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
3
votes
1answer
60 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?
1
vote
1answer
77 views
Definition of first normal space
Given an immersed submanifold $M$ of a Riemannian manifold $\overline{M}$, the first normal space of $M$ at a point $p \in M$ is defined as the linear subspace $N_{p}^{1}M$ of $N_{p}M$ spanned by the ...
2
votes
0answers
45 views
Must bounded, closed, smooth curves with long straights have sharp bends?
Consider the family of bounded, closed, and continous curves $\Gamma$, i.e. for all $\gamma \in \Gamma$, we have $\gamma : [0, 1) \to [0, 1]^2$. Within this family, I am interested in curves that ...
3
votes
1answer
94 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?
...
0
votes
1answer
61 views
Parametric Surface Equations for Orthogonal Projection of Torus Knot Tube onto Torus [closed]
What are the parametric equations for the orthogonal projection of the torus knot tube onto the torus surface?
For instance, if we have the equations for the torus knot
$$
\vec r(t)= (R+r\cos pt)\...
4
votes
1answer
162 views
Pairing on arithmetic surfaces
Let $f: X \to S$ be an arithmetic surface, where $S=\operatorname{Spec } O_K$ for a number field $K$. It is well known that if we want to introduce a reasonable intersection theory on $X$ we have to ...
-2
votes
1answer
430 views
Is the conjecture true for n-sphere $(n>2)$? [closed]
This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here:
PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...
3
votes
0answers
175 views
Shapes defined by points
Can shapes determined by number of points?
From an amazing theorem in plane curves geometry we know that vertices of triangles similar to arbitrary triangle $T$ is dense on every closed jordan curve ...
0
votes
1answer
100 views
Rotation invariant of surface
Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four ...
0
votes
1answer
127 views
Do line bundles with enough sections on surfaces have generic divisors which are irreducible?
Let $L$ be a line bundle on a smooth connected complete complex algebraic surface $X$. Assume that $L$ has enough sections i.e. that $H^0(L,X)$ has dimension $> 1$. A nonzero section $s$ of $L$ ...
2
votes
0answers
96 views
Pascal theorem for three dimensions
A year ago I found the Pascal theorem for three dimentions as follows:
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...
3
votes
1answer
213 views
Some curves on the Jacobian of a genus $2$ curve and their image under certain maps (char $p$)
I hope this question belongs here. The situation in this question is quite particular and specific.
I am trying to weak some of theory to measure the degree of some function on the Jacobian of a ...
26
votes
5answers
2k views
If a triangle can be displaced without distortion, must the surface have constant curvature?
Suppose $S$ is a Riemannian 2-manifold (e.g. a surface in $\mathbb{R}^3$).
Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics.
If $T$ can be moved around arbitrarily on $S$ ...
6
votes
2answers
155 views
Sliding through a curvature-bounded tube: Maximum volume?
My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view.
Q1. Is it the case that the maximum convex volume body inside a ...
2
votes
1answer
76 views
Classifying transverse curves to a surface foliation carried by a train track
Suppose that a foliation $\cal F$ on a surface $F$ is carried by a train track $\tau$. Is it possible to classify all $\cal F$-transverse multi-loops in $F$ in terms of a combinatorial data on $\tau$ (...
0
votes
1answer
77 views
The radius of an interval mapped through a space-filling path
Take $f:[0,1]\to [0,1]^n$ a continuous tour through $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper ...
11
votes
1answer
196 views
Rigidity of doubled convex caps
Suppose that we have a convex cap, i.e., a convex surface in $R^3$ homeomorphic to a disk whose boundary lies in a plane. Reflect the cap through the plane of its boundary and glue it back to the ...
0
votes
0answers
46 views
Is there any simple expression for $(\dot{F}\phi_u)\times(\frac{\partial \dot{F}}{\partial u} \phi_v)$
$F$ is an isometry of two dimensional regular surfaces $A$ to $B$. $\phi$ and $F\circ\phi$ are the coordinate chart for $A$ and $B$ respectively.$\phi_u$ and $\phi_v$ are unit tangent vector of $A$ ...
9
votes
2answers
582 views
A necessary and sufficient condition for a space curve to lie on a ellipsoid
Any (arc-length parametrized) space curve is uniquely determined (up to rigid motion) by its curvature and its torsion.
For instance we know that a necessary and sufficient condition for a space ...
2
votes
1answer
79 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 ...
1
vote
1answer
71 views
Equation for a geometrical half surface from folding a flat curved in one direction surface in half [closed]
I cannot not formulate the problem as i do not know how to model this idea. This is an open question not an mathematical exercise so if anybody has a good proposition i'm happy to use it.
Sorry in ...
6
votes
1answer
143 views
Obstructions to realizing a balanced presentation as a 3-manifold group
I am thinking of 3-manifolds as arising from Heegaard splittings which I am thinking about in terms of Heegaard diagrams. I know that 3-manifold groups are rather special in the class of all finitely ...
6
votes
0answers
68 views
Is there a convex three-dimensional body with constant width and only finitely-many equilibria? Or: do spheroform gömböcök exist?
Mathematical questions. The mathematical (and 'gravity'-free) formulation of the question in the title is given by the following questions:
Q1. Does there exist $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...
3
votes
1answer
79 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 ...
2
votes
0answers
60 views
Which surfaces embedded in $\mathbb{R}^3$ have only axially-symmetric sections?
Dmitry Ryabogin and I considered the following question some time ago, but got nowehere:
Let $M$ be a (smooth or algebraic) surface in $\mathbb{R}^3$. Suppose that for every section $S$ (an ...
8
votes
1answer
244 views
Degeneration of curves inside a family of surfaces
We are interested in understanding how a higher genus curve in a smooth surface of general type can degenerate into the non-normal locus of the limit of a degeneration of surfaces. More precisely:
...
3
votes
1answer
263 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)\}...
1
vote
0answers
113 views
Connected component of Picard group of a genus $2$ curve and its Jacobian over an imperfect field
Let $H/k$ be a genus $2$ curve. Consider the function field of $H$ given by $k(H)$. Take the base extension of $H$ to $k(H)$, namely $\hat H:=H\otimes_k k(H)$. Consider the Jacobian $J$ of the curve $...
0
votes
0answers
67 views
Is there a curve in d dimensions whose intersection (contact) with every hyperplane is of order less than d?
Can a curve $\phi:[a,b] \to \mathbb{R}^d$ be of finite type $k < d$?
The type $k$ of $\phi$ is the smallest $k \geq 1$ (if it exists) such that $\phi$ is $C^k$ and for all $x \in [a,b]$ and all $u ...
6
votes
1answer
125 views
Estimate of number of boundary components of a compact Riemannian 2-surface
Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the ...
7
votes
1answer
166 views
Estimate of area of 2-dimensional surface
Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form ...
4
votes
0answers
72 views
Relation between point pushing pseudo-Anosov map and the minimum length
Let $S$ be a closed hyperbolic surface. Suppose $Mod(S)$ denotes the mapping class groups and $T(F)$ denotes the Teichmüller space.
By Birman exact sequence we get the point pushing map $Push:\pi_1(S,...
1
vote
1answer
69 views
Splines with bounded first derivative?
I have a set of points $(x_i,y_i)\in{\mathbb R}_+\times{\mathbb R}$, $i=1,...,n$, ($x_i$ are the independent variables and $y_i$ are the dependent variables or responses) that I want to fit using ...
1
vote
0answers
49 views
Regular surfaces with boundary and $C^1$ domains
I would like to ask about the equivalence between these two definitions for a $C^1$ domain. In the book Vector Analysis Versus Vector Calculus, we have:
Definition 8.2.1: Let $\mathbb{H}^k=\{(t_1,\...
5
votes
2answers
217 views
homogeneous surface in $\mathbb{R}^4$
It is well known that the only homogeneous surfaces in $\mathbb{R}^3$ are the spheres, cylinders or planes. My question is about other examples in dimension $4$. Such a surface should have "constant ...
7
votes
1answer
172 views
Uniformisation for non simple closed curves
Given a simple closed curve in the plane, there is a homeomorphism from the unit open disk to the interior of the curve. The homeomorphism can be taken conformal, this is the uniformisation theorem.
...
7
votes
0answers
267 views
Can generalization of a generalization Pascal theorem, Pappus theorem to Higher Dimensions? [closed]
Please see a chain of six circles associated with a conic. This is a generalization of Pascal theorem, Pappus theorem. I reformulate as following:
Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a ...
7
votes
3answers
396 views
Identity involving an improper integral (with geometric application)
Is it (for some reason) true that
$\lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt=\frac{\pi}{2}$?
Numerical evidence (from Mathematica):
when $c=1/5$, the integral is $\...
0
votes
0answers
25 views
Smoothness Conditions for Planar “Mock-parametric” Spline Interpolation
By "mock-parametric" interpolating curves I understand a class of curves that connect a discrete sequence of points with a predefined degree of smoothness and, that correspond to a non-parametric ...
4
votes
1answer
109 views
Name for Curves from Driving on Smooth Manifolds
Is there already name for the generalization of Clothoids to curves on smooth manifolds, i.e. where the curve's curvature depends linearly on the curve's length-parameter?
In the euclidean plane ...
2
votes
0answers
130 views
A gradient trajectory connecting boundary components in an annulus
In a course of writing a paper I realised that I need a lemma below, and found a half page proof. I have not seen this lemma before and wonder if maybe someone here knows this lemma or can prove it in ...
4
votes
1answer
116 views
Convex hull of a connected subset on a complete surface of non-positive curvature
Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...
1
vote
1answer
86 views
Polar coordinates of a set with different radius and angle
Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that
$$U=\lbrace{ (r,\theta): 0<...
2
votes
1answer
350 views
Castelnuovo and Artin contractibility criteria for families
In the course of a proof, I need some version of Castelnuovo and Artin's contractibility criteria for a family of surfaces. Say I have a family (flat is probably needed, in order to compare ...
2
votes
1answer
227 views
Regularity of the reparametrization map between curves [closed]
I am looking for a reference for the following kind of results.
Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm.
Let $B$ be a Borel subset of ...
7
votes
2answers
262 views
Most general version for the Gauss-Bonnet theorem for polygons
Suppose $M$ is a 2-dimensional smooth Riemannian manifold and $P\subset M$ is an open and connected subset with compact closure and a piecewise geodesic boundary.
My question is: What further ...
2
votes
1answer
88 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
0answers
111 views
intersections between closed curves on surfaces
I would like to find a result telling me that two simple closed curves $\alpha$ and $\beta$ (on a non-orientable surface $S$) are in minimal position if and only if there is not a disk in $S$ whose ...
15
votes
2answers
903 views
A variant of the Monge-Cayley-Salmon theorem?
Suppose one has a smooth non-degenerate curve $\gamma: [0,1] \to {\bf R}^n$ into Euclidean space (thus $\gamma'$ never vanishes), with the property that the velocity $\gamma'(t)$ and acceleration $\...
4
votes
1answer
190 views
Finding the shortest curve that is at distance $\epsilon$ of every point of a surface
Let $M$ be a compact connected (smooth) surface (possibly with boundary) in $\mathbb{R}^3$ and $\epsilon>0$ a constant.
Is there (and if there's not, what conditions on ($M$, $\epsilon$) should ...
