Today I will discuss the torsion tensor of a Koszul connection. It measures the deviation from being symmetric in a sense defined below.
Given a Koszul connection on the smooth manifold , define the torsion tensor by
It can be checked that is a tensor, using the identities for and those for the Lie bracket. For instance, here is the proof that for a smooth .
In view of the connection identities and , we get that this equals
When , we call symmetric. There is a useful motivation for this terminology: when we are in an open subset of , the condition is equivalent to for all , or equivalently
where the are the Christoffel symbols.
Vector fields along a surface
There is a useful consequence of symmetry that I will now discuss. If where is open, then we can call a surface. We can define a vector field along in the same way as a vector field along a curve—it is a smooth map with projecting down to .
One way to get a vector field along is to consider the partial derivatives . Equivalently, these are obtained by .
Given a vector field along , we can consider the “partial covariant derivatives” defined respectively at as the covariant derivative of the vector field along a curve or . These are also vector fields along . It can be checked that these operations preserve smoothness.
Another way to think of this is , and similarly for . There is a slight abuse of notation because is only a vector field along , but at least when is an immersion this is ok because you can locally extend vector fields along .
Here is a useful consequence of symmetry.
Proposition 1Let be a surface in , and let be a symmetric connection on . Then
So symmetry is just a version of Clairaut’s theorem.
One way to prove this is to compute in a coordinate system.
The following argument is incorrect. I will post a corrected proof later today.
Another, more conceptual way, to argue is as follows. Given an embedding , we can obtain a connection on by defining
Of course, there is some checking to be done here. aren’t really vector fields, but one can check (as with the definition of the covariant derivative) that it doesn’t matter, because you just extend them locally to vector fields. Then if are -related to , it follows that
Consider an immersion for open. Then
Now is symmetric, so and this turns into
We’ve now handled the case of an immersion. Now consider the general case. Fix an arbitrary with , and consider the two quantities in (1) at . If we choose an immersion where is very small (possible if , which we can assume), then is an immersion at if is very small. Thus
at , at least. Letting , the two quantities are equal.
So what’s next? First, I ought to do something like this for the curvature tensor, though it’ll go a bit more quickly since the pattern is similar. Then, I should talk about the Levi-Civita connection associated to a Riemannian metric and how geodesics turn out to be locally distance-minimizing paths. This will provide ample material for a couple of more posts, and then I have other differential geometry topics in mind.