Is the Levi-Civita connection on a Lorentzian manifold the same as that on a Riemannian manifold?

Question

The Levi-Civita connection is defined by the Koszul formula to be independent of the metric. This applies to both Riemannian and pseudo-Riemannian metrics. In the latter, this is known as the miracle of semi-Riemannian geometry (p60 of Semi-Riemannian geometry with applications to relativity O'Neil). This seems to conclusively prove that the Levi-Civita connection is the same in both geometries. Certainly, the connection coefficients are also defined the same. Am I misunderstanding anything with that conclusion?

Answer 1

The "miracle" only means that given a semi-Riemannian metric $g$ on a manifold $M$ there exists a unique torsion-free affine connection $\nabla=\nabla_g$ which preserves $g$. The word unique here means that if $\nabla'$ is another torsion-free affine connection which preserves $g$, then $\nabla=\nabla'$. The uniqueness here does not mean that if $h$ is another semi-Riemannian metric on $M$ then $\nabla_g=\nabla_h$. In fact, it is very seldom that two Riemannian metrics share the same connection. It is also seldom the case that if $g, h$ are semi-Riemannian metrics of different signature then $\nabla_g=\nabla_h$.

Edit: If you want an explicit formula for the L-C connection (in terms of the Christoffel symbols), you have $$\Gamma^l_{jk} = \tfrac{1}{2} g^{lr} \left( \partial _k g_{rj} + \partial _j g_{rk} - \partial _r g_{jk} \right).$$ This is equivalent to the Koszul formula, but easier to parse. The semi-Riemannian metric enters this formula in the most obvious way.