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.
280
questions
39
votes
1
answer
6k
views
Classification of surfaces and the TOP, DIFF and PL categories for manifolds
A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories TOP, DIFF and PL. Well known proofs (e.g. via ...
- 2,076
34
votes
2
answers
910
views
If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?
I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...
- 1,442
29
votes
1
answer
1k
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 ...
- 1,243
28
votes
12
answers
3k
views
Creating high quality figures of surfaces
I am not sure if this question is suitable for mo, it is more about visualization than math. Anyway, here it is:
What is the best way to visualize a 2-surface in Euclidean space with high quality?
...
26
votes
1
answer
823
views
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 ...
- 43.7k
22
votes
2
answers
902
views
Fixed-point free diffeomorphisms of surfaces fixing no homology classes
One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...
- 313
21
votes
1
answer
1k
views
Homeomorphism historically: When did it reach its modern formulation?
Q. When did the notion of homeomorphism reach its
modern formulation as a bicontinuous bijection, i.e., a
continuous bijection
between topological spaces whose inverse is also continuous?
Was ...
- 149k
21
votes
5
answers
1k
views
Is a rhombus rigid on a sphere or torus? And generalizations
If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a ...
- 149k
20
votes
3
answers
1k
views
All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid?
Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open:
Given $S\subseteq\mathbb{...
- 805
19
votes
2
answers
2k
views
Simple curves on non-orientable surfaces.
Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to ...
- 31.5k
19
votes
0
answers
286
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 ...
- 221
17
votes
4
answers
2k
views
Morse theory in TOP and PL categories?
Apparently there are topological and piecewise linear versions of Morse theory. I would like to know of references that treat these topics.
How is a Morse function defined for compact manifolds (with ...
- 2,076
17
votes
2
answers
2k
views
Geodesics on the twisted pseudosphere (Dini's surface)
I wonder how difficult it is to compute geodesics on Dini's Surface,
a twisted pseudosphere?
Here is one parametrization, from
Alfred Gray's Modern Differential Geometry of Curves and Surfaces, p....
- 149k
16
votes
3
answers
889
views
Maximal degree of a map between orientable surfaces
Suppose that $M$ and $N$ are closed connected oriented surfaces. It is well-known that if $f \colon M \to N$ has degree $d > 0$, then $\chi(M) \le d \cdot \chi(N)$.
What is an elementary proof of ...
- 965
16
votes
3
answers
1k
views
Does a random walk on a surface visit uniformly?
Let $S$ be a smooth compact closed surface embedded in $\mathbb{R}^3$ of genus $g$.
Starting from a point $p$, define a random walk as taking discrete steps
in a uniformly random direction,
each step ...
- 149k
16
votes
2
answers
448
views
$G$-action on the integral homology of a compact surface
Let $S$ be a compact connected orientable surface, and let $G$ be a nontrivial finite group acting freely on $S$ and preserving orientation (note the the action being free is a strong condition, since ...
- 4,933
16
votes
0
answers
631
views
Approximating homeomorphisms of 2-disk by diffeomorphisms
Any homeomorphism of a compact surface can be approximated by diffeomorphisms.
Is there a parametrized version of this result, where the parameter space is an $n$-disk?
In other words, if $S$ is a ...
- 28.5k
15
votes
3
answers
1k
views
Space of embeddings of circle in a surface
Let $S$ be a compact oriented surface of genus at least $2$ (possibly with boundary). Let $X$ be a connected component of the space of embeddings of $S^1$ into $S$.
Question : what is the ...
- 151
14
votes
1
answer
1k
views
Who proved that two homotopic embeddings of one surface in another are isotopic?
If $\Sigma_1$ and $\Sigma_2$ are two compact topological surfaces with boundary and $\phi, \psi : \Sigma_1 \hookrightarrow \Sigma_2$ are two orientation-preserving embeddings that are homotopic, then ...
- 10k
14
votes
1
answer
885
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 ...
- 43.7k
13
votes
2
answers
838
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 ...
- 7,047
13
votes
2
answers
3k
views
Connected sum of surfaces
I'm looking for a detailed reference about connected sums. I'd like it to contain a proof that a connected sum of connected surfaces is independent - up to homeomorphism - of the various choices ...
- 778
13
votes
0
answers
252
views
Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$
An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane.
The following questions are motivated by Anton Petrunin's Disc bounded ...
- 7,256
12
votes
3
answers
998
views
How to generate all triangulations of an orientable surface?
$\newcommand{\comb}{\mathrm{comb}}$Consider an orientable surface $S$ with punctures and boundaries (each boundary having at least a marked point).
A triangulation, up to orientation preserving ...
- 1,335
12
votes
1
answer
2k
views
Naive definition of surface area doesn't work?
A first stab at a definition of surface area might go like this:
Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...
- 12k
12
votes
2
answers
421
views
How the hyperbolic metric changes when we add a puncture?
Suppose we have a surface $S$ of a finite genus, without boundary with a finite number of punctures. Suppose that this surface comes equipped with a hyperbolic metric of curvature $-1$.
Question 1: If ...
- 4,761
11
votes
2
answers
5k
views
What is parameterization of the trefoil knot surface in R³?
What is a parameterization, say (x(u,v),y(u,v),z(u,v)), of the trefoil knot surface in R³ whose cross-section can be circular or, in general, elliptic?
Thanks!
- 199
11
votes
1
answer
758
views
When does the shape operator commute with a derivative?
Suppose we have a smooth map $\varphi : S\to H$ between two regular parametric surfaces in $\mathbb{R}^3.$ Then at any point $p\in S,$ we have following maps between corresponding tangent spaces:
$\...
- 1,019
11
votes
5
answers
3k
views
How can I sample uniformly from a surface?
Given an equation of a parametric surface, is there a general way to sample of points uniformly distributed on that surface?
I'm interested in this problem for purposes of visualisation - rather than ...
- 213
10
votes
3
answers
846
views
Homeomorphic but Non-Conjugate Mapping Tori
Suppose we fix a genus $g$ closed surface $S$. Let $f, g \in Map(S)$ be conjugate, for $Map(S)$ the mapping class group of $S$. Then I know that $M_f$ (the mapping torus of $M$ with monodromy $f$) is ...
- 561
10
votes
2
answers
1k
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 ...
- 305
10
votes
2
answers
377
views
Group of surface homeomorphisms is locally path-connected
I think the following is true and I need a reference for the proof. (Given a closed surface $S$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $S$ with a distance ...
- 1,517
10
votes
2
answers
2k
views
Do all combinatorially distinct fundamental polygons correspond to surfaces?
The topology of a closed surface can be constructed
by identifying edges of a fundamental polygon of an
even number $2n$ of edges.
Labeling the edges and using $\pm 1$ exponents to indicate
direction,
...
- 149k
10
votes
1
answer
679
views
Solids with constant surface area during "erosion"
Imagine a drug, a pill that you swallow, which is designed to dissolve in your
stomach at a constant rate. It must be shaped such that the surface area
remains constant when the volume is "eroded" ...
- 203
10
votes
0
answers
336
views
Surfaces with many (but not solely) closed geodesics?
Let $S$ be a closed surface embedded in $\mathbb{R}^3$,
let's say of genus zero.
I seek examples of $S$ with the following property:
If one selects a random any point $p$ on $S$, and a random
...
- 149k
9
votes
6
answers
2k
views
Classification of surfaces composed of circles
Define a circle as a geometric circle of positive, finite radius:
a set of points in $\mathbb{R}^3$ congruent to the
set $x^2 + y^2 = r^2$ in the $xy$-plane. [Edited as per BMann's comment.]
I am ...
- 149k
9
votes
2
answers
1k
views
Triple bubble conjecture: Natural candidate?
Is there a standard natural candidate surface for
the shape that encloses three given volumes
in $\mathbb{R}^3$ and has minimal surface area?
I know the planar triple bubble conjecture was proved
...
- 149k
9
votes
2
answers
557
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 ...
- 914
9
votes
1
answer
166
views
When do the lengths of simple closed curves determine a hyperbolic surface?
Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g$...
- 2,636
9
votes
1
answer
267
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 ...
- 1,379
9
votes
1
answer
223
views
Links and non-orientable surfaces
Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion.
Is the ...
- 443
9
votes
0
answers
206
views
Algebraic context for Mednykh's formula?
Let $S$ be a closed orientable surface and let $G$ be a finite group, then Mednykh's formula says that
$$
\sum_{V}d(V)^{\chi(S)} = |G|^{\chi(S) - 1} |\text{Hom}(\pi_1 S, G)|
$$
where the sum is over ...
- 5,319
8
votes
2
answers
449
views
Flat metric on compact surface minus a point
Let $T^2$ be a compact smooth surface and let $p\in T^2$. Suppose that $T^2$ admits a symmetric $\left(0,2\right)$-tensor which is a flat Riemannian metric restricted to $T^2-\{p\}$. Is it true that $\...
- 537
8
votes
2
answers
246
views
Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence
Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper
map $f\colon \Sigma\to \Sigma$ of degree $1$ and not homotopic to
any self-homotopy ...
- 265
8
votes
2
answers
205
views
Tangent fields spanning the distribution of principal directions on a surface
Suppose $S$ is an orientable regular surface in $\mathbb R^3$ without umbilical points (not necessarily compact, and with no boundary). There are two well-defined smooth $1$-dimensional tangent ...
- 46.8k
8
votes
1
answer
918
views
All non-compact simply connected $2$-manifolds with boundary
There are two corresponding posts MSE and MSE by me without any answers.
Problem: Let $\Sigma$ be a non-compact simply-connected $2$-dimensional manifold,
with boundary. Then, up to homeomorphism $\...
- 632
8
votes
2
answers
891
views
What is the homotopy type of the space of simple closed curves isotopic to a given one?
For surfaces there are many statements along the lines of: if two simple closed curves are homotopic, they are isotopic. I'm interested in such questions for families of curves.
More precisely, let $\...
- 7,933
8
votes
1
answer
437
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 ...
- 312
8
votes
3
answers
264
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 ...
- 2,636
8
votes
1
answer
250
views
Characterizing surface area
(This question is a variant of an unanswered question at math.stackexchange.)
The Definition section of Wikipedia's article on surface area currently starts as follows:
While the areas of many ...
- 24.6k