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**12**

votes

**1**answer

487 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) ...

**0**

votes

**1**answer

43 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 ...

**0**

votes

**1**answer

140 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 $R$ ...

**20**

votes

**2**answers

646 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 ...

**1**

vote

**0**answers

63 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 ...

**7**

votes

**0**answers

82 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 ...

**0**

votes

**0**answers

62 views

### Embedding curves in hypersurfaces

Consider a curve $C$ in $\mathbb{F}_q^m$, say. I am interested in the existence of curves not contained in any small degree hypersurface.
For instance, a helix is not contained (or non-embeddable) in ...

**2**

votes

**4**answers

576 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?

**0**

votes

**0**answers

34 views

### Angle at self-intersection points of a curve in hyperbolic surface

Let $F$ be a hyperbolic surface of finite type. Let $\alpha$ be a closed oriented geodesic with more than one self intersection. Suppose all the self-intersections are double points. Let $\angle_p$ ...

**6**

votes

**1**answer

119 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**

vote

**1**answer

129 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 ...

**0**

votes

**1**answer

77 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 ...

**3**

votes

**1**answer

293 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 ...

**4**

votes

**1**answer

229 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 ...

**0**

votes

**0**answers

39 views

### Unitary derivative and countable set

Let $\mathbf{r}:I\to\mathbb{R}^2$, where $I\subseteq\mathbb{R}$ is an open interval, be a continuous function that is not constant on any subinterval $J\subseteq I$ such that at each point $t\in I$ ...

**1**

vote

**1**answer

134 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 ...

**0**

votes

**0**answers

88 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 $\partial U \backslash p$ ...

**3**

votes

**0**answers

206 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 ...

**2**

votes

**0**answers

65 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

92 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 $C$?
...

**1**

vote

**1**answer

110 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 ...

**4**

votes

**0**answers

200 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 ...

**0**

votes

**0**answers

43 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**

vote

**0**answers

108 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 ...

**1**

vote

**1**answer

132 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:
$
...

**9**

votes

**0**answers

110 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 ...

**1**

vote

**0**answers

66 views

### General reparameterization of a b-spline

Say I have a bspline 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 ...

**0**

votes

**0**answers

77 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 ...

**3**

votes

**5**answers

485 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 ...

**4**

votes

**1**answer

169 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 ...

**0**

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**0**answers

221 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**

votes

**0**answers

293 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 ...

**2**

votes

**1**answer

140 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 ...

**5**

votes

**2**answers

426 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 ...

**3**

votes

**1**answer

88 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 ...

**0**

votes

**0**answers

217 views

### Understanding a proof of a lemma in elliptic surfaces

In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (Sung and Tian) in page 621, I am trying to understand the proof of part 1 of Lemma 3.4.
In the part 1 of ...

**3**

votes

**1**answer

141 views

### A query about Hatcher flow on arc complex

In the paper "Triangulations of Surfaces" Hatcher proved that the arc complex associated to a punctured surface is contractible. The main proof is divided into two parts. In the first part he assumes ...

**1**

vote

**1**answer

74 views

### On Severi's definition of the complementary correspondence

In Weil's short note entitled "On the Riemann hypothesis in function-fields"
he mentions the notion of the complementary correspondence associated to a given correspondence $T:C\rightarrow C$ where ...

**3**

votes

**2**answers

226 views

### Effective cone of C x C where C is a Fermat curve

A search of the literature reveals that for a curve $C$ of genus $\geq 2$, determining the effective cone of $C \times C$ is hard. My question is this: do we know a single example of a curve $C$ of ...

**2**

votes

**1**answer

206 views

### The class of uniformly accelerated curves and surfaces

Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math.
Let's consider two examples of curves:
1)The curve defined by the ...

**1**

vote

**0**answers

139 views

### Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR

Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...

**5**

votes

**1**answer

131 views

### Curvature and Failure to return to starting point

Assume I have a geodesic polygon $P$ in a Riemannian manifold $M$ that is given by the image of a piecewise geodesic closed curve $\gamma(t)$ (parametrized by arclength), with vertices $x_i = ...

**1**

vote

**0**answers

41 views

### Connectivity and contarctibility of complexes associated to curves and arcs

There are various complexes associated to a surface using the curves and arcs e.g. Curve complex, Arc complex, curve arc complex and so on (for a collection of such objects see This). Now to ...

**0**

votes

**1**answer

119 views

### A doubt from “Geometry of the complex of curves II: Hierarchical structure” by Masur and Minsky

In the paper "Geometry of the complex of curves II: Hierarchical structure" (Paper) there is a construction of curve complex for an Annular subdomain (2.4). The construction depends on the domain ...

**1**

vote

**1**answer

104 views

### Invariant of isotopy of curves in a surface.

Suppose $S_g$ is a sorface of genus $g>1$. Let $\gamma_1$ and $\gamma_2$ be two simple closed curves containing points $p_1, p_2$. Suppose $\gamma_1$ and $\gamma_2$ are isotopic. Now there can be ...

**2**

votes

**1**answer

134 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 ...

**0**

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**0**answers

50 views

### A question on the curvature of smooth and embedded curves in a plane

I am from China. This is my first question in this website.
Given a family of embedded, smooth and closed curves $\{X_i(s)\}_{i=1}^\infty$ in a plane, let us denote by $L_i$ the perimeter and ...

**2**

votes

**1**answer

209 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 ...

**4**

votes

**3**answers

342 views

### Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$

Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$:
$$S_2\times Q\rightarrow Q,\; ...

**14**

votes

**2**answers

783 views

### How useful/pervasive are differential forms in surface theory?

Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...