# 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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 ...

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### 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)(\...

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### 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}(...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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:
$\...

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### 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 ...

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### 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'(...

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### 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 $p$.
...

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### 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 ...

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### 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\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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 ...

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### 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{\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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^{...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 \...

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### 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 ...

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### 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],...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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 $U \cap V = \varnothing$,...

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### 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 ...