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.

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10 votes
3 answers
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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 ...
Krishna's user avatar
  • 561
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 ...
Victor's user avatar
  • 2,076
26 votes
1 answer
820 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 ...
Anton Petrunin's user avatar
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....
Joseph O'Rourke's user avatar
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 ...
Wlodek Kuperberg's user avatar
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!
QHLIU's user avatar
  • 199
0 votes
0 answers
81 views

Let $S$ be a surface, $K$ compact in $S$ with finitely many components. Does the frontier of a component of $S-K$ have finitely many components?

Let $S$ be a connected surface and $K$ a compact subset of $S$ with finitely many connected components. Let $U$ be a connected component of $S-K$. Does the frontier of $U$ in $S$ have finitely many ...
Fernando Oliveira's user avatar
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 ...
Joseph O'Rourke's user avatar
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{...
Alufat's user avatar
  • 805
16 votes
3 answers
886 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 ...
Andrey Ryabichev's user avatar
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 ...
Baptiste Calmès's user avatar
13 votes
2 answers
836 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 ...
Mohammad Ghomi's user avatar
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 ...
Steven Gubkin's user avatar
11 votes
1 answer
748 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: $\...
Bumblebee's user avatar
  • 1,019
9 votes
1 answer
263 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 ...
Paul's user avatar
  • 1,379
8 votes
1 answer
206 views

Minimal surface enclosing two congruent balls

Let $B_1$ and $B_2$ be two unit-radius balls in $\mathbb{R}^3$ whose centers are separated by a distance $d \ge 2$. Q. For sufficiently small $d$, is the minimal area surface enclosing $B_1$ and $B_2$...
Joseph O'Rourke's user avatar
7 votes
4 answers
832 views

Lagrangian Kleinian bottles

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in ${\mathbb C}^2$ for the standard symplectic structure, mentioning that this were the only compact ...
ThiKu's user avatar
  • 10.3k
7 votes
2 answers
886 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 $\...
skupers's user avatar
  • 7,923
6 votes
1 answer
334 views

Immersion in $\mathbb R^3$ of a Klein bottle with Morse-Bott height function without centers

Can the Klein bottle be immersed in $\mathbb R^3$ so that the associated height function be of Morse-Bott type and have no centers? That is, the height function would have only Bott-type extrema and ...
Alexander Gelbukh's user avatar
5 votes
1 answer
226 views

Untangling two simple closed curves on a surface

Let $S$ be a smooth surface and $\gamma_1, \gamma_2$ be two transversal simple closed curves on it. Suppose moreover that there exists a simple closed curve $\gamma_1'$ on $S$ isotopic to $\gamma_1$ ...
aglearner's user avatar
  • 14k
4 votes
1 answer
451 views

Immersion of non-orientable surface in $\mathbb R^3$ with conditions on the height function

EDIT: The answer is trivially positive; the question arose from my misunderstanding of the figure below. Can a non-orientable closed surface of odd genus be immersed in $\mathbb R^3$ so that the ...
Alexander Gelbukh's user avatar
4 votes
1 answer
146 views

Given $f$ from the cylinder $C$ to the interval constant on one boundary, is there a $r:C\to C$ constant on a boundary with $f\circ r = f$?

My question might be trivial, but my lack of knowledge of this particular subject has not enabled me to find the answer. What I want to know is the following. Let $I=[0,1]$ and $C=S^1\times I$ be the ...
Mathieu Baillif's user avatar
4 votes
1 answer
481 views

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 ...
Arielle Leitner's user avatar
3 votes
1 answer
373 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 ...
Thomas's user avatar
  • 511
2 votes
1 answer
207 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 ...
user2367663's user avatar
1 vote
1 answer
324 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 connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least $C$....
user45673's user avatar
1 vote
1 answer
225 views

Continuity of $r\mapsto\int_{\Sigma\cap B_r(x)}f^2d\mu$

Let $\Sigma$ be an embedded smooth surface in $\mathbb{R}^3$, and let $f:\Sigma\to\mathbb{R}$ be a smooth function. Suppose $f$ is square-integrable on $\Sigma$, with \begin{align} 0<\int_{\Sigma}f^...
Anonymous amateur's user avatar
1 vote
1 answer
160 views

Minimal surface enclosing balls

(This question is tangentially related to an earlier question I posed: Minimal surface enclosing two congruent balls.) Let $B_1,\ldots,B_k$ be unit-radius balls in $\mathbb{R}^3$, with pairwise ...
Joseph O'Rourke's user avatar
1 vote
0 answers
135 views

End space of non-compact 2-manifolds described with proper rays

I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
Carlos Adrián's user avatar
0 votes
3 answers
928 views

Names of certain surfaces

Are there any generally used names for the following algebraic and nonalgebraic surfaces? Any references to literature where the surfaces are studied are also appreciated. Surface I. Implicit ...