Questions tagged [curves-and-surfaces]

A surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line

Filter by
Sorted by
Tagged with
1 vote
1 answer
98 views

Requirement of parametrization of surfaces

If I have a smooth surface $M$ (2D embedded in 3D), under what conditions I can assure that there exists a finite collection of charts $\{U_i, \phi_i \}_{i}$, with $\phi_i : U_i \to M$, such that its ...
user3646987's user avatar
3 votes
0 answers
106 views

Spin structures on surfaces in terms of homology classes

It is well known that the spin structures on an oriented surface (with boundary) $M$ are in bijection with the set of cohomology classes $H^1(M,\mathbb{Z}/2)$. By Lefschetz duality, these correspond ...
Tanny Sieben's user avatar
2 votes
0 answers
43 views

Efficiently determining surface intersections along a line segment

Background In general, I know how to determine the points of intersection between a surface and a line. In my case, I may have a large number of defined surfaces that may (or may not) intersect each ...
Sterling Butters's user avatar
1 vote
0 answers
123 views

Motivation of Zariski–Van Kampen theorem

The Zariski–Van Kampen theorem gives the presentation of the fundamental group of the complement of the plane curve of degree $d$. But what's the motivation of this theorem? More generally, why are ...
Katherine's user avatar
  • 123
6 votes
2 answers
121 views

Generate $\mathrm{Mod}(S_g)$ by two Dehn twists

Let $S_g$ be a closed orientable surface of genus $g>1$. How can one prove that its mapping class group $\mathrm{Mod}(S_g)$ is not generated by two Dehn twists? A pair of simple closed curves in $...
Andrey Ryabichev's user avatar
7 votes
2 answers
244 views

Does the surface area of the unit Lp ball go to zero for all $p < \infty$?

We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...
usul's user avatar
  • 4,260
0 votes
1 answer
103 views

Conditions for surface area of surface of revolution to be product of arclengths

Given a circle $C$ in the xz-plane which does not intersect the $z$-axis, we can build a smooth 2-torus with surface area $(2\pi a)(2\pi b)$ where $a$ is the radius of the circle $C$ and $b$ is the ...
locally trivial's user avatar
0 votes
0 answers
132 views

Implicit function theorem on curves

I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\...
Matthias Himmelmann's user avatar
4 votes
1 answer
166 views

Closed-form examples of CMC surfaces

Besides the trivial cases of cylinders and spheres, are there any other known examples of non-zero constant mean curvature surfaces which can be represented explicitly in a closed form? I am ...
Jiří Minarčík's user avatar
2 votes
0 answers
54 views

Irreducible components over a singular divisor

Setup. Let $K$ be an algebraically closed field of characteistic zero, let $X/K$ be a smooth projective surface and let $Z \subset X$ be an integral curve which is nonsingular except for a finite set ...
Jackson Morrow's user avatar
0 votes
0 answers
209 views

Geometric meaning of cusps/component labels in Katz-Mazur book

In Katz-Mazur book "Arithmetic moduli of elliptic curves" there is a very short section (see image below) regarding the cusp-labels and component-labels. The set of cusps labels intuitively ...
manifold's user avatar
  • 169
1 vote
0 answers
37 views

Reference for preimage of boundary of spacefilling curve

Given a continuous map $\gamma$ from $[0,1]$ onto a bounded contractible subset $S$ of $\mathbb R^2$ such that $S$ contains an open subset of $\mathbb R^2$ which is dense in $S$, the preimage $\gamma^{...
Roland Bacher's user avatar
2 votes
1 answer
230 views

Planar curves in $M^{m}$ vs curves in $M^{2}$

Following Anton Petrunin’s suggestion, I revise the question to make it less vague. Let $M^{m}$ be an $m$-dimensional Riemannian manifold, and let $\gamma$ be a unit-speed curve $I \to M^{m}$. We say ...
MK7's user avatar
  • 823
0 votes
0 answers
77 views

If $\kappa^{2} > 0$ and $\tau^{2} >0$, how small can $\kappa_{n}^{2} + \tau_{g}^{2}$ be?

Let $\gamma$ be a unit-speed curve lying on a surface $S$ in $\mathbb{R}^{3}$, and suppose that both the curvature $\kappa$ and the torsion $\tau$ of $\gamma$ are everywhere nonzero. Since $\gamma$ is ...
MK7's user avatar
  • 823
22 votes
0 answers
451 views

Sphere with bounded curvature

Let $V$ be a body in $\mathbb{R}^3$ bounded by a smooth sphere with principal curvatures at most 1 (by absolute value). Is it true that $$\mathop{\rm vol} V\ge \mathop{\rm vol} B,$$ where $B$ denotes ...
Anton Petrunin's user avatar
8 votes
0 answers
160 views

Intuition for the volume form - combinatorial definition?

I apologize that this is short of research level but I have realized that I am not happy with my understanding of the volume form on an oriented Riemannian manifold and I was hoping to find some ...
Sprotte's user avatar
  • 933
1 vote
1 answer
58 views

Cross product of two infinitesimal bendings

Let $M$ be a smooth (embedded or immersed) surface in $\mathbb{R}^3$. Let $Z_1,Z_2$ be two vector fields along $M$, thought of as $\mathbb{R}^3$-valued functions, satisfying the following differential ...
Dmitry K's user avatar
  • 1,414
1 vote
1 answer
176 views

Singularities of arithmetic surface

I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves. The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
Matthias's user avatar
  • 113
6 votes
1 answer
327 views

Can every smooth space curve be realized as an origami curved crease?

Many years ago, Ron Resch told me that he proved that every smooth simple space curve $C$ could be realized as a curved crease $\gamma$ in the interior of a piece of paper. He never published this (as ...
Joseph O'Rourke's user avatar
3 votes
1 answer
138 views

Smoothness of ruled surface (asymptotic) parameterisations

A ruled surface $S$ shall be defined as surface consisting of straight line segments. It is commonly known (cf. [BER, p.362] or [STR, p.93] - bibliography at the end) that a ruled surface allows for a ...
Benjamin Bauer's user avatar
2 votes
0 answers
186 views

Why is $H$ the standard notation for mean curvature?

I am curious about the origin of the notation $H$ to denote the mean curvature of a surface in $\mathbb{R}^{3}$. I suppose that the symbol $K$, which is commonly used to denote the Gaussian curvature, ...
MK7's user avatar
  • 823
1 vote
1 answer
101 views

Smoothness of the asymptotic parametrization of a ruled surface

Let $S$ be a smooth developable surface in $\mathbb{R}^{3}$. It is well known that, if $S$ is free of planar points, then it admits a local parametrization of the form $$\begin{align} \sigma \colon I \...
MK7's user avatar
  • 823
2 votes
0 answers
163 views

rational curves over K3 surfaces over $\mathbb{Q}$

There are many partial results towards the following conjecture: Every projective K3 surface over an algebraically closed field contains infinitely many integral rational curves. My question is: is ...
did's user avatar
  • 449
3 votes
1 answer
241 views

Where's the negative section of a deformation of a Hirzebruch surface?

As in Deformations of Hirzebruch surfaces and toric action, the Hirzebruch surface $F_n$ can be deformed into $F_{n-2m}$ ($0<2m\leq n$) under the fibration given by $$ M=\{([x_0:x_1],[y_0:y_1:y_2],...
yuki swou's user avatar
-1 votes
1 answer
427 views

Essential simple closed curves in a torus [closed]

Definition: By a closed curve in a surface $S$ we will mean a continuous map $S^1 \to S$. We will usually identify a closed curve with its image in $S$. A closed curve is called essential if it is not ...
T566y65tt's user avatar
1 vote
1 answer
213 views

About isotopy of simple close curve

In the Primer mapping class group by farb Margalit. We have : Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\alpha$ is isotopic to $\beta$ if ...
T566y65tt's user avatar
3 votes
0 answers
134 views

The classification of developable surfaces: Are these statements equivalent?

This is a cross-post from MSE (https://math.stackexchange.com/q/4330772/242708). I thought to know very well the answer to the classification problem for developable surfaces, so I sought for some ...
MK7's user avatar
  • 823
3 votes
1 answer
355 views

Is the Moebius strip Riemannian homogeneous?

Let $ M $ be the Moebius band. In other words, the total space of the nontrivial line bundle over the circle. Can we equip $ M $ with a metric such the the isometry group acts transitively? My ...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
281 views

Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$?

Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$ and geometrically which things it represents?
MAS's user avatar
  • 776
8 votes
1 answer
489 views

What is the status of Jordan's theorem in constructive mathematics in the language of locales?

By constructive mathematics in this matter we mean intuitionistic ZF (*). In the language of locales, the Jordan curve can be defined as $f\colon S^1 \to X$ such that "if $U \cap V = \varnothing$,...
Arshak Aivazian's user avatar
1 vote
0 answers
23 views

Separability of graph component embeddings

I have an undirected graph. I also have an embedding of the graph in $\mathbb{R}^3$. Assume that the graph has 2 connected components. I want to know whether there exists a plane (or better - any ...
user1747134's user avatar
2 votes
0 answers
89 views

Can we define surface integral on 'bad surface'?

We can define a surface integral on a piecewise smooth surface, but if the surface is not piecewise smooth can we use measure theory to generalize the definition of surface integral? And does Stokes ...
YuerWu's user avatar
  • 405
0 votes
1 answer
75 views

How can I find the area of a rectangle created by the spiral r=theta at a certain theta? [closed]

I would like to have a function that gives the area of a rectangle at a certain theta of the spiral r=theta. The height of the rectangle is the y value of the point on the spiral and the base of the ...
Elan SK's user avatar
2 votes
0 answers
95 views

understanding the definition of subgroup of the Grothendieck-Teichmuller group

Definition. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an ...
1200785626's user avatar
0 votes
0 answers
88 views

Pushing figures into holes

Let $\gamma_1,\gamma_2:[0,1]\to \mathbb{R}^2$ - smooth curve, $\gamma_i(0)=\gamma_i(1)$, $X_1$ and $X_2$ are the areas bounded by the corresponding curves. . Suppose we have an $X_1 $-shaped hole, and ...
Ben Tom's user avatar
  • 107
2 votes
1 answer
292 views

What's the name of this surface: $z = \exp(xy)$ [exponentialoid?] [closed]

When studying the real value exponential, I encounter the surface $z = e^{x\cdot y}$ but I don't know if it has a name. I've created a 3D applet to explore it. When I cut it by the plane $$ (x-x_0)\...
Harusada's user avatar
2 votes
1 answer
370 views

Continuity of the perimeter of level sets w.r.t. level function

Working with the level set method introduced by Osher & Sethian in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the ...
Bogdan's user avatar
  • 1,154
5 votes
2 answers
698 views

Continuity of Hausdorff measure on level sets

Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that: $\bullet$ $\phi^{-1}(0)\neq\emptyset$; $\bullet$ $\nabla\phi(x)\...
Bogdan's user avatar
  • 1,154
4 votes
1 answer
107 views

Classes of curves closed under Minkowsky sum

Let $C$ be a class of plane curves, regarded as subsets of $\mathbb{R}^2$ (parametrization won't matter), I'm thinking for example of splines or algebraic subsets. Let $D$ be a class of topological ...
Stefan Witzel's user avatar
3 votes
1 answer
315 views

When does the Hirzebruch surface have a nef anticanonical divisor?

Let $\mathcal H_r=\mathbb P (\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(r))$ be a Hirzebruch surface for some $r\in\mathbb Z$. As a toric variety, the fan structure is spanned by $(-1,0)$,...
Hang's user avatar
  • 2,661
6 votes
2 answers
311 views

Necessary and sufficient curvature condition for a regular planar curve to be simple and closed

Given a smooth, $2\pi$-periodic function $\kappa(s)$, the associated planar curve $\gamma(s)$ for which $\kappa(s)$ is the (signed) curvature, is uniquely determined up to Euclidean invariance: a ...
Frits Veerman's user avatar
5 votes
1 answer
205 views

When does a spherical curve equal its tangent indicatrix?

Given a smooth regular curve $\gamma$ in $\mathbb{R}^{3}$, one defines the tangent indicatrix of $\gamma$ to be the spherical curve $\gamma'/\lVert \gamma'\rVert$. It is then natural to look for ...
MK7's user avatar
  • 823
3 votes
0 answers
105 views

Existence of developable ribbonization of a surface

Let $S$ be a smooth compact surface embedded in $\mathbb{R}^{3}$. It is well-known that there exists a triangulation of $S$. I am considering an alternative way of approximating $S$, where instead of ...
MK7's user avatar
  • 823
6 votes
1 answer
443 views

When is the cut locus a finite tree?

Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for ...
Leo Moos's user avatar
  • 4,536
1 vote
0 answers
135 views

Understanding sheaves on normalisation of a curve: $v_* \mathcal{O}_{\tilde{C}} / \mathcal{O}_C$

Let $(C, \mathcal{O}_C)$ be a reduced irreducible curve and $(\tilde{C},\mathcal{O}_{\tilde{C}})$ its normalisation with $v : \tilde{C} \rightarrow C$. Then we have an imoprtant skyscraper sheaf $v_* \...
XT Chen's user avatar
  • 1,034
0 votes
1 answer
147 views

"Arc" length parametrization for surfaces

If we have a function $\phi:\Omega\subseteq\mathbb{R}^2\to\mathbb{R}$, $\phi\in C^2(\Omega)$ is there a way to find a function $d:\Omega\to\mathbb{R}, d\in C^2(\Omega)$ so that: $|\nabla d(x,y)|=1,\ \...
Bogdan's user avatar
  • 1,154
2 votes
1 answer
157 views

Hyperbolic length of curve that does not enter a collar

Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that ...
user158773's user avatar
9 votes
3 answers
685 views

Is there the longest geodesic?

Given a closed 2-surface $M$ together with a Riemannian metric $g$. We pick a free homotopy class $\gamma \in \pi_1(M)$ and consider the set $C(\gamma)$ of all closed geodesics homotopic to $\gamma$. ...
Enumerator's user avatar
2 votes
0 answers
150 views

Degree $4$ curves on K3 double covers of Del-Pezzo surfaces

Let $S$ be a smooth del-Pezzo surface and $\pi : X \longrightarrow S$ be the double cover of $S$ ramified in a smooth section of $-2K_S$. Going through the classification of del-Pezzo surfaces, one ...
Libli's user avatar
  • 6,756
13 votes
1 answer
334 views

Are any of these complex surfaces ever projective?

Let $C$ and $T$ be compact connected Riemann surfaces (or: smooth projective connected curves over $\mathbb{C}$) of genus at least two and let $X:=C\times T$. Let $(c,t)$ be a point of $X$, and let $...
Ariyan Javanpeykar's user avatar

1
2 3 4 5
7