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Questions tagged [torus]

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38
votes
2answers
2k views

3D Billiards problem inside a torus

I have been trying to simulate the behavior of a light particle being reflected inside of a torus (essentially a 3D billiards problem). I have found that after a few thousand bounces, it converges on "...
13
votes
1answer
278 views

Homogeneous spaces that are homotopy tori

Let $G$ be a compact Lie group, and let $H$ be a closed subgroup such that $G/H$ is homotopy equivalent to a torus. Is it true that $H$ is normal and $G/H$ is isomorphic to a torus as a Lie group? ...
10
votes
1answer
237 views

Foliations by circles on the 3-torus

Let $T=S^1\times S^1 \times S^1$ be the 3-torus. Let us denote by $\alpha$ the isotopy class of the loop $S^1\times pt\times pt$ and let $\mathcal F_\alpha$ be the set of all smooth oriented ...
5
votes
3answers
547 views

Constructing a vector field with given zeros on a torus

By the Hopf-Poincaré theorem, the sum of the indices of the zeros of a vector field on the d-dimensional torus must equal zero. Given an even number of points $x_i$ on a d-dimensional torus, and ...
5
votes
1answer
117 views

Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?

In a nutshell: Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form? Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...
5
votes
1answer
252 views

Topology of connected subsets of the $3$-torus

Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$. We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded. I am ...
4
votes
1answer
170 views

Special coordinates for periodic metrics

This question is a follow-up to that one. Given a $\mathbb{Z}^n$-periodic metric $g$ on $\mathbb{R}^n$ (with $n>2$), is it possible to find a periodic diffeomorphism $\varphi$ such that $\varphi^*...
3
votes
2answers
198 views

Study of Hex on the Torus

Hex is usually played on a parallelogram shaped board. What if you play it on a Torus? One thing I notice is that the idea of connecting opposite sides doesn't make much sense anymore, since a torus ...
1
vote
0answers
71 views

On some finiteness properties of cohomological algebras of complex tori

Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module. ...
1
vote
0answers
511 views

How can one define “punctured torus” in Homotopy Type Theory? Is its fundamental group the free product of the integers with themselves?

Questions. Has the beautiful old idea, in part already known to Gauß, of a punctured torus surface (take, if you will, the classical set-theoretic definition as the meaning of the latter three words)...
1
vote
0answers
29 views

Low bound approximation of a Torus Knot length

Is there a formula for approximating (lower bound) the torus knot length ? The torus knot parameters are (p, q, R, r) where (p,q) are co-primes and R is major axis and r is minor axis of the torus.
0
votes
1answer
72 views

Parametric Surface Equations for Orthogonal Projection of Torus Knot Tube onto Torus [closed]

What are the parametric equations for the orthogonal projection of the torus knot tube onto the torus surface? For instance, if we have the equations for the torus knot $$ \vec r(t)= (R+r\cos pt)\...
0
votes
0answers
93 views

Open subsets of the n-torus containing no nontrivial loops

Let $T^n$ denote the $n$-dimensional torus. Suppose there is an open subset $U\subset T^n$ not containing any nontrivial loop. Does this imply that the inclusion $U\hookrightarrow T^n$ is ...
-2
votes
0answers
44 views

Let $V$ and $W$ be real representations of a torus $T$ s.t. $\dim V^H=\dim W^H$, $\forall H<T$. Show that $V\simeq W$

$V^H:=\{v\in V:hv=v,\,\forall h\in H\}$ is the fixed point set. One has that $$V=\bigoplus_jV(\chi_j)\oplus V^T,$$ with $\chi_j:T\to U(1)$ a character of some nontrivial irreducible subrepresentation ...