Questions tagged [torus]
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31 questions
5
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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) ...
- 51
1
vote
0
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53
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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), ...
- 146
4
votes
1
answer
419
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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: ...
- 385
17
votes
1
answer
644
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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 ...
- 19.3k
5
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0
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238
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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. ...
- 3,317
2
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0
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43
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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 ...
- 767
1
vote
0
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375
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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 $(\...
- 51
3
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1
answer
190
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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?
- 673
16
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2
answers
758
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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 ...
- 169
5
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1
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830
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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?...
- 1,516
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0
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62
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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}\...
- 61
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 ...
- 2,095
4
votes
0
answers
83
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 ...
- 2,095
4
votes
1
answer
441
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 \...
- 85
14
votes
1
answer
1k
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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.
...
- 253
1
vote
1
answer
190
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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 ...
- 3,031
5
votes
1
answer
421
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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 ...
- 153
4
votes
2
answers
690
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 ...
- 16.2k
7
votes
1
answer
504
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 ...
- 181
5
votes
1
answer
162
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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 ...
- 6,741
1
vote
0
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74
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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.
...
- 1,227
0
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1
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263
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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)\...
- 119
0
votes
0
answers
116
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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 ...
user83520
4
votes
2
answers
424
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 ...
- 6,353
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 "...
- 589
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)...
- 6,051
10
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1
answer
359
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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.
- 183
13
votes
1
answer
323
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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?
...
- 56.9k
6
votes
3
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1k
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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 ...
- 3,065
4
votes
1
answer
208
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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^*...
- 14.4k