[Math] Product of Riemannian manifold and product metric

Question

according to wikipedia the product metric between 2 metrics is the metric given by:
$d(x,y)=\sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}$

Now if $(M,g_m)$ and $(N,g_n)$ are 2 Riemannian manifolds we can construct the product $M\times N$ equipped with the riemannian metric $g_m+g_n$.

Is there a link between the "product metric" and the natural metric on $M\times N$ or is it two different things ?

Thanks

Answer 1

Yes. If you endowed $M = \mathbb{R}^1$ as a manifold with the usual metric $g = g_{ij}dx^i \ dx^j = 1.dx\ dx = dx^2$, then $M \times M$ has induced metric

$$ds^2 = dx^2 + dy^2$$ which gives Euclidean distance.

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Relatedly all Riemannian metrics induced by Euclidean metrics.