# 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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### 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 very 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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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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)\...

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### 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 ...

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### 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)\...

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### 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 ...

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### 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 $(-1,0)$,...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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_* \...

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### "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,\ \...

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### 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 ...

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### 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 $\gamma$.
...

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### 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 ...

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### 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 $...