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—what does torsion measure intuitively?



What is torsion in differential geometry intuitively?
The torsion is a notoriously slippery concept. Personally I think the best way to understand it is to generalize past the place people first learn about torsion, which is usually in the context of Riemannian manifolds. Then you can see that the torsion can be understood as a sort of obstruction to integrability. Let me explain a little bit first. 


The torsion really makes sense in the context of general [G-structures](http://en.wikipedia.org/wiki/G-structure). Here $G \subseteq GL\_n(\mathbb{R}) = GL(V)$ is some fixed Lie group. Typical examples are $G = O(n)$ and $G = GL\_n(\mathbb{C})$. We'll see that these will correspond to Riemannian metrics and complex structures respectively. Now given this data, we have an exact sequence of vector spaces,


$$0 \to K \to \mathfrak{g} \otimes V^\ast \stackrel{\sigma}{\to} V \otimes\wedge^2 V^\ast \to C \to 0 $$ 


Here $\sigma$ is the inclusion $\mathfrak{g} \subseteq V \otimes V^\ast$ together with anti-symmetrization. K and C are the kernel and cokernel of $\sigma$. 


If we are given a manifold with $G$-structure, we then get four associated bundles, which fit into an exact sequence:


$$ 0 \to \rho\_1P \to ad(P) \otimes T^\*M \to \rho\_3P \to \rho\_4P \to 0$$ 


Now the difference of two connections which are both compatible with the G-structure is a tensor which is a section of the second space $\rho\_2P = ad(P) \otimes T^\*M$. This means that we can write any connection as $$\nabla + A$$
where $A$ is a section of $\rho\_2(P)$. 


Now the torsion of any G-compatible connection is a section of this third space. Suppose that we have two compatible connections. Then their torsions are sections of this third space. However since we can write the connections as $\nabla$ and $\nabla + A$, the torsion differ by $\sigma(A)$. Thus they have the same image in the fourth space $\rho\_4(P)$. 


The section of this fourth space is the *intrinsic torsion* of the G-structure. It measures the failure of our ability to find a torsion free connection. If this obstruction vanishes, then the torsion free connections form a torsor over sections of the smaller bundle $\rho\_1P$. Now some examples:


1. $G = O(n)$. This is the case of a Riemannian structure. In this case $\sigma$ is an isomorphism so that the there is always a unique torsion free connection. The Levi-Civita connection.
2. $G = GL\_m(\mathbb{C})$. This is the case of a complex structure. More precisely a $GL\_m(\mathbb{C})$-structure is the same as an almost complex structure. In this case the intrinsic torsion can be identified with the Nijenhuis tensor. So it vanishes precisely when the almost-complex structure is integrable (i.e. a ordinary complex structure).
3. $G = Sp(n)$. Having an $Sp(n)$-structure on a manifold for which the intrinsic torsion vanishes is equivalent to having a symplectic manifold.


From these examples you can see that the vanishing of torsion can be viewed as a sort of integrability condition. In these latter two cases the space of torsion free connections consists of more then a single point. There are many such connections. That's one reason why we don't see them popping up more often. 


