I calculated some properties of the Levi-Civita Connection on a semi-riemannian manifold. But I'm not sure, whether my results are correct. Can you please tell me, when something below is wrong or if everything is right?:
1) The Levi-Civita Connection on $\mathbb{R}^n$ furnished with a covariant metric tensor of type (0,2) of this form: $g(X,Y):=-X^1Y^1-...-X^jY^j+X^{j+1}Y^{j+1}+...+X^nY^n$. can be written as:
$\nabla_X Y(p):=dY_p(X_p)$. I checked all the properties of Levi-Civita connection. Since the Connection is unique, this should be the Levi-Civita-Connection!
2) Within my proof I recognized that this Connection is always symmetric, i.e. $\nabla_X Y -\nabla_Y X=[X,Y]$.
Regards