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
355
questions
2
votes
1
answer
446
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 $X_0$...
- 1,949
2
votes
0
answers
78
views
Largest disk inside a spherical domain
It is known (Pestov-Ionin theorem) that if $k_{max}$ is the maximum curvature of a smooth planar loop $\gamma$, then there is a disk of radius $1/k_{max}$ inside $\gamma$. I wonder is there any ...
- 175
15
votes
1
answer
709
views
A cubic and six conics problem
I am an electrical engineer system. I live in Viet Nam. I am not a Mathematician. I construct and found a problem as follows:
Let a cubic, and five conics $(C_1)$, $(C_2)$, $(C_3)$, $(C_4)$, $(C_5)$. ...
- 1,034
6
votes
1
answer
295
views
Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)
In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid.
Could you ...
- 845
4
votes
1
answer
339
views
Umbilic points on Euclidean hypersurfaces
Every smooth embedding of $S^2$ into $\mathbb{R}^3$ has at least one umbilic point (in fact, the recent proof of the Caratheodory conjecture yields two such points). The usual proof of this is to use ...
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 $...
1
vote
1
answer
207
views
Marginally Trapped surfaces with constant Gaussian curvature
By marginally trapped surface I mean a spacelike surface in a 4-dimensional Lorentzian manifold such that the mean curvature vector is lightlike.
In my research I have stumbled across marginally ...
- 403
37
votes
1
answer
2k
views
Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid)
Let me start with the context. This is definitely not a "research level" question, but I'm hoping that the research community will be able to settle for me whether or not a particular construction ...
- 26.4k
3
votes
0
answers
301
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 $...
- 31
0
votes
0
answers
172
views
Explicit form of certain polynomials and intersection of curves
Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ two effective divisors on $X$ intersecting at finitely many points. Is it true that if $C$ and $D$ intersect in ''low'' number ...
- 2,205
7
votes
3
answers
401
views
In the $\mathbb{H}^3$ upper half space model, is a hemiellipsoid perpendicular to the plane at infinity a minimal surface?
Given a Jordan curve on the $\mathbb{H}^3$ boundary at infinity, there is a minimal surface (topological disk) in $\mathbb{H}^3$ with the curve as its asymptotic boundary (page.mi.fu-berlin.de/...
- 103
2
votes
1
answer
221
views
Relation between curves in a complete linear system contained in another
Let $X$ be a projective surface over $\mathbb{C}$, let $x\in X$ be the only singular point of $X$. Let $L$ be an ample line bundle on $X$. Consider the blow up $Y$ of $X$ along $x$, $f:Y\...
- 643
6
votes
1
answer
175
views
Self-avoiding/reflecting geodesics on a convex surface
Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$.
For me $S$ is a convex polyhedron,
but I am happy to view $S$ as a smooth body with positive Gaussian curvature
at each point, or ...
- 149k
2
votes
1
answer
153
views
On conflicting descriptions for tor of a local cohomology group
Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1_C(\mathcal{O}_X)$ the sheaf associated to the presheaf $U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ...
- 823
0
votes
1
answer
405
views
What does the action of a 2-torsion line bundle on $Pic^d(C)$ do to the number of sections?
Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion ...
- 643
2
votes
1
answer
106
views
Is the length function associated with the twist parameter an increasing function?
Let $S$ be a closed hyperbolic surface and $x$ be an oriented simple closed curve in $S$. Let $y$ be an oriented closed curve such that the geometric intersection number between $x$ and $y$ is ...
- 1,703
4
votes
0
answers
289
views
Was this particular case of the tube formula known before Weyl and Hotelling?
The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...
- 4,033
7
votes
0
answers
288
views
Harmonic map heat flow in positive curvature
Suppose I wish to relax/smooth a map $\phi:M\rightarrow N$ between two surfaces $M,N$ embedded in $\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ...
- 695
25
votes
1
answer
3k
views
Why is it so hard to prove Toeplitz' conjecture?
I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) ...
- 161
1
vote
1
answer
2k
views
Finding t vlaue in Bezier curve [closed]
According to this question, I'm looking for some method to find the t value in Quadratic bezier curve equation:
$$
B(t)=P_0+t(1-t)P_1+t^2P_2 \space \space where \space 0 ≤ t ≤ 1
$$
In this ...
- 111
1
vote
1
answer
357
views
Model over DVR for smooth projective curves
Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ ...
- 503
29
votes
2
answers
1k
views
Is every closed curve in 3D a geodesic on a genus-0 surface?
Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.
Q. Does there always exist a smooth, embedded, genus-zero surface
$S \subset \mathbb{R}^3$
such that $\gamma$ is a (...
- 149k
1
vote
0
answers
308
views
Can a cylinder be regarded as a Riemannian manifold? [closed]
Consider the surface of a bounded cylinder consisting of a top,bottom and side part together with the metric induced by the euclidean norm on $\mathbb{R}^3$. Can this space be regarded as a Riemannian ...
- 2,286
7
votes
0
answers
160
views
Kinematics of rolling knots
It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes.
(An example:https://www.youtube.com/...
- 403
3
votes
4
answers
1k
views
Intrinsic definition of arc length [closed]
Is there an intrinsic way of defining the arc length of a curve in $\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve?
- 6,882
6
votes
1
answer
301
views
Reduction of self-intersections without reducing the geometric intersection
Let $F$ be a hyperbolic surface. Given a closed curve $a$, let $\bar{a}$ denotes the free homotopy class of $a$. Let $i(\bar{a},\bar{b})$ denotes the geometric intersection number and $i(\bar{a})$ ...
- 1,703
1
vote
1
answer
158
views
Image of any curve can be parametrized without zero derivative?
Let $\gamma :[a,b]\to\mathbb{R}^2$ be a $C^{1} ([a,b])$ injective application. Is it true that there is another continuous parametrization $\rho:[c,d]\to\mathbb{R}^2$ such that the following two sets ...
- 1,330
0
votes
1
answer
201
views
Double coset separability and the existence of vanishing sequences for surface group
Definition: Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite ...
- 1,703
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 ...
- 1,573
7
votes
2
answers
894
views
Ivanov's metaconjecture on surface homeomorphisms
In Fifteen problems about MCG Ivanov stated the following metaconjecture:
Every object naturally associated to a surface S and having
a sufficiently rich structure has $Mod(S)$ as its groups of ...
- 1,703
2
votes
1
answer
389
views
How to find isothermal coordinates equivalent to circles in far limit?
I am trying to find the most general rotational coordinate systems for Euclidean 3-space, with the following two defining characteristics: 1) being equivalent to spherical coordinates in the limit of ...
- 99
0
votes
0
answers
113
views
Positive curvature of the boundary away from a point implies regularity?
In a paper I'm refereeing, the authors make use of the following geometric fact:
Let $U$ be an open subset of $\mathbb{R}^2$. If there is a point $p\in \partial U$ so that $\partial U \backslash p$ ...
- 1,119
3
votes
0
answers
269
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 ...
- 4,761
2
votes
0
answers
86
views
Topological/numerical constraints for the existence of more than one pencil
A famous theorem of Castelnuovo and de Franchis tells us that for $S$ a smooth projective complex algebraic surface that for $b \geq 2$, pencils $f : S \to B$ of genus $b :=g(B)$ are in bijective ...
0
votes
2
answers
128
views
Planar curves identical to their inverses
Is the right strophoid
the only planar curve $C$ whose inverse curve w.r.t. some circle (in this case: centered on the origin)
is identical to $C$?
&...
- 149k
1
vote
1
answer
177
views
Characterization of $d$-gonal curves on a K3 surface
Let $X$ be a K3 surface and $C$ a curve on $X$. We say that $C$ is $d$-gonal if it admits a pencil of degree $d$ (and none of smaller degree).
I am wondering if there exist characterizations of $d$-...
- 761
4
votes
0
answers
248
views
Are there Zoll pancakes?
How flat (flat in pancake-style, not in curvature 0-style), in some extrinsic intuitive measure, can a Zoll surface of revolution (embedded in Euclidean three-space) be?
I don't want to impose a ...
- 13.2k
1
vote
0
answers
125
views
Is triple point intersection 'generic' in Teichmuller space?
Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...
- 1,703
1
vote
0
answers
124
views
Is there a unique solution? [closed]
Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a given continuous function and $t_0\in (a,b)$ a fixed point. Is it true that the following problem has a unique continuous solution $\mathbf{r}:(a,b)\to\...
- 1,330
1
vote
1
answer
138
views
Another type of derivative, and the associated primitive
Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so that:
$
\...
- 11
14
votes
1
answer
957
views
Area of the minimal surface of a non-planar quadrilateral in 3d
Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of ...
- 12.3k
1
vote
1
answer
376
views
General reparameterization of a B-spline
Say I have a B-spline function (or curve) of order $k_1$, defined over some knot vector
$\mathbf{t} = \{ t_i\}_1^{n_1}$, i.e. $$f(x) = \sum_i a^i B_{i,k_1}(x).$$
Do you know of a process of finding ...
- 175
1
vote
0
answers
107
views
Intersection points of closed curves inscribed in a convex polygon
Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...
- 31
8
votes
5
answers
3k
views
Variation of curvature with respect to immersion?
Let $M$ be a smooth surface and let $f: M \to \mathbb{R}^3$ be a family of immersions given by
$$ f(t) = f_0 + tuN_0, $$
where $f_0$ is some initial immersion, $N_0$ is the associated Gauss map, and ...
- 141
4
votes
1
answer
759
views
Angle between geodesics in hyperbolic surface
Let $F$ be an oriented surface of finite type with $\chi(F)<0$. Let $\gamma_1$ and $\gamma_2$ are two oriented closed curves which intersect transversally in double points. Given a hyperbolic ...
- 1,703
0
votes
0
answers
422
views
"Spreading out" locally free sheaves
Let $Y$ be a regular surface flat, projective over $R$, where $R$ is complete DVR. Let $X$ be the generic fiber and $F$ be a locally free sheaf on $X$. We know that if the rank of $F$ is $1$ then we ...
- 2,022
2
votes
0
answers
372
views
Symmetry on a sphere
Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
- 51
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-...
- 1,703
10
votes
2
answers
1k
views
Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?
There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
- 305
5
votes
1
answer
131
views
How to approximate volumes of m-dimensional manifolds with m-dimensional polyhedra
The areas of a sequence of polyhedra approaching a surface need not approach the area of the surface, but there are theorems guaranteeing that this be so. (T. Rado, On the Problem of Plateau, Chapter ...
- 151