# Questions tagged [torus]

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31
questions

5
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1
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240
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### Topology of windings on the two-torus

In short my question is: what can we say about the quotient topology induced by the linear flow on a two-torus?
I know that an irrational slope leads to a dense winding and hence (if I'm not mistaken) ...

1
vote

0
answers

52
views

### Dyck's Theorem via a Birational Transformation

In our paper, at Journal of Geometry and Graphics, v. 20, n. 1 (2016), Danny Maienschein and I showed that some well-known transformations of the real projective plane (defined almost everywhere), ...

4
votes

1
answer

407
views

### Faithful locally free circle actions on a torus must be free?

Is it true that every faithful and locally smooth action $S^1 \curvearrowright T^n$ is free?
I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$.
Another related question is: ...

16
votes

1
answer

584
views

### Can the product of an exotic torus and a circle be the standard torus?

As discussed in this question from last week, if $M$ is a closed manifold such that $M\times S^1$ is homeomorphic to the torus $T^{n+1}$, then $M$ is homeomorphic to $T^n$. Is the corresponding ...

5
votes

0
answers

235
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### Is a circle of circles necessarily a 2-manifold?

Let $X$ be a continuum (a compact connected metric space).
Suppose that there is a collection of simple closed curves $\mathcal C$ which partitions $X$ into pairwise-disjoint, nowhere dense sets. ...

2
votes

0
answers

42
views

### Is a fixed subgroup of a compact Lie group cotorally included in finitely many conjugacy classes?

Let $G$ be a compact Lie group, in the next discussion we consider only its closed subgroups without specifying it. We say that a subgroup $L$ is a cotoral subgroup of $K\leq G$ if $L \trianglelefteq ...

1
vote

0
answers

345
views

### Flat metrics on twisted torus

I think I figured out how to induce a flat metric on a 2-torus, by defining a parallelogram in $\mathbb{R}^2$ with base of length $\lambda$ along the x axis, and top left corner of coordinates $(\...

3
votes

1
answer

181
views

### Stein fillable tight contact structures on the 3-torus

Kanda classified tight contact structures on the 3-torus. Which of them is Stein fillable? Is there any good reference?

16
votes

2
answers

751
views

### What happens to a closed manifold to ensure it is homeomorphic to a torus $T^{n}$?

If $M$ is a smooth connected closed $n$-dimensional manifold, its universal covering space is homeomorphic to Euclidean space $R^{n}$, and its fundamental group is $Z^{n}$, then is it homeomorphic ...

5
votes

1
answer

782
views

### Categorical-geometric Langlands for tori

Fix a "nice" curve $X$ (smooth, projective, proper, geometrically connected, what-have-you) and an algebraic torus $G$, both over a field of characteristic $0$ (possibly algebraically closed?...

0
votes

0
answers

60
views

### Criterion for the existence of roots of a system of polynomial equations on a torus

Given a system of polynomial equations in $n$ indeterminates $z_{1}%
,\ldots,z_{n}$:
\begin{align*}
p_{1}\left( z_{1},\ldots,z_{n}\right) & =0\\
& \vdots\\
p_{m}\left( z_{1},\ldots,z_{n}\...

2
votes

0
answers

74
views

### Is this family of minimal tori compact?

Let $\Sigma$ be a smooth $2$-sphere and let $M = \Sigma \times \mathbb{S}^1$. Fix an integer $n \geq 0$. Is there generic set $\mathcal{S}$ of Riemannian metrics on $\Sigma$ such that the following ...

4
votes

0
answers

82
views

### Infinitely many distinct minimal tori

Let $M = \Sigma_g \times \mathbb{S}^1$ be endowed with the product metric, where $\Sigma_g$ is a compact orientable surface of genus $g$ with an arbitrary fixed metric. Is it true that there are ...

4
votes

1
answer

427
views

### Smallest subgroup of unitary group, containing diagonal matrices and a fixed unitary matrix is the whole group

Suppose that $U_n(\mathbb{C})$ is the group of unitary matrices of dimension $n$ over complex numbers. Fix a unitary matrix $A \in U_n(\mathbb{C})$ and consider the smallest closed subgroup $K \...

14
votes

1
answer

974
views

### Minimal good cover of the torus

Recall that an open cover $\mathfrak{U} = \{ U_\alpha \}$ of a manifold $M$ is called a good cover if all possible finite intersections $U_{\alpha_1} \cap ... \cap U_{\alpha_n}$ are contractible.
...

1
vote

1
answer

188
views

### What information about the lattice $\Lambda$ can be recovered from the metric space $\mathbb{R}^n/\Lambda$?

Let $\Lambda\subset\mathbb{R}^n$ a lattice, i.e., a discrete subgroup that spans $\mathbb{R}^n$. Now we can look at the torus $T=\mathbb{R}^n/\Lambda$ which naturally carries the metric $d_T$ induced ...

5
votes

1
answer

412
views

### Poisson summation formula and its implication for the spectrum of the flat torus

I usually ask questions on math.stackexchange but I figure this one is more suited to being asked here. I should preface that I am a complete novice undergraduate, and unlikely to understand answers ...

4
votes

2
answers

656
views

### Hörmander-Mikhlin theorem on the torus

Let me first recall a particular case of the classical Hörmander-Mikhlin multiplier theorem: Let $m$ be a bounded function on $\mathbb {R} ^{n}$ which is smooth except possibly at the origin, and ...

7
votes

1
answer

501
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 ...

5
votes

1
answer

158
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 ...

1
vote

0
answers

74
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.
...

0
votes

1
answer

258
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### 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

0
answers

116
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 ...

4
votes

2
answers

407
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 ...

39
votes

2
answers

3k
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 "...

2
votes

0
answers

2k
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### 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)...

10
votes

1
answer

354
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 ...

1
vote

0
answers

35
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.

13
votes

1
answer

317
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?
...

6
votes

3
answers

1k
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 ...

4
votes

1
answer

204
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^*...