Local Expression for Metric of Product Riemannian Manifold

Theorem

Let $\struct {M_1, g_1}$ and $\struct {M_2, g_2}$ be Riemannian manifolds.

Let their smooth local coordinates be $\tuple {x^1, \dots, x^n}$ and $\tuple {x^{n + 1}, \dots, x^{n + m}}$ respectively.

Let $M_1 \times M_2$ be the product Riemannian manifold with smooth local coordinates $\tuple {x^1, \dots, x^{n + m}}$.

Then the product Riemannian metric $g = g_1 \oplus g_2$ has the local expression $g = g_{ij} dx^i dx^j$ where $\paren {g_{i j} }$ is the block diagonal matrix:

$\paren {g_{i j} } = \begin{pmatrix}

\paren {g_1}_{ab}  & 0 \\
0 & \paren {g_2}_{cd}

\end{pmatrix}$

and the indices $a,b$ run from $1$ to $n$, while $c,d$ run from $n + 1$ to $n + m$.

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