# Tagged Questions

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. It is a topological space (called the underlying space) with an orbifold structure.

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### Can the hyperbolic orbifold 2*55 be smoothly and isometrically embedded in 3-space?

Grow a square in the hyperbolic plane until its vertex angles become $\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is $-1$, the sides of the resulting hyperbolic ...
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### A space with “interchangeable” coordinates, $\mathbb{R}^n / S_n$

(I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best): I have a space that is similar to $\mathbb R^n$ ...
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### Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$?

The first question is Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$? In other words, is it obvious? The question stems from a theoretical physics ...
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### points which are fixed points of a finite group action

consider an open set $\tilde{U}\subset\mathbb{R}^n$ and a finite Lie-group $G$, which acts smoothly on $\tilde{U}$, i.e. we have a smooth map $G\times \tilde{U}\rightarrow\tilde{U}$. Suppose further, ...
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### Ref. Request — Non-Transitive Lie Group Actions, Applications to Orbifolds/Groupoids

I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some ...
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### Prove existence of local orbifold chart

Let $(U, G, \phi)$ be a $n$-dimensional complex orbifold chart over $x \in X$, i.e., $x \in \phi(U)$ I want to know if there is a a subset $V$ of $U$ such that $(V, G_x, \phi|_V)$ is also an orbifold ...
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### Is the Groupoid of germs associated to an orbifold a Hausdorff proper Lie groupoid?

I was studying the book Introduction to Foliations and Lie Groupoids by I. Moerdijk and J. Mrcun and I have a doubt. On page 140 they give the following definition for a proper Lie groupoid. A ...
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### Orbifold chart.

i'm trying to define an orbifold chart for the teardrop $R^2/Z_2$, where $Z_2$ acts via rotations. My advisor gave a tip: to use stereographic projections. But I'm a little stuck. Any help?
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### Weighed projective space charts

I'm studying weighed projective spaces and I found this reference http://arxiv.org/pdf/math/0510331v1.pdf* where it describes its orbifold charts (starts at page 53 of the PDF). My doubt is very ...
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### Set of non fixed points of an automorphism

I am trying to prove the following "For an orbifold chart $(\tilde{U},G,\phi)$ the set of non fixed point of $g : \tilde{U} \rightarrow \tilde{U}$ where $1 \neq g \ \in G$ is dense in $\tilde {U}$"...
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### Does Stokes' Theorem hold on spaces with singular points?

I have come across the question whether Stokes' theorem holds also on orbifolds. Let us take the simple case of $T^2/Z_2$ with a one-form $A$, then the question becomes: For a region $\Gamma$ with ...
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### Singular locus of orientable 3-orbifolds

Any reference (or any hints if the proof is easy) for the proof that the singular locus of a 3-dimensional orientable orbifold is a trivalent graph with each edge labelled by integers $a,b,c>1$ and ...
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### Quotient of Poincare dodecahedral space-example of spherical orbifold

Let $\mathcal{O}$ the orbifold with underlying space $S^3$ and singular locus the trefoil knot with local groups of order five. Then how can we see that it is the quotient space of the Poincare ...
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### General introduction to orbifolds?

Where should I go to learn about orbifolds? I am interested in a general introduction that gives precise definitions and clear explanations. I have a fair background in topological and smooth ...
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### Reference - Riemannian Orbifolds

I am looking for papers or textbooks talking about the various analog theorems of Riemannian Geometry of Manifolds to Riemannian Orbifolds like Toponogovs Theorem, Bonnet-Myers, Gauss Bonnet etc. So ...
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### Does this orbifold embed into $\mathbb{R}^3$?

Let $X$ be the space obtained by gluing together two congruent equilateral triangles along corresponding edges. Note that $X$ has the structure of a Riemannian manifold except at the three cone ...
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### Orbifolds and singular points

If I understand correctly, we roughly define the singular locus $\Sigma_O$ of an orbifold $O$ to be the set of points where the orbifold fails to be a manifold. In particular, if $x \in O$ has a ...
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### Connected sum while keeping curvature bounded.

Is it possible to perform a connected sum of two Riemannian Manifolds or Orbifolds while keeping curvature bounded from below? More explicitly, If $M_1$ and $M_2$ are two Riemannian manifolds (or ...
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### What is this orbifold?

Consider the non-negative real line with identification $x\sim 2x$. Is there a special name for this quotient space; what's known about it?
Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...