Semi-Riemannian manifold

Signature of a semi-Riemannian manifold

Let (𝑀,𝒜,𝑔) be a Semi-Riemannian manifold. At any point, 𝑔𝑎𝑏 is congruent to a matrix of the form

diag⁡(1,…,1⏟𝑠,−1,…,−1⏟__⏟__⏟𝑡)

where by Sylvester’s law of inertia the quantity (𝑠,𝑡) is uniquely determined continuous function of points on the manifold. Thus if 𝑀 is connected, we have a uniquely determined signature (𝑠,𝑡) for the entire manifold. diff

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