Questions tagged [torsion]

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What is torsion in differential geometry intuitively?

Hi, given a connection on the tangent space of a manifold, one can define its torsion: $$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$ What is the geometric picture behind this definition&...
Jan Weidner's user avatar
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27 votes
4 answers
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Rolling without slipping interpretation of torsion

This is, in a sense, a follow up to this question. Hehl and Obukhov try to give an intuitive description of torsion. I am confused about their description. I am looking at the following paragraph on ...
David E Speyer's user avatar
23 votes
4 answers
4k views

Why is it important that partial derivatives commute?

I am asking this in the context of differential geometry (specifically Riemannian). When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates ...
R S's user avatar
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21 votes
2 answers
4k views

Torsion and parallel transport

There's a close relationship between curvature and the holonomy group; the holonomy theorem of Ambrose and Singer, for example. It seems to me that there should be an analogous result for torsion. I ...
Oliver Jones's user avatar
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15 votes
1 answer
422 views

Can a torsion-free group be quasi-isometric to a torsion group?

I have looked around in the literature on group theory and geometric group theory and this looks to be an open question as far as I can tell (by torsion group, I mean as usual a group in which every ...
Hollis Williams's user avatar
8 votes
1 answer
244 views

Is there an elliptic curve over a number field with a point of order 64 and Mordell-Weil rank zero?

It seems to me that there ought to be elliptic curves over number fields with arbitrarily large torsion subgroups but Mordell-Weil rank zero. But I'll settle for a point of order 64. Does anyone ...
David McKinnon's user avatar
7 votes
1 answer
347 views

Torsion-free Cartan connections

Let $M$ a differentiable manifold and $H\subset G$ a Lie group with a closed subgroup such that $G/H$ is connected. A $H\subset G$-Cartan connection on $M$ can be defined by A principal $G$-bundle on ...
jpdm's user avatar
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6 votes
3 answers
923 views

Torsion-free tensor powers

Let $R$ be an integral domain. If $M$ is an $R$-module such that every tensor power of $M$ over $R$ is $R$-torsion-free, then is $M$ necessarily flat as an $R$-module? If not, then does this ...
Jesse Elliott's user avatar
6 votes
3 answers
968 views

Why torsion is only defined for linear connection on TM?

The concept of curvature is defined for any linear connection on any vector bundle $E \to M$, but the concept of torsion is only defined for connection on the tangent bundle $TM$ of a manifold $M^n$, ...
ychemama's user avatar
  • 1,326
6 votes
2 answers
346 views

The action of a subgroup of the torsion group of elliptic curves on integral points?

Let $E$ be an elliptic curve given in long Weierstraß form with all coefficients $a_1,a_2,a_3,a_4,a_6 \in \mathbb{Z}$. It is known that the rational points $E(\mathbb{Q})$ form a group which has a ...
user avatar
6 votes
1 answer
222 views

Cohomology of toric blowup

Let $n\geq2$. Let $G$ be a linear automorphisms group of prime order on $\mathbb{C}^n$. We assume that $0$ is the unique fixed point of $G$. I consider the quotient $\mathbb{C}^n/G$. It is a toric ...
Kovalevskaia's user avatar
5 votes
2 answers
1k views

Torsion and Non-metricity Tensor on a Surface

In differential geometry of surfaces, how can one define a non-zero Torsion tensor? It seems that the connection you provide has always to be symmetric since, by definition, $$\Gamma^{\gamma}_{\alpha\...
Ayan's user avatar
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5 votes
1 answer
572 views

First Chern class of torsion sheaves

Let $X$ be a smooth projective variety, $\mathscr T$ a torsion sheaf with irreducible support of codimension $1$, say $Z$. Then the first Chern class $c_1(\mathscr T)$ is of form $r[Z]$. Is there ...
Makimura's user avatar
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5 votes
1 answer
215 views

Commutative/ symmetric second covariant derivative

Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$. Is it possible to have an affine connection, possibly with non-zero ...
Khaled T.'s user avatar
5 votes
0 answers
193 views

Torsion-free sheaf cohomology over discrete valuation rings

Let $R$ be a Henselian discrete valuation rings with algebraically closed residue field and $X$ be a regular, flat, proper $R$-scheme. Assume that the generic fiber to the natural morphism from $X$ to ...
Ron's user avatar
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4 votes
3 answers
577 views

Possible $p$-torsion subgroup of $E(\mathbb{Q}_p)$, and if there is a theorem to say which case happens when?

What is the possible $p$-torsion subgroup of $E(\mathbb{Q}_p)$ for an elliptic curve $E$ over $\mathbb{Q}_p$, and if there is a theorem to say which case happens when?
Dipendra Prasad's user avatar
3 votes
2 answers
2k views

Interpretation of Curvature and Torsion

Dear all, When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields $[\nabla_\mu,\nabla_\nu]V^\rho = R_{\...
Dox's user avatar
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3 votes
1 answer
198 views

Torsion submodules of non-noetherian modules

Let $R$ be a commutative ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. The $\mathfrak{a}$-torsion submodule of $M$ is defined as $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\...
Fred Rohrer's user avatar
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3 votes
0 answers
273 views

Torsion-free subgroups in arbitrary lattices

Let $\Gamma $ be a lattice in a semi-simple Lie group $G$. If the Lie group $G$ is linear (that is, it has a faithful finite-dimensional linear representation), then $\Gamma$ contains a torsion-free ...
Vladimir47 's user avatar
2 votes
2 answers
516 views

Why is $M$ torsion-free?

I am studying the following article https://www.math.nagoya-u.ac.jp/~takahashi/tc9.pdf The main theorem is the Theorem 3.3. Howewer, I have the following questions about the proof: How does it help ...
Danimenru's user avatar
2 votes
1 answer
510 views

Torsion theory for quasi-coherent sheaves?

In a category $\mathcal C$, we will say that $(\mathcal T,\mathcal F)$ is a torsion theory if it satisfies: (1) $Hom(T,F)=0$ for all $T\in \mathcal T$ and $F\in \mathcal F$. (2) If $Hom(T,F)=0$ for ...
stupidq75's user avatar
2 votes
0 answers
87 views

Poincaré connection encode torsion and curvature

I'm trying to understand something that is written in Baez & Wise paper "Teleparallel Gravity as a Higher Gauge Theory". In section 4, they discuss Poincaré connection and, first of all, split ...
BVquantization's user avatar
2 votes
0 answers
372 views

Alternate definition for the torsion tensor

I would be pleased to have some information about an alternate definition for the torsion tensor. Let us consider a smooth manifold $\mathcal{M}$ together with an arbitrary connection $\nabla$. The ...
user avatar
2 votes
0 answers
246 views

Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$

Hi, overflowers. I have a question concerning the torsion of elliptic curves over number fields. Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can ...
Chema Tornero's user avatar
1 vote
0 answers
65 views

Bertini type result for torsion-freeness

Let $R$ be a local, regular $\mathbb{C}$-algebra and $\mathfrak{m}$ be the maximal ideal. Let $M$ be a finitely generated torsion-free $R$-module. Suppose there exists $f \in \mathfrak{m}$ such that $...
Chen's user avatar
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1 vote
0 answers
121 views

Base change of cohomology when the cohomology is a torsion

Let $(R,m,k)$ be a discrete valuation ring, where $k=R/m$. Let $X\rightarrow \mathrm{Spec}R$ be a projective, integral and flat $R$-scheme. Let $\mathscr F$ be a coherent sheaf such that $H^i(X,\...
Pickle Liobe's user avatar
1 vote
0 answers
47 views

Torsion order on Prym variety

Consider two hyperlliptic curves $C_1,C_2$ over $\mathbb{Q}$, and a morphism $\phi:C_1 \rightarrow C_2$. Lifting this morphism on the Jacobians of $C_1,C_2$ and taking its kernel defines a Prym ...
T. Combot's user avatar
  • 231
1 vote
0 answers
92 views

Torsion functors and weak assassins

Let $R$ be a commutative ring, and let $\mathfrak{a}\subseteq R$ be an ideal. For an $R$-module $M$, we set $\Gamma_{\mathfrak{a}}(M)=\{x\in M\mid\exists n\in\mathbb{N}:\mathfrak{a}^nx=0\}$, and we ...
Fred Rohrer's user avatar
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1 vote
0 answers
400 views

connection between non-orientable manifolds and torsion in 1D (co) homology

I'm interested in understanding the probability that given a prime $p$, $p$ divides the order of the torsional part of $H^k(X,Z)$, where $X$ is a finite simplicial complex. Lets say you have a ...
rhl's user avatar
  • 133
0 votes
1 answer
376 views

Torsion and submanifolds [closed]

EDIT: Let me modify the question then: for what submanifolds $N$ does the torsion $T$ preserve tangent vectors to $N$? If $\nabla$ is a connection on a manifold $M$, then torsion is defined to be the ...
Oliver Jones's user avatar
  • 1,368
0 votes
0 answers
337 views

What is the physical meaning of torsion

The torsion tensor in 4 dimensions $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-...
Eris's user avatar
  • 1
0 votes
0 answers
215 views

Torsion in cohomology

Suppose to have a short exact sequence of chain complexes of $\mathbb{Z}$-modules: $$0\to A^\bullet\to B^\bullet\to C^\bullet\to 0$$ such that $A^k,B^k,C^k$ are non zero for $k=0,1,2$. Moreover, ...
user53075's user avatar