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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. $\nabla_X Y -\nabla_Y X=[X,Y]$.


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Try consulting Barrett O'Neill's Semi-Riemannian Geometry. – Neal Apr 16 '12 at 18:08
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It seems like you are considering singular semi-Euclidean space. See Example $2.3$ in the following link to the arXiv.

On Singular Semi-Riemannian Manifolds

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