# Description of the difference tensor between a non-metric torsion-free connection and the Levi-Civita connection

A known result is the fact that a connection on a manifold $$M$$ is not a tensor, but the difference between any two is. I am interested in knowing a little more about that tensor in particular cases, for example if we take $$\nabla_{LC}$$ the Levi-Civita connection (the unique metric torsion-free connection on $$M$$) and $$\nabla$$ any other metric connection the difference between them is known as the contorsion tensor (https://en.wikipedia.org/wiki/Contorsion_tensor). What can be said about such a tensor if $$\nabla$$ is not metric, but it is torsion-free? Or more general, what can be said about the difference tensor of any two torsion-free connections (not necessarily metric ones)?

Thanks.

• You question is not quite clear. what properties you looking for? Commented Feb 16, 2022 at 15:44
• For example, some description in terms of Christoffel symbols like in the case of the contorsion tensor. I just want to know if the torsion-free condition help us to say something relevant about the tensor. Sorry. Commented Feb 16, 2022 at 16:08

Let $$\nabla$$ and $$\nabla'$$ be two affine connections on the same manifold. As you point out, there is a $$(1,2)$$ tensor $$T$$ such that $$\nabla'_XY=\nabla_XY+T(X,Y)$$. Direct computation shows that their respective torsion tensors $$\tau,\tau'$$ are related by $$\tau'(X,Y)=\tau(X,Y)+T(X,Y)-T(Y,X)$$ Thus, if $$\nabla$$ is torsion free, $$\nabla'$$ is also torsion free iff $$T$$ is symmetric (in its covariant components).