pseudo-Riemannian connection on $\mathbb{L}^3$

Question

I want know more about the pseudo-Riemannian connection in the Lorentz space $\mathbb{L}^3$. Does exist any relation between the usual Riemannian connection in $\mathbb{R}^3$ and the pseudo-Riemannian connection in $\mathbb{L}^3$? If existed, can you prove it for me?

Answer 1

All pseudo-Euclidean spaces $\Bbb R^n_\nu = (\Bbb R^n, \langle\cdot,\cdot\rangle_\nu)$, where $$\langle x,y\rangle_\nu = x_1y_1+\cdots+x_{n-\nu}y_{n-\nu} - x_{n-\nu+1}y_{n-\nu+1} -\cdots -x_ny_n,$$have the same Levi-Civita connection, characterized by having all Christoffel symbols vanish relative to the standard rectangular coordinates. This is trivial from the formula $$\Gamma_{ij}^k = \frac{1}{2}\sum_r g^{kr}(\partial_ig_{jr}+\partial_jg_{ir}-\partial_rg_{ij})$$with all components of the metric tensor being constant. If you're particularly interested in $3$-dim Lorentz-Minkowski space $\Bbb L^3$, this book might be useful.