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
354
questions
3
votes
2
answers
238
views
Is the radial projection map area increasing?
Let $S$ be a hypersurface enclosed inside the unit sphere in $R^n$. Assume that every ray $\{t x: t \geq 0 \}$ intersects $S$ at most once.
Is it always true that ${\rm Area}(S) \leq {\rm Area}(P(S))$...
3
votes
1
answer
915
views
How to prove a developable surface must be ruled surface?
A developable surface is a smooth surface whose Gaussian curvature vanishes everywhere. A ruled surface is a surface where for each point there must be a line passing through the point lying on the ...
3
votes
1
answer
308
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
$$
M=\{([x_0:x_1],[y_0:y_1:y_2],...
3
votes
1
answer
547
views
Intersection of closed geodesics in hyperbolic surface
This question may be easy but I could not come up with a proof.
Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed non-...
3
votes
1
answer
738
views
How to find the action of an automorphism on the 27 lines on a cubic surface?
Assume $C\subset \mathbb{P}^2$ is a smooth cubic curve. Then there is a cyclic triple cover $\pi: S\rightarrow \mathbb{P}^2$ ramified in $C$. Let $\sigma$ be the covering automorphism (sheet ...
3
votes
1
answer
177
views
Smoothness of ruled surface (asymptotic) parameterisations
A ruled surface $S$ shall be defined as surface consisting of straight line segments. It is commonly known (cf. [BER, p.362] or [STR, p.93] - bibliography at the end) that a ruled surface allows for a ...
3
votes
1
answer
437
views
When does the Hirzebruch surface have a nef anticanonical divisor?
Let $\mathcal H_r=\mathbb P (\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(r))$ be a Hirzebruch surface for some $r\in\mathbb Z$. As a toric variety, the fan structure is spanned by $(-1,0)$,...
3
votes
1
answer
1k
views
Geometric/Algebraic intersection numbers of curves on surfaces
I have the following problem, and struggling to find some references.
Suppose I start with a homology class of a curve on a closed genus $g$ surface $$h=(a_{1},b_{1},\dots,a_{g},b_{g})\in H_{1}(\...
3
votes
2
answers
565
views
Curve of 3-secant lines
Let $C\subset\mathbb{P}^{3}$ be a smooth, non-degenerate curve over an algebraically closed field of characteristic zero. Let $d$ be the degree of $C$ and $g$ be its genus.
Consider the variety $S_{3}...
3
votes
2
answers
276
views
Points of a linear system on a cubic surface
Let $S$ be a generic cubic surface and let $C$ be its intersection with a generic quadric surface.
In the linear system of hyperplane sections of $S$, how many points represent the planes $H$ tangent ...
3
votes
1
answer
396
views
Defining ideals for rational curves in space
A rational normal curve $C_d\subset\mathbb{P}^d$ is defined by quadrics. I guess, for the generic projection $\mathbb{P}^d\stackrel{\pi}{\to}\mathbb{P}^n$ the image $\pi(C_d)$ is still defined by ...
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 ...
3
votes
1
answer
141
views
An algorithm to tell if two cut systems are handle slide equivalent?
Let $\Sigma$ be a genus $g$ closed orientable surface and let $\alpha = \{ \alpha_1,...,\alpha_g\}$ and $\beta = \{ \beta_1 ,..., \beta_g \}$ be two cut systems (i.e. nonintersecting homologically ...
3
votes
1
answer
119
views
Are isotopic transversal curves on a foliated surface transversally isotopic?
Let $F$ be an orientable surface (possibly with boundary) with a foliation $\cal F$ with $k$-prong saddle singularities only for $k\geq 3,$ (as in figure borrowed from Farb-Margalit book). Suppose ...
3
votes
1
answer
205
views
Quotient of product of curves
Let $C_1,C_2$ be smooth, projective curves of genera $g_1,g_2 \geq 2$. Assume that a group $G$ of order $(g_1 - 1)(g_2 - 1)$ acts on $C_1$ and $C_2$ such that $C_1/G \cong \mathbb{P}^1$ and $C_2/G \...
3
votes
2
answers
694
views
Does this surface contain all perfect cuboids?
Probably this is wildly known, but the closest I found was a surface with additional restrictive conditions.
A perfect cuboid is a cuboid having integer side lengths, integer face diagonals and an ...
3
votes
3
answers
2k
views
Gaussian curvature radius
In the paper Surface sampling and the intrinsic Voronoi diagram (2008), Ramsay Dyer defines the Gaussian curvature radius at a point $x$ of a surface $S$ to be $\rho_K(x) = 1/\sqrt{K(x)}$ where $K(x)=\...
3
votes
1
answer
432
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 ...
3
votes
1
answer
318
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?
3
votes
1
answer
150
views
Intersecting geodesics on a surface from non-intersecting geodesics
Let $a$ and $b$ be non-intersecting closed geodesics on a hyperbolic surface. Can these curves be homotopied to transversely intersect but still be geodesics?
3
votes
1
answer
171
views
Atoric equation
I'm looking for a general equation/function z = f(x, y, radius1, radius2, p1, p2) for an atoric surface. p1 and p2 could be either eccentricity or conic constant values. Can anyone help me with that?
...
3
votes
1
answer
456
views
Surface curves equidistant from a simple closed geodesic
Let $S \subset \mathbb{R}^3$ be a surface embedded in $\mathbb{R}^3$,
let's say (to keep it simple) of genus zero.
Let $\gamma$ be a simple, closed, oriented geodesic on $S$.
Because $\gamma$ is ...
3
votes
1
answer
472
views
Bezier Curves question
Hi everyone
I have a fairly simple question about bezier curves: can you represent n bezier curves that have been continuously joined together by a single bezier curve of degree 3n?
My instinct is ...
3
votes
1
answer
273
views
Symmetry of functions on $S^2$
Let $f$ be a continuous function on $S^2$ and suppose there exists a constant $C>0$ such that for every $\mathcal{R} \in SO(3)$ the area of every connected component of $\{f(x)\geq f(\mathcal{R}x)\}...
3
votes
1
answer
369
views
On equations defining space curves
I am reading a text by Prof. Szpiro Tata lectures on equations defining space curves. In the proof of Proposition $1.2$ on page $12$ he gives explicit description of the defining equations of a local ...
3
votes
0
answers
114
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 ...
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 $...
3
votes
0
answers
126
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 ...
3
votes
0
answers
162
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 ...
3
votes
0
answers
108
views
Shape derivative of boundary integrals and differentiability of the integrand on a tubular neighborhood
Let $d\in\mathbb N$, $U\subseteq\mathbb R^d$ be open, $$\mathcal A:=\{\Omega\subseteq\mathbb R^d:\Omega\text{ is bounded and open},\overline\Omega\subseteq U\text{ and }\partial\Omega\text{ is of ...
3
votes
0
answers
163
views
The space of rearrangements of a plane curve
I learned of the paper "Closing curves by rearranging arcs" by L. Alese (arXiv link) by a post on Reddit and was curious about related questions and generalizations. Here a rearrangement of ...
3
votes
0
answers
138
views
Chord of fixed length traveling around a Jordan curve
Let $C$ be a Jordan curve with nice enough properties whenever necessary (e.g. smooth, or just rectifiable, perhaps). I am interested in knowing how long can a chord be that "traverses" the ...
3
votes
0
answers
134
views
Minimal extension of local systems
Let $M$ be a complex manifold of dimension $2$, $D \subset M$ be a connected, simple, normal crossings divisor and $L$ be a $\mathbb{C}$-local system defined over $M\backslash D$. Denote by $j: M\...
3
votes
0
answers
141
views
Reducing curves in surfaces by Dehn twists
Let $F$ be a compact, oriented surface.
A Dehn move, $D_\beta$, on a simple closed curve (scc) $\alpha$ is a Dehn twist applied to $\alpha$ along a scc $\beta$ which intersects $\alpha$ once.
Is ...
3
votes
0
answers
202
views
Does Leray Spectral sequence degenerates at $E_2$ over product of curves
Let $C$ be a smooth, projective curve (can assume to be rational) and $X:=C \times C$. Denote by $p:X \to C$ one of the two natural projections. Let $E$ be a vector bundle on $X$. Is it true that,
$$...
3
votes
0
answers
126
views
Behaviour of geodesics on surfaces as one of the two endpoints moves slightly
Let $u$ and $v$ be two points on a surface (I guess, a Riemann surface) $\Sigma$ such that there is a unique geodesic between $u$ and $v$ on $\Sigma$. Now let $l$ be an arbitrary line that passes ...
3
votes
0
answers
308
views
Does a rectangle exist on any Jordan curve?
Let $C$ be a Jordan curve in $\mathbb{R}^2$. Does there exist points $P,Q,R,S$ on $C$ such that quadrangle $PQRS$ is a non-degenerate rectangle?
3
votes
0
answers
137
views
Intersection patterns of loops on surfaces
Let $a,b$ be to simple closed loops on a surface $S$ with homologically trivial intersection (more generally I'm also interested in the case when $b$ is 1-codimensional). Denote their intersection on $...
3
votes
0
answers
300
views
Laplace-Beltrami of the Gauss map
Let $M$ be a surface in $\mathbb{R}^3$ given by a regular chart, say $X:M \longrightarrow \mathbb{R}^3$, with its first fundamental form $g$, Gauss map $N$, Gaussian curvature $K$ and mean curvature $...
3
votes
0
answers
268
views
Hypersurface with singularities
I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?).
It was said ...
3
votes
0
answers
267
views
Mapping One Curve to another using Dehn Twists
Let $M$ be an orientable surface with genus $g>1$. Let $\alpha$ and $\beta$ represent two different isotopy classes of essential curves on the surface. Is anyone aware of a technique or algorithm ...
3
votes
0
answers
259
views
The central fiber of this family of surfaces?
I have a question on a description of a central fiber of the following family of surfaces.
Let $B,C \subset \mathbb{P}^2$ be smooth curve of degree $2d$ and $d$ respectively. Let's consider the ...
3
votes
0
answers
334
views
What is the analog of "monotonic" for scalar functions on surfaces?
"monotonic" is well defined for functions $f(x)$, where e.g. $x\in[0,1]$ and $f(x)\in\mathbb{R}$. The quality I particularly care about is that if $f(x)$ is monotonic then it will not have any local ...
2
votes
1
answer
459
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 $p$.
...
2
votes
2
answers
599
views
Numerically negative exceptional divisor on a surface.
Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...
2
votes
1
answer
116
views
Construction of a linear Weingarten surface from a space curve
In Ivey and Landsberg's book Cartan for Beginners, the end paragraph of example 5.8.2 claims that linear Weingarten surfaces can be constructed by a space curve. They cite an older book from 1945 that ...
2
votes
1
answer
543
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 ...
2
votes
1
answer
444
views
Families of smooth projective varieties over dvr
Let $R$ be a discrete valuation ring with residue field $k$, an algebraically closed field of characteristic zero and $\pi:X\to \mbox{spec}(R)$ a smooth, projective family of surfaces. Denote by $X_0$...
2
votes
1
answer
205
views
winding number for outer-pointing normal
While trying to characterize the complexity of a closed differentiable curve (for a path planning application), I've been using a notion which is similar in spirit to the winding number of a curve. ...
2
votes
2
answers
409
views
Producing $(-2)$ curves on a smooth surface
We know that blowing up a point on a surface produces a $(-1)$ curve. Is there any such standard techniques to produce $(-2)$ curves in a smooth surface?