Questions tagged [surfaces]
A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.
3
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0answers
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fundamental domains in H^2 containing large balls
I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic ...
14
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1answer
645 views
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 ...
0
votes
1answer
139 views
Complex structures on topological surfaces
I am interested in the number of complex structures on a surface. More precisely, given a genus $g$ surface (topological manifold of real dimension 2) with $n$ punctures $X_{(g,n)}$, how many complex ...
3
votes
0answers
251 views
Is this a 2-cyclic cocycle ? Does it have a nontrivial geometric interpretation?
Let $S$ be a surface in $\mathbb{R}^3$.
Inspired by the $2$-cyclic cocycle defined in page 20 of the book "Non-commutative geometry" by Alain Connes, we consider the following $3$-linear map on the ...
4
votes
1answer
256 views
Thurston's preprint: “On the geometry and dynamics of diffeomorphisms of surfaces”
W. Veech on Teichmüller curves in moduli space, Eisenstein series and applications to triangular billiards says on the second paragraph of page 579:
"Thurston's original construction [8] corresponds ...
4
votes
1answer
129 views
Invariant lifts of a closed curve on a surface of genus > 1
I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question :
Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...
15
votes
0answers
439 views
Does the Gauss-Bonnet theorem apply to non-orientable surfaces?
I hesitated for a long time to ask such an elementary-seeming question on Math Overflow, but when I asked and bountied it on Math SE, I found that a few experts seem to disagree on the answer, and I ...
6
votes
2answers
146 views
Thrice intersecting closed geodesic on genus 2 orientable closed surface
Does there exist a closed geodesic on a closed genus 2 orientable surface (with hyperbolic metric) that self-intersects at only one point thrice?
2
votes
2answers
81 views
Flat R-bundles on surfaces
I have questions about the definition of representation variety. In François Labourie's book "Lectures on representations of surface groups", Section 3.5, the author gives four models of the ...
8
votes
1answer
371 views
The differential of the Gauss normal map from a Lie algebraic view point
Let $S\subset \mathbb{R}^3$ be a smooth surface with the Gauss normal map $N:S\to S^2$.
Then for every $x\in S$, the differential $(dN)_x:T_xS\to T_{N(x)}S^2$ can be considered as an ...
-2
votes
1answer
446 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, ...
2
votes
1answer
174 views
Projection of a ball in the ambient space to a manifold
Let $B_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$.
Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $...
1
vote
0answers
58 views
Does this “algebraic” method for the application of the constructive proof of the classification of closed & compact surfaces have any use? [closed]
Disclaimer: This question is cross-posted in here. I have never asked a question in mathoverflow before, so if the level of this question is not appropriate for this site, please just vote close it.
...
0
votes
0answers
32 views
Edgebreaker algorithm over 2-manifolds
Suppose I triangulate a 2-manifold and make a dual pseudograph over it. If I do the Edgebreaker compression algorithm over this graph to generate a spanning tree, can exist an edge with three or more ...
4
votes
1answer
146 views
How disconnected can a Seifert surface be?
Seifert surfaces
The standard definition of a Seifert surface for a link in $S^3$ is an oriented, compact surface embedded in $S^3$, bounding the link. Often, it is assumed to be connected, but given ...
5
votes
1answer
296 views
Formula for Goldman Lie bracket of surface
Let $\Sigma_{g}$ be a closed oriented surface of genus $g$, Goldman defined a Lie algebra structure on the free module generated by the free homotopy
classes of loops on $\Sigma_{g}$. Roughly speaking,...
8
votes
1answer
186 views
Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms
Let $\Sigma$ be an oriented compact surface with non-empty boundary that is not a disk or a cylinder (i.e. negative Euler characteristic). Let $\phi, \psi: \Sigma \to \Sigma$ be two orientation ...
6
votes
2answers
136 views
Handel's Theorem for surfaces with boundary
Handel's Theorem(Entropy and semi-conjugacy in dimension two, 1987): let $M$ denote a closed surface. Let $\vartheta$ be a pseudo-Anosov (orientation-presrv.) homeomorphism of $M$ and $g$ be an (...
3
votes
1answer
134 views
total mean curvature for singular surface
(a) The total mean curvature of a smooth oriented surface $S$ is defined as $$\iint_S H dA$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a ...
6
votes
1answer
166 views
Examples of normal subgroups of a surface group with no embedded elements
As a consequence of the loop theorem, if $F$ is a closed surface in the boundary of a 3-manifold and if the kernel $N = \ker(\pi_1(F) \to \pi_1(M))$ is nonempty then there is a nontrivial element of $...
2
votes
0answers
81 views
Lifts of geodesics on surfaces onto the universal cover [closed]
self-intersecting geodesic on hyperbolic surface of genus 2
Given a self intersecting geodesic on a hyperbolic surface of genus 2 as in the picture, how can I understand precisely what the lift to ...
0
votes
1answer
154 views
Nice decomposition of surface diffeomorphisms
The question is very vague therefore any kind of suggestions, reference, ideas are welcome.
Suppose $S$ is an oriented surface with or without boundary. Let $m$ be an area form. Let $f$ be a ...
3
votes
1answer
93 views
Generalizations of Dehn-Nielsen-Baer for topological branched cover?
For any manifolds $M$, a homotopy class of diffeomorphism gives rise to an automorphism of $\pi_1(M)$ (up to conjugacy since we are dealing with free homotopies). Moreover, in the specific case of ...
3
votes
1answer
145 views
Normal generating set for the intersection of two normal subgroups of a surface group
Let $G = \left< a_1,b_1, ... , a_g, b_g | [a_1,b_1] \cdots [a_g,b_g] \right>$ be the fundamental group of a surface of genus $g$. Let $N_1$ and $N_2$ be two normal subgroups of $G$ that are ...
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 $A$ ...
4
votes
0answers
79 views
Relative Poincaré 2-complexes
I have a couple of questions about (relative) low-dimensional Poincaré-Duality spaces, also known as Poincaré complexes.
From Eckmann-Müller and Eckmann-Linnell we know that a CW complex is a ...
3
votes
1answer
88 views
Is there a geometric interpretation of a Zariski dense surface subgroup?
Is there a geometric interpretation of a surface subgroup being Zariski dense? Or, conversely, given a $\Pi_1$ injective surface in a 3-manifold, is there a geometric or topological requirement on the ...
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 ...
1
vote
1answer
232 views
Dehn twist generators for mapping class group of a genus zero surface with boundary
Can you help me find a reference or explain how to find explicit Dehn twist generators for $MCG(S_{0,n})$, the mapping class group of a genus $0$ surface with $n$ boundary components, fixing the ...
3
votes
1answer
267 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)\}...
4
votes
0answers
155 views
Right-veering periodic automorphisms of surfaces that are not composition of right-handed Dehn twists
Basically the title of the question. For the sake of completeness I state an introduction to the question.
In "Right-veering diffeomorphisms of compact surfaces with boundary I" and II, the authors ...
5
votes
0answers
110 views
References on differential geometry and low-tech surveying
Apologies if this is a duplicate question, putting the word "surveying" into a search on this site is not very effective.
I'm interested in science education, and I was recently reminded of the old ...
1
vote
2answers
245 views
Does there exist a surface over a finite field which contains three skew lines?
Does there exist an irreducible surface, other than Hermitian surface, in $\mathbb{P}^3 (\mathbb{F}_q)$ containing three skew lines?
I know that this is true for Hermitian surface. In fact, at every ...
11
votes
2answers
430 views
Intrinsic vs Extrinsic geometry of convex surfaces
By Alexandrov's isometric embedding theorem, any locally convex metric prescribed on the sphere admits a realization as a convex surface in Euclidean 3-space, which, by Pogorelov's rigidity result, is ...
2
votes
0answers
92 views
Conditions of a quasiconvex codimension 1 subgroup of a hyperbolic group that determine it to be a surface subgroup
A subgroup $H$ of a group $G$ is called quasiconvex if in the Cayley graph of $G$, $\Gamma (G,S)$, there exists $k>0$ such that all geodesics that join a pair of points in $H$ are subsets of a $k$-...
4
votes
0answers
95 views
Positivity of the ramification divisor
Let $X$ be a non-normal surface such that $K_X$ is a pseudo-effective divisor
and ${\rm Bs}_{-}(K_X)$ (the diminished base locus of $K_X$) equals, at least set-theoretically, the non-normal locus of $...
1
vote
0answers
58 views
Characterizing a surface in $R^3$ with a given metric [closed]
Let $g$ be a Riemannian metric in $\mathbb{R}^2$. How can I find a surface in $R^3$ such that their curvature are the same? The shape of the surface in $3$ space is important. I mean I dont like to ...
2
votes
1answer
93 views
Constructing a Polyhedron given areas of its faces
I want to visualize a set of data as a polyhedron in 3d space. Imagine set A includes areas of such polyhedron's faces. I assume the first step is to check if there exist a polyehdron by making sure ...
2
votes
1answer
129 views
Is there a pair of pants decomposition analogue for orbifolds?
The pair of pants decomposition is a useful tool is surface theory. Is there an analogous decomposition for orbifolds?
Thanks
4
votes
0answers
41 views
Lower estimate on length of boundary of 2d Riemannian surface
Fix constants $\kappa\in \mathbb{R}, D>0,A>0$. Does there exist a constant $C>0$ depending on $\kappa, D, A$ only such that for any compact 2-dimensional Riemannian surface (or more generally ...
7
votes
2answers
314 views
Handle decompositions using only 1-handles
Let $\Sigma$ be an oriented, compact, connected 2-manifold with boundary. Assume that its boundary is equipped with a disjoint union decomposition into two non-empty parts:
$$\partial\Sigma=\partial_{...
8
votes
1answer
288 views
generalisation of umbilic surfaces
It is well known that if you have a complete surface in $\mathbb{R}^3$ with umbilic points, that is to say $k_1=k_2$ everywhere, where $k_1$ and $k_2$ are the principal curvatures, that is to say the ...
5
votes
2answers
221 views
homogeneous surface in $\mathbb{R}^4$
It is well known that the only homogeneous surfaces in $\mathbb{R}^3$ are the spheres, cylinders or planes. My question is about other examples in dimension $4$. Such a surface should have "constant ...
8
votes
3answers
212 views
Zone of negative curvature on surfaces embedded in $\mathbb{R}^3$
I consider the standard embedding of a compact oriented surface $\Sigma$ (say of genus 2) in the Euclidean space $\mathbb{R}^3$. I have coloured on the picture below the zone of this surface where the ...
4
votes
1answer
80 views
Bishop-Gromov type inequality for Jordan curve on 2-sphere
Let $\Sigma$ be a 2-sphere with a (smooth) Riemannian metric $g$ of nonnegative curvature. Let $\mathcal{C}$ be a simple closed smooth curve on $\Sigma$. Then $\mathcal{C}$ splits $\Sigma$ into two ...
3
votes
2answers
137 views
What is the homeomorphism from $\Gamma \backslash T_1 \mathbb{H}$ to $T_1(\Gamma \backslash \mathbb{H})$
Let $\mathbb{H}$ be hyperbolic plane, $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R}$) so that $\Gamma \backslash \mathbb{H}$ is a compact hyperbolic surface. Maybe it will be very simple to you ...
4
votes
1answer
167 views
The first eigenfunction of Dirac operator for surface
Let $M$ be a spherical oriented surface with Riemannian metric and with trivial spin structure. We know that the equation $$D \phi = \rho \phi$$, where $\rho: M\rightarrow \mathbb{R}$ is a real scalar ...
1
vote
1answer
102 views
Surfaces of $\mathbb{R}^3$ invariant by an affine map
I have a rather elementary question.
I would like to know what are the surfaces of $\mathbb{R}^3$ which are globally preserved by the action of a linear or affine map in a non trivial way. This ...
1
vote
1answer
86 views
Polar coordinates of a set with different radius and angle
Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that
$$U=\lbrace{ (r,\theta): 0<...
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votes
0answers
180 views
Modified Willmore energy and surfaces with infinitesimally narrow necks
Disclaimer: This is a copy of a question that I asked on the Mathematics Stack Exchange. It was suggested to me there that the question was worth asking over here.
There is an open problem in ...
