# Tagged Questions

**2**

votes

**0**answers

126 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 ...

**1**

vote

**2**answers

254 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,
...

**0**

votes

**0**answers

186 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

865 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 = ...

**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 ...

**75**

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 ...