A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.

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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 ...
21
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1answer
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? ...
18
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3answers
637 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\...
17
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4answers
1k 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 ...
16
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5answers
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 ...
16
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2answers
356 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 ...
15
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2answers
1k 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 ...
14
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1answer
2k 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 ...
13
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0answers
353 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 ...
12
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1answer
1k 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 ...
12
votes
3answers
634 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 ...
11
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1answer
851 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....
9
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2answers
706 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, ...
9
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2answers
1k 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 ...
8
votes
2answers
143 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 ...
8
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1answer
207 views

Foliation with leaves which are and are not dense

Do there exist a foliation on a closed surface (i.e. real dimension 2) which has a dense leaf and also a leaf which is not dense?
8
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1answer
148 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 ...
8
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0answers
235 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 ...
7
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2answers
554 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 ...
7
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1answer
212 views

Archimedean fibers “intersecting” curves on arithmetic surfaces

Let's fix a number field $K$ with its ring of integers $O_K$. Moreover consider an arithmetic surface $f:S\to \text{Spec } O_K$. For every archimedean place $\sigma$ in $K$, $K_\sigma$ is the ...
7
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1answer
336 views

Costa's minimal surface and the structure of lungs

Seeing this image of Costa's minimal surface        (MathWorld image) made me wonder if the fine-grained structure of the human lung is somehow composed of pieces of ...
6
votes
2answers
445 views

Do the following set of Dehn twists generate the mapping class group?

If $S$ is the surface illustrated below, do the Dehn twists about the red curves generate the mapping class group $\operatorname{MCG}(S,\partial S)$?
6
votes
2answers
228 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 ...
6
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2answers
2k 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!
6
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1answer
351 views

Which surfaces have only a finite number of connecting geodesics?

Q1. For a smooth, closed (compact) surface $S$ embedded in $\mathbb{R}^3$, under which conditions is it true that, for every pair of points $a,b \in S$, there are an infinite number of ...
6
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2answers
247 views

Geodesic flow on infinite surfaces

The geodesic flow on a compact hyperbolic surface (i.e. a surface with a riemannian metric of constant curvature $-1$) has been well-studied, in particular it has been known for a long time that it is ...
6
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2answers
415 views

How to compute the normals to Costa's minimal surface?

I am trying to draw Costa's minimal surface in high resolution using the PovRay raytracer. For this I need to compute points on the surface as well as the normals. It is relatively easy to compute the ...
6
votes
1answer
137 views

Can a surface group act on a finite-valence simplicial tree?

Question. Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree. ...
6
votes
0answers
140 views

Harmonic map heat flow in positive curvature

Suppose I wish to relax/smooth a map $\phi:M\rightarrow N$ between two surfaces $M,N$ embedded in $\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ...
5
votes
2answers
340 views

Action of $\pi_1(S)$ on its commutator subgroup

Let $G$ be a group. It acts canonically on its derived subgroup by conjugation. Can on describe the orbits of this action when $G$ is the fundamental group of a compact orientable surface of genus $g \...
5
votes
1answer
515 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 ...
5
votes
2answers
594 views

Pursuit-Evasion on a Manifold

I know pursuit-evasion has been studied in many contexts, including on a manifold (e.g., Melikyan, "Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds"), but I have not seen this version: ...
5
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1answer
159 views

Translation surfaces & integer multiples of $\pi$

Richard Schwartz, in Mostly Surfaces (Vol. 60. American Mathematical Soc., 2011), defines (on p.14) a translation surface as "a Euclidean cone surface, all of whose 'angle errors' are integer ...
5
votes
2answers
438 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 $\...
5
votes
4answers
432 views

Lagrangian Kleinian bottles

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for ...
5
votes
1answer
232 views

Fixing a proof of the systolic inequality for higher genus surfaces

I'm currently learning some stuffs about systolic inequalities. While reading the relevant sections (p329 to 340) in Berger's Panoramic View of Riemannian Geometry, I noticed a gap in one of the ...
5
votes
1answer
348 views

Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface

It is well known that Grayson's dumbbell neck-pinch1,2 separates into disconnected pieces under mean curvature flow:                     Image ...
4
votes
2answers
258 views

Some facts about cut-locus

Let $M$ be a 2-dimensional closed Riemmanian manifold diffeomorphic to $S^2$. S.B.Myers says "the cut-locus of every point $x\in M$ is a finite tree." How the set of point can be a tree? ...
4
votes
2answers
464 views

Surface Laplace-Beltrami without coordinates, exterior calculus?

Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator $\...
4
votes
1answer
259 views

Representing groups with two generators as graph automorphisms

Suppose we have a group $G$ which can be generated by two elements $x$, $y$. Call $H$, $K$, $L$ the subgroups of $G$ generated by $x$, $y$ and $y^{-1}x^{-1}$, respectively. With these data, we can ...
4
votes
1answer
732 views

intersection number

I vaguely recall the following fact that I'd like to use in my research. It should be easy to see that this holds (if it does) but I can't seem to prove it. Let $p:X\longrightarrow S$ be a (regular) ...
4
votes
1answer
186 views

Fair surfaces - general mathematical theory

Fairness measures for surfaces are, in general, functionals containing more complicated terms thatn the usual bending energy, and may depend not only on the mean curvature but also on principal ...
3
votes
3answers
2k views

Covering spaces of surfaces

Let $\Sigma_g$ be a surface of genus $g\ge 2$, and let $\Sigma_k$ be an $m$-sheeted covering space of $\Sigma_g$. It is known that $k=m(g-1)+1$. An example of such a covering space is a regular ...
3
votes
2answers
175 views

Random metrics on compact orientable surfaces

Hello everyone, Let $S_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S_g$ endowed with the topology of ...
3
votes
2answers
243 views

Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$

I asked this on math.stackexchange.com, but didn't get a single answer. Charles Weibel writes in his survey of homological algebra Riemann defined a surface $S$ to be $(n + 1)$-fold connected ...
3
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2answers
339 views

Moving a canonical divisor on a normal surface away from the singular locus

In a previous question Moving a Weil divisor on a normal surface away from a finite set of closed points I probably asked for too much. As J.C. Ottem pointed out, it is not always possible to move a ...
3
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1answer
82 views

Homotopy classes of homeomorphisms of a multiple pointed space

Let $M$ be a multiple pointed space, i.e. $M$ is a topological space and there is a finite point set $M\supset P=\{p_1,...,p_k\}, k<\infty$. Such a $p_i$ is called a marked point. A map $$\varphi:M,...
3
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1answer
99 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 ...
3
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1answer
266 views

Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial

Let $f:X\longrightarrow B$ be a family of curves, with $f$ relatively minimal, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all ...
3
votes
1answer
257 views

Removing intersections of curves in surfaces

Let $C_1, \dots, C_n$ be a family of disjoint simple curves in a surface $\Sigma$. If $C$ is any simple curve in $\Sigma$, it turns out that we can map $C$ to a curve $C'$ (via a homeomorphism of $\...