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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

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Equivariant disk theorem in dimension 2

All groups I'll consider are finite. An important part of equivariant differential topology in dimension 3 is the equivariant disk theorem, which says that for a $G$-action on a compact $3$-manifold ...
Evan Scott's user avatar
0 votes
1 answer
49 views

Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K? [closed]

Let $S$ be a compact connected surface. Let $K$ be a compact subset of $S$ and suppose that $K$ has a finite number of connected components. Let $U$ be a connected component of $S \setminus K$ and ...
Fernando Oliveira's user avatar
0 votes
0 answers
25 views

Is it possible to find the intersection of this involute and roulette, given their parametric equations?

Background I have two parametric curves, and I want to find the parameter values of their intersection point closest to zero under certain conditions. The first curve is an involute of a circle with ...
Lawton's user avatar
  • 105
2 votes
0 answers
52 views

Is a $C^1$ surface with $C^1$ boundary and uniformly continuous normal a manifold with boundary?

Consider an oriented $C^1$ surface ${\cal S}$ whose closure is a $C^0$ surface with boundary whose boundary is a $C^1$ curve. If the normal to ${\cal S}$ is uniformly continuous, so that it has a ...
Brian Seguin's user avatar
1 vote
0 answers
80 views

Instantaneous rotation field in relation to a developable surface

I have a ruled surface, let it be given by $\Sigma: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ parametrized by $(u,v)$ with the rulings along the $u$-lines. Now, let $X: U \subset \mathbb{R}^2 \...
RWien's user avatar
  • 193
4 votes
1 answer
132 views

Characterization of convexity by connectedness of hyperplane sections

Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$. Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...
Dmitrii Korshunov's user avatar
1 vote
1 answer
336 views

Topological degree of differentiable map using line integrals?

Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$ I can then define $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...
António Borges Santos's user avatar
9 votes
2 answers
554 views

Is a local system on a surface determined by simple closed loops?

Let $\Sigma$ be a closed oriented topological surface of genus $g\geq 2$, and let $\mathfrak{X}_n$ denote the $\mathrm{SL}_n$-character variety of $\pi_1(\Sigma)$, i.e. $$ \mathfrak{X}_n= \mathrm{Hom}(...
Josh Lam's user avatar
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1 vote
1 answer
195 views

Question on ideal triangulation and geodesic lamination

Q1. Does a closed hyperbolic surface admit an ideal triangulation? Here, an ideal triangulation of a surface means a partition of a surface by geodesics such that each component of the complement ...
one potato two potato's user avatar
3 votes
0 answers
116 views

Divide Euclidean space by surfaces

It is well known that $n$ hyperplanes in $\mathbb{R}^k$ can divide $\mathbb{R}^k$ into at most $p$ regions where $p$ is \begin{equation} 1 + n + C^2_n + \cdots + C^k_n \end{equation} Is there similar ...
Hao Yu's user avatar
  • 185
4 votes
2 answers
334 views

Holomorphic Gauss normal map

Let $S$ be a Riemann surface smoothly embedded in $\mathbb{R}^3$. Is there necessarily a smooth embedding of $S$ in $\mathbb{R}^3$ such that the Gauss normal map $n:S \to S^2$ would be a holomorphic ...
Ali Taghavi's user avatar
0 votes
0 answers
101 views

Under what condition can a smooth map be factored through the Gauss normal map

Inspired by this question entitled When does the shape operator commute with a derivative? we ask the following question: Assume that $S,H$ are two surfaces whose corresponding Gauss maps are denoted ...
Ali Taghavi's user avatar
3 votes
0 answers
160 views

A higher-dimensional "line of curvature"?

Let $M$ be a submanifold of a Riemannian manifold $Q$, and let $\Gamma$ be a $d$-dimensional submanifold of $M$, i.e., $\Gamma^{d} \subset M \subset Q$. Suppose that, for all (unit) normal vectors of $...
Matteo Raffaelli's user avatar
2 votes
1 answer
104 views

Parametrization of $k$-ruled submanifold: can we choose the base to be orthogonal to the rulings?

A smooth $m$-dimensional submanifold of $\mathbb{R}^{d}$ is said to be $k$-ruled if it is foliated by $k$-dimensional planes, called rulings. Let $M$ be a $k$-ruled submanifold. Then $M$ can be ...
Matteo Raffaelli's user avatar
11 votes
1 answer
764 views

When does the shape operator commute with a derivative?

Suppose we have a smooth map $\varphi : S\to H$ between two regular parametric surfaces in $\mathbb{R}^3.$ Then at any point $p\in S,$ we have following maps between corresponding tangent spaces: $\...
Bumblebee's user avatar
  • 1,019
2 votes
1 answer
403 views

What are the best definitions for smoothness of a 2D curve (real-valued function)?

Sounds like a trivial question, but could not find any answer other than the fact that there are many ways to define it. My problem is this: I look at different elevation maps, some with sharp ...
Vincent Granville's user avatar
1 vote
0 answers
119 views

Does there always exist a regular curve connecting two points in an open connected subset of $\mathbb{R}^n$? [closed]

As the title says, given $A\subseteq \mathbb{R}^n$ open and connected and $x, y\in A$, I am looking for a continuous curve $\gamma:[0, 1]\rightarrow A$ which is differentiable in $(0,1)$ with $\gamma'(...
roxingby's user avatar
2 votes
1 answer
460 views

Number of points on a surface modulo p

I am guessing that the number of solutions $(x_1,x_2,\cdots ,x_s)$ modulo $p$ of the system of polynomials $$x_1x_2\cdots x_s=1,$$ $$(x_1-1)(x_2-1)\cdots (x_s-1)=u$$ where $u$ is non-zero modulo $p$. ...
user avatar
1 vote
1 answer
166 views

Identifying a curve on a closed surface of genus 4

The notation is the one used in the attached picture. Take a closed, orientable surface $\Sigma_4$ of genus $4$, obtained as the identification space of a polygon with $16$ sides in the usual way. The ...
Francesco Polizzi's user avatar
4 votes
0 answers
54 views

Kernel of the geometric intersection form

Let $\Sigma$ be a closed surface and $\mathcal C$ the set of all free homotopy classes of closed (may be nonsimple) curves in $\Sigma$. Consider the geometric intersection form $i$ on $\mathbb Z\...
nim's user avatar
  • 357
1 vote
1 answer
131 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
131 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
50 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
2 votes
1 answer
243 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 ...
Ktt's user avatar
  • 197
6 votes
2 answers
175 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
8 votes
2 answers
273 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
  • 4,489
0 votes
1 answer
149 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
144 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
191 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
65 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
253 views

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

In Katz-Mazur book "Arithmetic moduli of elliptic curves" there is a short section (see image below) regarding the cusp-labels and component-labels. The set of cusps labels intuitively ...
manifold's user avatar
  • 299
1 vote
0 answers
43 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
3 votes
1 answer
251 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 ...
Matteo Raffaelli's user avatar
22 votes
0 answers
520 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
242 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
  • 1,065
1 vote
1 answer
72 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 ...
Dmitrii Korshunov's user avatar
1 vote
1 answer
225 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
  • 203
6 votes
1 answer
408 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
184 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
215 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, ...
Matteo Raffaelli's user avatar
1 vote
1 answer
129 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 \...
Matteo Raffaelli's user avatar
2 votes
0 answers
188 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
  • 595
3 votes
1 answer
314 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
707 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
  • 119
1 vote
1 answer
319 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
  • 119
3 votes
0 answers
172 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 ...
Matteo Raffaelli's user avatar
3 votes
1 answer
446 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
319 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
  • 870
8 votes
1 answer
513 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
29 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

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