Riemannian manifold
Products

Let (M, g) {\displaystyle (M,g)} and (N, h) {\displaystyle (N,h)} be two Riemannian manifolds, and consider the product manifold M × N {\displaystyle M\times N}. The Riemannian metrics g {\displaystyle g} and h {\displaystyle h} naturally put a Riemannian metric g ~ {\displaystyle {\widetilde {g}}} on M × N, {\displaystyle M\times N,} which can be described in a few ways.

For example, consider the n {\displaystyle n} -torus T n = S 1 × ⋯ × S 1 {\displaystyle T^{n}=S^{1}\times \cdots \times S^{1}}. If each copy of S 1 {\displaystyle S^{1}} is given the round metric, the product Riemannian manifold T n {\displaystyle T^{n}} is called the flat torus. As another example, the Riemannian product R × ⋯ × R {\displaystyle \mathbb {R} \times \cdots \times \mathbb {R} }, where each copy of R {\displaystyle \mathbb {R} } has the Euclidean metric, is isometric to R n {\displaystyle \mathbb {R} ^{n}} with the Euclidean metric.
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