On page 79 of David Bleecker's book Gauge Theory and Variational Principles, the author states a theorem:
6.2.5 Theorem. There is a unique connection $\theta$ (called the Levi-Civita connection) on $F(M)$ with vanishing torsion form (i.e. $D^{\theta}\phi=0$).
However, the whole book does not mention the concept of metric-compatibility at all, which I believe is a defining property of Levi-Civita connection. So my question is: how is the metric-compatibility of this unique connection $\theta$ encoded?
In fact, I think this theorem is wrong, and the correct one should be:
My Claim. For any affine connection $\theta$ on $F(M)$, there is a unique connection $\sigma_\theta$ on $F(M)$ such that $\theta+\sigma_\theta$ is torsion-free. For different $\theta$, $\theta+\sigma_\theta$ may differ. Thus, on any manifold there may be several torsion-free affine connections. However, if the manifold is pseudo-Riemannian and we further require $\theta$ to be compatible with the metric, there is only a unique one left among those torsion-free affine connections.
Following is my argument. First of all, the proof of Theorem 6.2.5 in the book seems to prove only my statement, i.e. for any given $\omega$ there is a unique $\sigma_\omega$ such that $\omega+\sigma_\omega$ is torsion-free, but not prove that for different choice of $\omega$ those two are the same.
Furthermore, if, as stated in the original Theorem, there is a globally unique torsion-free connection, in the affine sense, regardless of metric, there is no way to guarantee that this connection is metric-compatible when the manifold is equipped with a Riemainnian metric, contradicting the fundamental theorem of pseudo-Riemannian geometry.
Following are some searches I have done. I found an entry named Metric connection on wiki. It has a section Riemannian connection in which it is stated that:
For every Riemannian connection, one may write a (unique) corresponding Levi-Civita connection. The difference between the two is given by the contorsion tensor.
Then I goto the Contorsion tensor wiki page. It has a section Affine geometry, which shows
given an Ehresmann connection $\omega$, there is another connection $\omega+\sigma_\omega$ that is torsion-free.
although this section uses Bleecker's book as a reference and the proof is almost exactly the same as Theorem 6.2.5 mentioned above. I believe the editor has modified the conclusion of that theorem.
Am I right?