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?
2026-03-25 17:26:18.1774459578
Semi-Riemannian Manifold
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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!