I don't understand the principal idea of Semi-Riemannian Manifold. Why is that if I have a metric tensor $g$ on a smooth manifold $M$ that is a symmetric nondegenerate $(0, 2)$-tensor field on $M$ of constant index (metric tensor), I can define a scalar product on every tangent space?
This is going to be a generalized scalar product $g(X,Y)$ that may take negative values. This kind of thing is useful in relativity theory, for example. But it is not a familiar Euclidean scalar product!