Smooth Local Coordinates for Product Manifold

Theorem

Let $M_1, M_2$ be Riemannian manifolds.

Let $\tuple {x^1, \dots, x^n}$ be the smooth local coordinates for $M_1$.

Let $\tuple {x^{n + 1}, \dots, x^{n + m}}$ be the smooth local coordinates for $M_1$.

Let $M_1 \times M_2$ be the product manifold.

Then the smooth local coordinates for $M_1 \times M_2$ can be written as $\tuple {x^1, \dots, x^{n + m}}$.

Proof

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Sources

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