Editing Riemannian manifold (section)

=== Products ===
{{multiple image | image1=Grid for torus.svg  |  image2=Flat torus stereographic.svg | alt1=A 2x2 square grid | alt2=A torus embedded in Euclidean space | footer=A [[torus]] naturally carries a Euclidean metric, obtained by identifying opposite sides of a square (left). The resulting Riemannian manifold, called a [[flat torus]], cannot be isometrically embedded in 3-dimensional Euclidean space (right), because it is necessary to bend and stretch the sheet in doing so. Thus the intrinsic geometry of a flat torus is different from that of an embedded torus.}}

Let <math>(M,g)</math> and <math>(N,h)</math> be two Riemannian manifolds, and consider the [[Manifold#Cartesian_products|product manifold]] <math>M\times N</math>. The Riemannian metrics <math>g</math> and <math>h</math> naturally put a Riemannian metric <math>\widetilde{g}</math> on <math>M\times N,</math> which can be described in a few ways.
* Considering the decomposition <math>T_{(p,q)}(M\times N) \cong T_pM \oplus T_qN,</math> one may define
*: <math>\widetilde{g}_{p,q} ((u_1, u_2), (v_1, v_2)) = g_p(u_1, v_1) + h_q(u_2, v_2).</math>{{sfn|Lee|2018|p=20}}
* If <math>(U,x)</math> is a smooth coordinate chart on <math>M</math> and <math>(V,y)</math> is a smooth coordinate chart on <math>N</math>, then <math>(U \times V, (x,y))</math> is a smooth coordinate chart on <math>M \times N.</math> Let <math>g_U</math> be the representation of <math>g</math> in the chart <math>(U,x)</math> and let <math>h_V</math> be the representation of <math>h</math> in the chart <math>(V,y)</math>. The representation of <math>\widetilde{g}</math> in the coordinates <math>(U \times V,(x,y))</math> is
*:<math>\widetilde{g} = \sum_{ij} \widetilde{g}_{ij} \, dx^i \, dx^j</math> where <math>(\widetilde{g}_{ij}) = \begin{pmatrix} g_U & 0 \\ 0 & h_V \end{pmatrix}.</math>{{sfn|Lee|2018|p=20}}

For example, consider the [[N-torus|<math>n</math>-torus]] <math>T^n = S^1\times\cdots\times S^1</math>. If each copy of <math>S^1</math> is given the round metric, the product Riemannian manifold <math>T^n</math> is called the ''[[flat torus]]''. As another example, the Riemannian product <math>\mathbb R \times \cdots \times \mathbb R</math>, where each copy of <math>\mathbb R</math> has the Euclidean metric, is isometric to <math>\mathbb R^n</math> with the Euclidean metric.