# 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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### Minimal area of Seifert surfaces

Let $K$ be a knot smooth knot in a 3-manifold $M$ and fix a metric on $M$. Let $F$ be a orientable surface of genus $g$ with one boundary component. Then we can consider the family of all maps $\...

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**1**answer

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### Hadamard theorem about embedding

The following theorem is commonly attributed to Jacques Hadamard.
Assume $\Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $\Sigma$ is embedded and bounds a convex ...

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**1**answer

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### Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$

Is there any Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ in the literature?
More precisely I would like to know if there is an answer to the following
QUESTION: Let $f : \...

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**1**answer

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### Explaining patterns in modular multiplication graphs

Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$.
These graphs often look ...

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**2**answers

782 views

### A problem on Gauss--Bonnet formula

While teaching a course in differential geometry, I came up with the following problem, which I think is cool.
Assume $\gamma$ is a closed geodesic on a sphere $\Sigma$ with positive Gauss curvature.
...

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**1**answer

122 views

### Length functions on Teichmuller space with constant difference

Let $S$ be a closed oriented surface of genus $g\geq 2$. Let $\mathcal{T}$ be the corresponding Teichmuller space. Given a free homotopy class of closed curve $[\gamma]$ we can define the length ...

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### Syzygies in Steinberg Module of genus 2 mapping class group $MCG(\Sigma_2)$

Consider the mapping class group $MCG(\Sigma_2)$ of the closed genus 2 oriented surface $\Sigma_2$. The algebraic-duality theory of $MCG_2:=MCG(\Sigma_2)$ is explicitly described by Nathan Broaddus' ...

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**1**answer

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### A variation on four-vertex theorem

Is it true that, if a closed, strictly convex curve has exactly four vertices (extrema of curvature), then any circle has at most four points of intersection with it?

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### Existence of smooth principal direction near umbilics

Given a hypersurface $M^n\hookrightarrow \mathbb{R}^{n+1}$, $n\geq 3$, and a point $p\in M^n$ does there exist a smooth orthonormal frame of principal directions in a neighbourhood of $p$?

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### 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,
$$...

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### 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?

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### 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 ...

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### 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 ...

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**1**answer

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### 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?
...

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**1**answer

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### 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)\...

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**1**answer

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### 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 ...

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### 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, ...

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**0**answers

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### 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 ...

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**1**answer

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### 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 ...

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**1**answer

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### 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 $L$ ...

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### 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 $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...

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**1**answer

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### 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 ...

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votes

**5**answers

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### 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 $S$ ...

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**2**answers

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### 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 ...

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**1**answer

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### 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 $\tau$ (...

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**1**answer

78 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 ...

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### 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 ...

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**0**answers

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### 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 $A$ ...

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### 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 ...

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**1**answer

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### 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 ...

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**1**answer

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### 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 ...

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**1**answer

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### 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 ...

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### 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 $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...

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### 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 ...

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### 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 ...

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**1**answer

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### 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:
...

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**1**answer

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### 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)\}...

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### 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 $...

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### 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 ...

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### 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 ...

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**1**answer

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### 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 ...

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### 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,...

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**1**answer

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### 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 ...

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### 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,\...

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### 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 ...

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### 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.
...

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### 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 ...

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**3**answers

399 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 $\...

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**1**answer

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### 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 ...

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**1**answer

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### 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 ...