The product riemannian metric

Given: Finite-dimensional smooth Riemannian manifolds $(M,g),(N,h)$, with their product smooth structure.

Coordinate criterion for a riemannian metric: A tensor $g={\sum }_{i,j}{g}_{ij}d{x}^i\otimes d{x}^j$ is Riemannian exactly when its coordinate matrix $G=(g_{ij})$ has smooth entries and is symmetric positive definite. Under $J=\mathrm{\partial }x\mathrm{/}\mathrm{\partial }y$ it transforms by $G_y=J^T{G}_{x}J$.

Products of smooth manifolds have a canonical product smooth structure: Let $(M,\mathcal{S})$ and $(N,\mathcal{T})$ be smooth manifolds of dimensions $m$ and $n$. Then $M\times N$ with the product topology is a topological $(m+n)$-manifold. If $\mathcal{A}$ and $\mathcal{B}$ are smooth atlases with $[\mathcal{A}]=\mathcal{S}$ and $[\mathcal{B}]=\mathcal{T}$, then the set of product charts $\mathcal{A}\times \mathcal{B}:=\left\{\right(V\times W,\phi \times \psi ):(V,\phi )\in \mathcal{A},(W,\psi )\in \mathcal{B}\}$ is a smooth atlas on $M\times N$, and the maximal atlas it generates is independent of the presenting atlases: it depends only on $\mathcal{S}$ and $\mathcal{T}$. This maximal atlas is the product smooth structure of $M\times N$.