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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.

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  • $\begingroup$ You question is not quite clear. what properties you looking for? $\endgroup$
    – C.F.G
    Feb 16, 2022 at 15:44
  • $\begingroup$ 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. $\endgroup$
    – Santiago
    Feb 16, 2022 at 16:08

1 Answer 1

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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).

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