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.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics