Is torsion a property of a manifold or of a connection?

Question

In the standard presentation of general relativity, one adopts the Levi-Civita connection and the Christophel symbols; one has $\Gamma^a_{bc} = \Gamma^a_{cb}$ and the torsion tensor is zero.

But of course one could equally adopt some other connection and get a different set of connection coefficients, and then the torsion tensor need not be zero. (Einstein-Cartan and teleparallel approaches do this, for example). So this makes it seem that torsion is not a property of a manifold, it is a property of the way one chooses to relate different tangent spaces to one another.

However it is often stated that such-and-such a manifold "is torsionless", or that spacetime is assumed to be torsionless in general relativity. Also, one can give geometric pictures of manifolds with torsion, such as one where one considers the continuous limit of a crystal with the appropriate type of dislocation. So this makes it seem that the torsion is there "in the manifold", as it were.

Which is right? Both? Neither?

To be precise,

(i) is there a (reasonably sensible and non-pathological) manifold which has to have torsion no matter what connection is adopted? Or is that question meaningless because torsion is all about connection and manifold together?

And (just to check), I think it is the case that as soon as we have a metric then we also have the possibility of adopting the Levi-Civita connection.

(ii) Does that imply that approaches to gravity in which there is torsion must either be abandoning the concept of a spacetime metric, or else they are treating an interaction which could equally well be treated by asserting that spacetime is torsionless and they are proposing a new field which couples to spin and mass in some sort of universal way that can be captured through the use of a suitably defined connection? (When I look at Einstein-Cartan theory, I see something called a metric popping up, so it clearly has not been abandoned altogether.)

Added note. I edited the above in order to highlight the two more precise questions.

Answer 1

The torsion is indeed defined from the connection, independently of the manifold and metric. By synedoche, people sometimes refer to the structure composed of the manifold, the differential structure, the metric and the connection, $(M, \mathfrak{A}, g, \nabla)$, as "the manifold", even though those are somewhat independent objects.

Given a reasonable manifold, it is always possible to find a torsionless connection on it, since every (metrizable) manifold admits a metric tensor, and every manifold with a metric admits a Levi-Civita connection. As a general rule, the difference between two connections is

\begin{equation} \nabla_a \omega_b = \tilde{\nabla}_a \omega_b - {C^c}_{ab} \omega_c \end{equation}

If we have a connection with a torsion tensor ${T^c}_{ab}$, we can in particular define $C$ to be the torsion tensor, so that any connection with torsion can give rise to a torsionless connection.

Similarly, if you have a torsion-free connection, and you add a tensor field ${C^c}_{ab}$ that is not symmetric in $a$ and $b$ (you can always do this by picking the zero tensor plus a non-zero antisymmetric tensor in a small neighbourhood), then this will give rise to a connection with torsion.

Answer 2

Formally what you do is the following. Spacetime is a set of data $(M,\mathcal{O},\mathscr{A},\nabla,g)$ where $(M,\mathcal{O},\mathscr{A},g)$ is a smooth Lorentzian manifold and $\nabla$ is a connection. Let $G$ be some Lie group, the idea is to consider a principal $G$-bundle $(P,\pi,M)$ with associated fibre bundle $P_V$ with $V$ the representation space of $G$. The idea is then to look at so called solder forms, which are elements $\theta \in \Omega^1(P) \otimes C^\infty(P,V)$ under certain conditions. A torsion is defined as $\Theta = D\theta$ where $(D\phi)(X_1,...,X_{k+1}):= (d\phi)(\mathrm{Hor}(X_1),...,\mathrm{Hor}(X_{k+1}))$ for $X_1,...,X_{k+1} \in \Gamma(TP)$ ($\mathrm{Hor}$ is the horizontal part of the vector(field)) where $\phi \in \Omega^k(P,V)$. In order words you choose some underlying Lie group, consider spacetime as the underlying base manifold of some principal $G$-bundle. The meaning is that in this way you let the vectors transform under e.g. Lorentz transformations $O(3,1)$ at each tangent space. Thus I would say that torsion depends on the choice of the Lie group and the underlying manifold or spacetime.

Answer 3

Depends what kind of manifolds you're dealing with. I think the reason for much confusion is fundamentally a flawed foundational understanding of what manifolds are, meaning what their defining properties should be. That being said, As long as you have a metric you can define a metric compatible connection. If you require that connection to also be torsion free that it is called Levi civita and it is then uniquely defined. Torsion free just means that the commutator of vector fields [a, b] is equal to ∇a(b) - ∇b(a).