Signature of a semi-Riemannian manifold

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

d i a g (1, \dots, 1 ⏟ 𝑠, - 1, \dots, - 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. #m/def/geo/diff

#state/tidy | #lang/en | #SemBr