The torsion tag has no wiki summary.

**2**

votes

**0**answers

142 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 ...

**0**

votes

**1**answer

292 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 ...

**1**

vote

**2**answers

292 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,
...

**8**

votes

**2**answers

1k 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 ...

**1**

vote

**0**answers

131 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 ...

**0**

votes

**0**answers

199 views

### About the parallel transport and choice of connection

Thought Experiment
Consider a 2-sphere, $S^2$, and let $p$ be a point at the equator.
Case 1
Let us parallel transport a vector, $V$ from $p$ using the recipe:
Move one unit of length East.
Move ...

**17**

votes

**4**answers

1k 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 ...

**3**

votes

**2**answers

940 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 = ...

**6**

votes

**3**answers

580 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 ...

**1**

vote

**0**answers

259 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 ...

**18**

votes

**4**answers

2k views

### 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 ...

**79**

votes

**15**answers

9k views

### 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 ...