. It says that S is a closed surface and H is a region $$\int_S \frac{\textbf{r.n}}{r^2} dS\, = \int_H \frac{dH}{r^2} \,$$ My approach ...
0
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2answers
41 views

Flux outward Sphere-Cylinder

$D=\{x^2+y^2+z^2\le 25,y^2+z^2\le 9\}$ $F=\{y^2,x^2,z\}$ I need to calculate the flux outward the boundary of $D$. I think I can use the divergent theorem, but How can I define the triple integral ...
1
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1answer
28 views

Generalisation of elementary result for embedded surfaces

If we have a triangle $T$ in $\mathbb{R}^3$ and we consider the 3 canonical projections $$\pi_{xy}(x,y,z)=(x,y)$$ $$\pi_{xz}(x,y,z)=(x,z)$$ $$\pi_{yz}(x,y,z)=(y,z)$$ Then we have a pythagoras-type ...
2
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1answer
35 views

find flux outward a sphere cutted with $y\le-4$

$$D=\{x^2+y^2+z^2\le 25,y\le -4\}. \qquad F=\{z^2,y^2,x^2\}$$ In order to find the total flux going outward D I need to evaluate two fluxes(or maybe not ? ): the first one is the flux of the ...
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0answers
55 views
+100

Surface with mean curvature proportional to height $z$

This curve, described with the inverse hyperbolic secant $\text{arsech}(x) = \text{arcosh}(1/x)$, $$y = \text{arsech}\frac x2 - \sqrt{4-x^2}$$ has its curvature equal to the linear coordinate $x$: $...
0
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1answer
24 views

Given rotor and curve find circulation of a vector field

Given the curve $C$ of equation $$\vec X=(3\cos t,3\sin t,6\cos t),\qquad0\leq t\leq2\pi$$ oriented according imposes this parameterization, find the circulation of $\vec f$ along $C$ if $\vec f\in\...
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0answers
40 views

Degree $deg(f)$ of a Fibration

Let $S$ be a surface (so a $2$ dimensional, proper $k$-scheme) and $B$ a curve (" $1$ dim " ). Consider a finite fibration $f:S \to B$ (so we have $\mathcal{O}_B = f_*\mathcal{O}_S $). My question ...
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0answers
20 views

Curve tracing : paraboloid [closed]

I know the equation $ x^2+y ^2=-z $ is paraboloid along negative z axis passing through origin...but what if there is absolute constant term in this equation... eg. $ x^2+y ^2=-z+2 $ what type of ...
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0answers
29 views

Examples of smooth implicit curves and surfaces

I am currently constructing a method to approximate implicitly given plane curves and surfaces, which are smooth and single-sheeted. Now I have finished writing a Matlab function doing all the ...
3
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1answer
62 views

How can one generalize the Gauss map to higher dimensions? More specifically, bi-dimensional manifolds in $\mathbb{R}^4$

It's easy to define the unitary tangent fields of a $2$-dimensional surface $S: I \times J \to \mathbb{R}^4$, but since I don't have the cross product anymore, an unitary normal field is harder to ...
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0answers
25 views

Locally Euclidean Surfaces

In Stillwell's Geometry of Surfaces, he defines a locally euclidean surface $S$ as a set equipped with a distance function $d_S(A,B)$ defined for every $A,B \in S$ such that for every $A \in S$, there ...
0
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1answer
51 views

parameterization of a part of a sphere

I have to parametrize $D=\{x^2+y^2+z^2\le 25,y\le -4\}$. I can see the I have to parametrize 2 surfaces : ($S_1$) the intersection between the plane $z=-4$ and the sphere: ($x^2+z^2\le 9$) ($S_2$) ...
1
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1answer
51 views

How can one best visualize two dimensional manifolds in $\mathbb{R^4}$ (more specifically, $\mathbb{S}^2 \times \mathbb{R})$?

I'm trying to "get a picture", so to speak, of hypersurfaces in $\mathbb{S}^2 \times \mathbb{R}$. One example would be $\left(\dfrac{\cos(u)}{\sqrt{1+u^2}}, \dfrac{\sin(u)}{\sqrt{1+u^2}},\dfrac{u}{\...
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0answers
61 views

Does the globally shortest path between two points on a surface in 3D satisfy the geodesic equation except for countably often?

Let $P:[0,1]\to S$ be the shortest path between two points on a compact regular surface. Will $P$ always be geodesic except for countably many turns, or could it be that there is a whole subinterval ...
9
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2answers
225 views

Whilst There Are Three Characteristic Equations, Only Two of Them Are Linearly Independent?

Take the general quasi-linear equation $$a(x, y, u)u_x + b(x, y, u)u_y - c(x, y, u) = 0. \tag{1}$$ We assume that there exists a solution of the form $u = u(x, y)$. We can define a solution ...
1
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2answers
32 views

Gradient vector $\nabla F = (z_x, z_y, -1)$ is normal to the integral surface?

I have an integral surface $z = z(x, y)$. Writing this integral surface in implicit form, we get $$F(x, y, z) = z(x, y) - z = 0$$ I am then told that the gradient vector $\nabla F = (z_x, z_y, -1)$ ...
2
votes
1answer
58 views

Parametric Surface Equations for Orthogonal Projection of Torus Knot Tube onto Torus

What are the parametric equations for the orthogonal projection of the torus knot tube onto the torus surface? For instance, if we have the equations for the torus knot $$ \vec r(t)= (R+r\cos pt)\...
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0answers
16 views

inverse function theorem and mapping of surface

The discussion of regular mappings in Section 7 of Chapter 1 translates easily to the case of a mapping of surfaces $F: M \rightarrow N$. $F$ is regular provided all of its derivative maps $F_{*p}: ...
5
votes
1answer
47 views

Decomposing surfaces into pairs of pants

I apologise if this question is badly worded/doens't make sense - if I knew how to formulate this question well, I'd probably be half-way to answering it myself. Let $F_g$ be a closed surface of ...
0
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1answer
30 views

Shape operator and path connected surface (global theory of surface)

3.1 Theorem (O'neill) If its shape operator is identically zero, them M is part of a plane in $R^3$ The first part of Proof. By the definition of shape operator, $S = 0$ means that any unit normal ...
2
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2answers
68 views

General definition of Surfaces in $\mathbb{R}^{n}$

What is a definition of a sphere that contains only topological information? And that of a torus? I can only Think of definitions like "points in $\mathbb{R}^{n}$ that satisfies: $f(x) = 0$" but is ...
2
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0answers
35 views

Reference request fundamental group of surface of genus $g$ and $n$ boundary components

Let $\Sigma_{g,n}$ be the compact, oriented surface of genus $g$ with $n$ disjoint open discs removed, where the boundary circles are called $\partial_i$. I would like to find a reference for the ...
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0answers
57 views

Different definitions of ruled surface

Let $X$ be a projective smooth surface. I read two different definitions of $X$ to be a ruled surface, namely: $X$ is birational equivalent to some $B\times \mathbb P^1$, where $B$ is projective ...
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1answer
53 views

Mapping on a surface [closed]

How can I map a 2D flat picture on a curved surface whilst maintaining distances and angles between points of the flat picture as much as possible?
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2answers
52 views

surface fitting with least square

I want to fit a polynomial surface to some 3d points. First in a loop, matrix R is calculated as follows: ...
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0answers
46 views

Does this “algebraic” method for the application of the constructive proof of the classification of closed & compact surfaces have any use?

In a typical (at least from what I have seen) geometric topology course, when the classification of closed & compact surfaces is introduced, what has been done first is to find equivalent surfaces ...
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1answer
47 views

Pencil of Curves on a Surface

Let $X$ be a projective surface. A pencil of curves with base $B$ is a dominant rational map $\pi: X \dashrightarrow B$ such that $k(B)$ is algebraically closed in $k(X)$. But we also have a ...
1
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1answer
198 views

how can we parametrize the swept surface by a curve on a 2D plane?

for simplicity let's assume the curve is just a straight line of length 1. say that curve is put on a plane. and assume we apply some affine transformations to it that doesn't change its length (like ...
0
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1answer
29 views

Directional Derivative on a Surface

The terms used in the following context follow the book Differential Geometry by Do Carmo. Let $S$ be a regular surface. Let $X(u_1,u_2)=X:U \rightarrow S$ be a parametrization of $S$. Given two ...
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0answers
31 views

Smoothness of Surfaces in $\Bbb R^4$

Let $U=\{(x,y,z): x>0,y>0,z>0\}$ be the first octant, and let $g:U\to\Bbb R^4$ be given by $g(t,u,v)=(tu,tv,uv,tuv)$. Determine whether the image of g defines a smooth surface. I know how to ...
2
votes
1answer
43 views

Uniformization theorem for Riemannian 2-manifolds with boundary? Specifically the closed disc.

Suppose that $D = \{ x \in \mathbb{R}^2 | ||x||_2 \leq 1 \}$ is the closed disc, and let $g$ be any Riemannian metric on $D$. I'm wondering about the following version of uniformization: Is $(D,g)$ ...
1
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1answer
69 views

First variation of mass

Reference: this paper page 15 Let $(M,g)$ be a three-manifold and consider a two-sided compact surface $\Sigma \subset M$. The mass is defined by $$m(\Sigma) =\ \bigg( \frac{|\Sigma|}{16\pi} \...
2
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2answers
26 views

Finding image of normal Gauss map on paraboloid.

Describe the region of unit sphere covered by normal Gauss map on the paraboloid $z = x^2+y^2$. I did this way: Consider the parametrization $X(u,v) = (u,v,u^2+v^2)$. Then $X_u = (1,0,2u)$ and $X_v = ...
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0answers
26 views

Why does the gradient of a level surface represent the differential tangent plane?

I understand that the gradient is the direction fastest rate of change and why this is true, but just because its direction is orthogonal to the surface, doesn't mean its magnitude is that of the ...
0
votes
1answer
17 views

Surface area of a cone under a plane

How do I find the surface area of of the cone $z^{2}=x^{2}+y^{2}$, $z\geq0$ under the plane $x+y+z=2a$, $a>0$? I tried to use cylindrical coordinates as follows: $$\left\{\begin{array}{l} x=r\cos\...
1
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2answers
57 views

Rotation of ellipsoid(quadric)

Consider $$φ(x, y , z) = x^2 + 2y^2 + 4z^2 −xy −2xz −3yz$$ find the coordinate transformation (translation or rotation) to eliminate $xy$, $xz$ and $yz$. In $\mathbb R²$, with conic sections, I would ...
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1answer
22 views

How can I show that $\langle v , w \rangle = \lambda \langle d {\Phi}_p(v) , d {\Phi}_p(w) \rangle$ for any $\lambda \in \mathbb{R}$?

Let $S$ and $S'$ be two surfaces and let $\Phi : S \to S'$ be a diffeomorphism. We suppose that $\Phi \circ \alpha$ is a geodesic in $S'$ for all geodesic $\alpha$ in $S$. If $S$ is connected and $\...
1
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1answer
166 views

Prove $\int_{\Sigma_r} |\nabla\varphi|^2 d\sigma_r \ge \frac{2}{u(r)^2} \int_{\Sigma_r} (\varphi - \bar{\varphi})^2 d\sigma_r $

Reference: this paper Given the deSitter-Schwarzschild metric with mass $m > 0$ and scalar curvature equal to $2$ is the metric $$\bigg( 1 -\frac{r^2}{3}-\frac{2m}{r} \bigg)^{-1} dr^2 + ...
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0answers
32 views

The B$\ddot{\text{o}}$chner-Weitzenb$\ddot{\text{o}}$ck identity on 2-surface $\Sigma_r$

Reference: this paper page 7. The B$\ddot{\text{o}}$chner-Weitzenb$\ddot{\text{o}}$ck identity applied on $\Sigma_r$ \begin{align} \frac{1}{2} \Delta |\nabla\phi|^2 =\ & |\nabla^2 \phi|^2 + \...
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0answers
22 views

Can we find a curve that covers the unit sphere s.t any point $x$ on the unit sphere $S^2$, the curve passes through $x$ at least once?

Can we find a curve such that it covers the unit sphere in a way that given any point $x$ on the unit sphere $S^2$, the curve passes through $x$ at least once ?
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2answers
26 views

Surface as an intersection of a sphere and a cone

I have to evaluate the density of the solid bounded by the surface $$(x^{2}+y^{2}+z^{2})^{2}=x^{2}+y^{2}$$ How can I see this surface? Is it a intersection of a sphere and a cone?
2
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0answers
52 views

Calculate $\phi \langle \nabla H , \nabla \phi \rangle$

Let $H$ be the mean curvature of a hypersurface $\Sigma \subset M$ and $\phi \in C^\infty(\Sigma)$. I want to calculate $\phi \langle \nabla H , \nabla \phi \rangle$. Here I have two options: $$\...
1
vote
1answer
31 views

Curve of Intersection between a Surface and a Plane

The following terms are defined as in the book Differential Geometry by Do Carmo. Let $S$ be a regular surface, $p \in S,$ $N$ be a normal vector at $p$ and $v \in T_pS$. Let $P$ be the plane ...
0
votes
1answer
18 views

Homeomorphism between sequence of edges

I have a problem of classification of topological surfaces. Let S be the surface given by the sequence of identifications A B C B E where A,B,C,E are sequences of edges. Let $\beta$ be a letter which ...
3
votes
0answers
46 views

What Makes a Ruled Surface Rational

I am new to the study of algebraic geometry and have decided to study it for a personal project of mine. As such I have been trying to understand what differentiates a rational and an irrational ruled ...