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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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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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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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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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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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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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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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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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-...
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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 ...
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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 ...
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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 ...
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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<...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 ...
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"Slope Analogue" of Clothoids
It is well known, that the characterizing property of Clothoids is, that their curvature is proportional to length; that is also the reason, why they are used as design elements e.g. in road design.
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Examples of spiraling geodesics?
Does there exist a closed, bounded surface $S$ embedded in $\mathbb{R}^3$
that has a geodesic $\gamma$ that spirals around a point $x$, getting closer
and closer, but never reaching $x$?
Here I ...
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Disc bounded by a plane curve
Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$.
Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve?
It is easy to find an open disc ...
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Is there an analog of Reidemeister's theorem for braids in a surface?
Reidemeister's classical theorem describes the set of links in $\mathbb R^3$ up to isotopy as the set $\{ \textrm{diagrams in } \mathbb R^2\textrm{ with crossings}\}$ modulo certain local relations on ...
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Gage-Grayson-Hamilton curve-shortening flow, at an angle
The Gage-Grayson-Hamilton curve-shortening flows along the normal to the curve:
&...
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Moving curves in minimal position to geodesics on hyperbolic surfaces
Let $\Sigma$ be a hyperbolic surface possibly with non-empty boundary and punctures.
A closed curve $\gamma$ if filling, if any simple closed curve intersects any representative of free homotopy ...
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Universal chord theorem for curves
Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve.
Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's ...
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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 ...
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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?
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a continuous version of axiom of choice?
Let $L_t(s):S^1 \rightarrow \mathbb{R}^2(t\in [0,1])$ be a Jordan curve, $O(t)$ be its interior and $H(t,s)=L_t(s)$.
If $H$ is a homotopy from $L_0(s)$ to$L_1(s)$, does there exist a continuous ...
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Isometric and conformally equivalent surfaces in $\mathbb R^d$ with the same boundary
Let $X$ be a compact smooth surface in $\mathbb R^d$ with smooth boundary $\partial X$. Is it possible to find another smooth surface $Y$ in $\mathbb R^d$ such that $\partial X = \partial Y$ and $Y$ ...
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Convexity of length function for surfaces with boundary
In the paper "The Nielsen realization problem" (here), Kerckhoff proved that the length function on the Teichmüller for closed surface is convex. In his paper "Geodesic length functions and the ...
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How to descend a line bundle from the normalization of a surface?
Let $\tilde S \overset{\nu}{\to} S$ be the normalization of a projective surface $S$ over a field $k$. Assume for now that $S$ is obtained from $\tilde S$ by gluing together two disjoint curves $C_1$ ...
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Research topics in Curves and Surfaces [closed]
I advance that I'm not a mathematician but I'm an undergraduate student of mathematics. In my courses at university I have studied a bit of Differential Geometry, in particoular differential geometry ...
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Curvature of plane curves on a surface
Let $S$ be a surface and $\gamma$ a curve on $S\subseteq \mathbb{R}^3$ obtained cutting $S$ with a plane. I wuold an upper bound for the curvature of $\gamma$. Are there papers for this topic?
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Are there some intrinsic invariants of surfaces other than Gaussian curvature?
The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$.
Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
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Trefoil Knot Seifert Minimal Surface Equation
I am not very familiar with knot theory nor with minimal surfaces, so I already apologize if my question appears too naive or simple :). I am trying to do the following:
Starting from a real ...
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Surfaces contained in a ball
In this Paper there is a proof that a closed plane curve of length
$L$ and curvature bounded by $K$ can be contained inside a circle of radius
$L/4 - (\pi - 2)/2K$. Are there similar results for ...
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Change of length of curve when Fenchel-Nielsen length coordinate increase
Let $F$ be a hyperbolic surface of finite type. Suppose $\alpha$ is a simple closed geodesic and $\beta$ is any closed geodesic intersecting $\alpha$. Consider a Fenchel-Nielsen coordinate of the ...
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Exact derivation of Von Kármán relation of Gauss curvature
Using relations for surface deformations (in structural mechanics notation)
$$ u,v,{\epsilon _x, \epsilon_y, \gamma_{xy}}$$
Notations {u,v } have same meaning as displacements in surface theory.
...
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decomposition of codim 1 currents
Let $T$ be an codimension one rectifiable mass-minimizing current. If $\partial T\llcorner B_R(a)=0$, then it is possible to write $T$ as an infinite sum of oriented boundaries: $T\llcorner B_R(a)=\...
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A third degree surface and a touching sphere [closed]
Let consider a surface $z=1/(xy)$ and a sphere defined by $(x-1.5)^2+(y-1.5)^2+(z-1.5)^2=3/4$. The sphere touches the surface at (1,1,1). Is it possible to prove that point (1,1,1) is the only ...
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Polars of algebraic curves and surfaces
I asked this on Math.StackExchange, but received no response, so trying here ...
A paper I'm reading says the following ...
With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x})...
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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 $X_0$...