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Results tagged with gt.geometric-topology
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user 25510
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
2
votes
Is it always possible to connect the endpoints of a smooth injective path, so the resulting ...
Solution of your original question: A smooth curve cannot intersect infinitely many times lines in all directions.
Let $\gamma: T\to \Gamma$ be the parametrization
of your curve, where $T$ is the unit …
- 91.8k
6
votes
Accepted
Is a simple closed curve always a free boundary arc?
The answer is positive. For any Jordan curve, there is a homeomorphism of the Riemann sphere, which sends this curve
to a circle. Your statement immediately follows.
One proof of this statement involv …
- 91.8k
4
votes
Accepted
Finding a hyperbolic metric with geodesic boundary on a given Riemann surface
A good reference is
W. Abikoff, The real analytic theory of Teichmuller space, Springer, 1980. (Chap. II section 1).
The idea is that you construct the double: it is the result of gluing of your surfa …
- 91.8k
2
votes
Accepted
Existence of covering isomorphism
I suppose that "non-compact complex algebraic curve" means complex affine curve.
The following counterexample was proposed by my friend Fedor Pakovich.
Let $D=\mathbf{C}\backslash\{-1,1\}$.
Consider t …
- 91.8k
6
votes
Accepted
Nonexistence of sphere with one conical point [reference request]
The proof is very simple. Let $f$ be the developing map (take an isometry of some small disk on your surface to a region in the plane with constant curvature metric, and then perform analytic continu …
- 91.8k
32
votes
How would a topologist explain "every Riemann surface of genus $g$ is hyperelliptic if and o...
A 19th century topologist would explain this by dimension count. By Riemann-Hurwitz, a surface of genus $g$ covering the sphere
with $2$ sheets has $2g+2$ ramification points which gives $2g-1$ free c …
- 91.8k
3
votes
Accepted
Number of curves in an admissible system of Jordan curves on a surface
It comes from the so called "pants decomposition". A "pair of pants" (or simply pants) is a sphere with $3$ holes. Every compact surface of genus $g\geq 2$ can be decomposed into such pants. The $3g-3 …
- 91.8k
2
votes
Maximum of a sum of Gaussian functions
Yes, and this has nothing to do with Gaussian: you can take $\phi_i(x)=g_i(|x-x_i|)$
where $g_i$ are any strictly decreasing functions.
Lemma. If all $x_j$ are all on one side of a hyperplane $H$ (o …
- 91.8k
3
votes
Is there a mathematical book on general relativity that uses exclusively a coordinate free l...
R. Penrose, Structure of space-time (Benjamin, NY, 1968).
- 91.8k
5
votes
Are there any books/articles that apply abstract coordinate free differential geometry to ba...
There is a calculus textbook which does this:
Bamberg and Sternberg A course in mathematics for students in physics, vol. II,
Chap. 22, "Thermodynamics".
- 91.8k
2
votes
Teichmuller uniqueness theorem with marked points
Yes. See, for example, W. Abikoff, Real analytic theory of Teichmuller space, Springer, 1980, Chap. II, section 1.5 Theorem 2.
- 91.8k
4
votes
Accepted
Construction of self-covering map of any surface
First of all, the Riemann-Hurwitz formula with $\chi<0$ implies that
for every self-covering $d=1$ so it is an automorphism. The only punctured surfaces with $\chi\geq 0$ are torus, sphere, and sphere …
- 91.8k
27
votes
Is every rational realized as the Euler characteristic of some manifold or orbifold?
The answer for connected 2-dimensional orbifolds is no. Euler characteristic is
$$\chi(O)=\chi(M)-\sum\left(1-\frac{1}{q}\right)-\frac{1}{2}\sum\left(1-\frac{1}{p}\right),$$
where $p,q\geq 2$ are inte …
- 91.8k
15
votes
Accepted
How many metrics of constant curvature exist on a Riemannian surface?
First on terminology. "Riemannian surface" is a surface already equipped with a Riemannian metric. So the question "how many metrics of constant curvature exist on a Riemannian surface" makes sense on …
- 91.8k
2
votes
Illumination of a convex body
The answer depends on the point $p$ and on the body. Here is a simple counterexample in $R^3$. Take two regular hexagons in parallel planes, so that the line $L$ connecting their centers is perpendicu …
- 91.8k