Product Manifold of Pseudo-Riemannian Manifolds is Pseudo-Riemannian Manifold

Theorem
Let $\struct {M_1, g_1}$ and $\struct {M_2, g_2}$ be pseudo-Riemannian manifolds of signatures $\tuple {r_1, s_1}$ and $\tuple {r_2, s_2}$ respectively.

Then $\struct {M_1 \times M_2, g_1 \oplus g_2}$ is a pseudo-Riemannian manifold of signature $\tuple {r_1 + r_2, s_1 + s_2}$, where $\times$ denotes the cartesian product and $\oplus$ stands for the direct sum.