20
votes
Conceptual proof of classification of surfaces?
I guess the most conceptual proof is the one using Morse theory:
Take a Morse function on the (closed, orientable) surface S. If it has no saddle points, then (using the gradient flow) $S\cong S^2$. ...
- 10.4k
19
votes
A necessary and sufficient condition for a space curve to lie on a ellipsoid
There is a straightforward way to deduce necessary conditions for a space curve to lie on an ellipsoid, and it's really a matter of calculation to make these conditions explicit in terms of the ...
- 108k
18
votes
Accepted
When does the shape operator commute with a derivative?
The question is essentially equivalent to the following classical question: Given a smooth surface $S$ and a bundle map $L:TS\to TS$, when does there exist an immersion $x:S\to \mathbb{R}^3$ such ...
- 108k
14
votes
Accepted
A variant of the Monge-Cayley-Salmon theorem?
Setting aside the assumption that $\phi$ be a polynomial mapping for the moment (however, see below for a construction of a large family of polynomial solutions), if one makes the 'nondegeneracy' ...
- 108k
14
votes
If a triangle can be displaced without distortion, must the surface have constant curvature?
Already Riemann in his famous "On the Hypotheses Which Lie at the Bases of Geometry" concludes that the spaces of constant curvature are precisely those in which figures can move without distortion. ...
- 16.5k
12
votes
Text on old-fashioned differential geometry
Chern explains moving frames in a masterful 4 pages: (1990, pp. 210–213).
Further nice book-length treatments using them:
Valiron (1950, Chap XIII–XIV) — last of the classic Cours, translated, very ...
- 31.5k
12
votes
Conceptual proof of classification of surfaces?
The proof of Zeeman described in this note is by a substantial margin the easiest and most conceptual proof I know. To simplify the exposition I restrict to orientable surfaces in the note, but it is ...
- 44.8k
12
votes
Conceptual proof of classification of surfaces?
This is more an extended comment than an answer to the question. The first thing to note is that there are different strenghts of the classification theorem for surfaces. Of course, there are the ...
- 12.1k
11
votes
Accepted
Hadamard theorem about embedding
I think the relevant location is item 23, page 352, but what Hadamard aims to is stated as follows:
A smooth, co-orientable surface of $\mathbb{R}^3$ with Gauss curvature bounded below by some $\...
- 14.4k
11
votes
Accepted
To find a point in Teichmüller space or measured foliation, how many lengths of curves do you need?
As indicated in my comments, the Teichmüller question is a duplicate of this question.
For the measured lamination case, the fact that $6g-5$ curves suffice was shown by Hamenstädt.
Hamenstädt, ...
- 68.8k
11
votes
Number of points on a surface modulo p
For $s \geq 6$ this is elementary as one can use the Weil bound for Jacobi sums, which predates Weil. By orthogonality of characters, we can express the number of points as $$ \frac{1}{ (p-1)^2} \sum_{...
- 148k
10
votes
Accepted
Minimal area of Seifert surfaces
In question (1), if you allow $g$ to vary, then this is answered positively by Hardt and Simon (see also).
The answer to question (2) is no. Almgren and Thurston construct unknots which do not bound ...
- 68.8k
10
votes
An abstract characterization of line integrals
I don't know if it's exactly what you're looking for, but line integration is the unique way to assign a real number $I(\omega,c)\in\mathbb{R}$ to every pair of a smooth $1$-form $\omega$ on a smooth ...
- 1,790
9
votes
Text on old-fashioned differential geometry
There is a very general discussion of moving frames in Clelland's book, From Frenet to Cartan: The Method of Moving Frames, and this in particular includes discussion of Darboux's use of frames on ...
- 26.3k
9
votes
Accepted
Examples of complicated parametric Jordan curves
Some examples and references are mentioned here Examples of plane algebraic curves. You can find many Jordan curves in the family $e^{it}+re^{int}, 0\leq t\leq 2\pi,$ by choosing parameters properly.
...
- 91.8k
9
votes
Accepted
Are any of these complex surfaces ever projective?
Here is a simple method for constructing projective examples:
Assume there exist maps $f:C \to \mathbb{P}^1$ and $g:T \to \mathbb{P}^1$ of the same degree which are totally ramified at $c$ and $t$. ...
8
votes
Accepted
homogeneous surface in $\mathbb{R}^4$
I'm rearranging my answer a little bit because I realized that I overlooked an apparent possibility (that turns out not to occur), and I didn't want my answer to be misleading:
If the surface in ...
- 108k
8
votes
Accepted
Most general version for the Gauss-Bonnet theorem for polygons
There are no other conditions, and in fact a more general statement is true.
The standard reference is the survey of Reshetnyak, Two-dimensional surfaces of bounded curvature,
in the book:
MR1263963
...
- 91.8k
8
votes
A necessary and sufficient condition for a space curve to lie on a ellipsoid
Robert describes the differential equations which one can write in terms of $\tau$ and $\kappa$, and the inherent limitations in this local approach. But maybe one can find more reasonable or useful ...
- 7,149
8
votes
Accepted
Realizing Morse functions on $S^2$ as height functions
Any Morse function on $S^2$ may be realized by an embedding $S^2\hookrightarrow \mathbb{R}^3 \to \mathbb{R}$. For a Morse function $F:S^2\to \mathbb{R}$, take the equivalence relation with equivalence ...
- 68.8k
8
votes
Accepted
Generate $\mathrm{Mod}(S_g)$ by two Dehn twists
In
Humphries, Stephen P.
Generators for the mapping class group. Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44–47,
Lecture Notes in Math., 722, ...
- 44.8k
7
votes
Accepted
Construction of a linear Weingarten surface from a space curve
You'll find a discussion of the analysis of linear Weingarten surfaces via exterior differential systems in these lecture notes of mine. Particularly look at Section 5.1, where it is discussed at ...
- 108k
7
votes
Geodesics on the twisted pseudosphere (Dini's surface)
Since Joseph asked for pictures, I will add to Robert's excellent response with the following images. I will stick to Robert’s notation (which includes the unconventional choice of using the $y$-axis ...
- 326
7
votes
Identity involving an improper integral (with geometric application)
Your guess is correct. Let's write $t=cs$, so the integral becomes
$$
\int_1^{\pi/(2c)} \sqrt{\frac{1+c^2 s^2}{s^2-1}}\, \frac{ds}{s} .
$$
Fix a small $\epsilon>0$, and consider first the integral ...
- 24.2k
7
votes
Accepted
Identity involving an improper integral (with geometric application)
Since the main contribution to the integral comes from $t<<1$, analytically one has
\begin{align}
\lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt&=\lim_{c\to 0^+}\int_c^{...
- 5,624
7
votes
Accepted
Estimate of number of boundary components of a compact Riemannian 2-surface
I think it follows from Gauss-Bonnet. Suppose $X$ has genus $g$ and $n$ boundary components. Gauss-Bonnet says that
$$\int_X K\;dA+\int_{\partial X}k\;ds=2\pi\chi(X)=2\pi(2-2g-n),$$
where $K$ is ...
- 1,100
7
votes
If a triangle can be displaced without distortion, must the surface have constant curvature?
This is sort of an answer and sort of not. I'll let you be the judge:
Suppose you formulate the question, not in terms of 'motion' (which you left vague) but terms of 'freely copying' a triangle $T$,...
- 108k
7
votes
A variation on four-vertex theorem
Yes, this is true. More generally, if a smooth closed strictly convex curve intersects some circle in $2n$ points, then it has at least $2n$ vertices. This is stated in Blaschke's book "Kreis und ...
- 6,337
7
votes
Accepted
Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$
This was solved in a series of articles in the 1960s.
De Giorgi, Almgren, and Simons have shown that in $\mathbb{R}^{\le 8}$ every CMC graph is a hyperplane. Then Bombieri - De Giorgi - Giusti have ...
- 6,337
7
votes
Accepted
Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?
I missed the real analyticity condition (my comment makes perfect sense for $C^\infty$ though), so let's move points in a fancy way to satisfy it.
First, observe that if $a_0,a_1,a_2$ are positive ...
- 61.9k
Only top scored, non community-wiki answers of a minimum length are eligible