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4
votes
2answers
99 views

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 ...
1
vote
0answers
19 views

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 ...
3
votes
1answer
48 views

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 ...
2
votes
1answer
68 views

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?
2
votes
0answers
51 views

How do we check if a covering of an orbifold is a manifold?

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 ...
1
vote
2answers
150 views

What's the difference between an orbifold with a conical singularity and a conifold?

In Becker, Becker, Schwarz's book 'String Theory and M-Theory: A Modern Introduction', page 360 they explain how an orbifold of $\mathbb{C}/\mathbb{Z}_{2}$ (which is equivalent to ...
2
votes
1answer
70 views

“[T]ransversely isotropic and mirror-symmetric (space group:$D_{\infty h}$)”, its orbifold notation?

I am trying to understand this frieze pattern $D_{\infty h}$ aka its orbifold notation. This describes spider's silk. The authors call it a space group, some sort of generalization from orbifolds. ...
5
votes
1answer
149 views

Magic theorem for cylinders? Symmetry classes according to Conway's notation?

My teacher Kirsi of Mat-1.3000 in Aalto University stated 17 symmetry classes for planes and 14 for spherical things (some lecture slides here). She used Conway Thurston's notation to classify ...
3
votes
1answer
170 views

Quotient Riemann surfaces

Let $\mathbb{H}$ be an upper half plane (this is a Riemann surface), then $PSL(2,\mathbb{Z})$ acts on $\mathbb{H}$ and it is well-know that $$ \mathbb{H}/PSL(2,\mathbb{Z})\cong \mathbb{C} $$ is again ...
2
votes
1answer
85 views

Etale groupoid and Morita equivalence

Let $\mathcal{G}=(G_{0},G_{1})$ be a groupoid, where $G_{0}$ is the space of objects and $G_{1}$ is the space of morphisms. $\mathcal{G}$ is called etale if both the source and target maps $$ ...
1
vote
0answers
60 views

Definition of Homolgy for an orbifold

I am moving to the study of moduli of curves and I am looking at these notes where $\mathcal{M}_{g,n}$ is described as an orbifold. In order to define the tautological ring the notion of homology and ...
3
votes
1answer
168 views

The teardrop and the spindle are bad orbifolds

I should prove that the teardop $S^2(p)$ (the orbifold with underlying surface $S^2$ and a single cone point of order $p>1$) and the spindle $S^2(p,q)$ (the orbifold with underlying surface $S^2$ and ...