## Tagged Questions

0answers
148 views

### Torsion product Tor^R_1(,) [closed]

We know that: Torsion product Tor$^R_1(,)$, which maps two modules, $M_1$ over $R$ and $M_2$ over $R$, to a third module $M_3$ over $R$: $M_3$ = Tor$^R_1(M_1,M_2)$. See for examp …
4answers
899 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 l …
0answers
99 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 o …
13answers
6k 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 behin …
2answers
478 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,\nab …
4answers
1k 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 fol …
3answers
462 views

### Torsion-free tensor powers

Does there exist an integral domain $R$ and an $R$-module $M$ that is not flat over $R$ such that every tensor power of $M$ over $R$ is nonzero and $R$-torsion-free? (If such a do …
0answers
221 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 s …
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