Editing Riemannian manifold (section)

== Examples ==
=== Euclidean space ===

Let <math>x^1,\ldots,x^n</math> denote the standard coordinates on <math>\mathbb{R}^n.</math> The (canonical) ''Euclidean metric'' <math>g^\text{can}</math> is given by{{sfn|Lee|2018|pp=12–13}}
: <math>g^\text{can}\left(\sum_i a_i \frac{\partial}{\partial x^i}, \sum_j b_j \frac{\partial}{\partial x^j} \right) = \sum_i a_i b_i</math>
or equivalently
: <math>g^\text{can} = (dx^1)^2 + \cdots + (dx^n)^2</math>
or equivalently by its coordinate functions
: <math>g_{ij}^\text{can} = \delta_{ij}</math> where <math>\delta_{ij}</math> is the [[Kronecker delta]]
which together form the matrix
: <math>(g_{ij}^\text{can}) = \begin{pmatrix}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1
\end{pmatrix}.</math>
The Riemannian manifold <math>(\mathbb{R}^n,g^\text{can})</math> is called ''Euclidean space''.

=== Submanifolds ===
{{Main|Riemannian submanifold}}

[[File:Sphere filled blue.svg|thumb|The [[N-sphere|<math>n</math>-sphere]] <math>S^n</math> with the round metric is an embedded Riemannian submanifold of <math>\mathbb R^{n+1}</math>.]]

Let <math>(M,g)</math> be a Riemannian manifold and let <math>i : N \to M</math> be an [[immersed submanifold]] or an [[embedded submanifold]] of <math>M</math>. The [[Pullback (differential geometry)|pullback]] <math>i^*g</math> of <math>g</math> is a Riemannian metric on <math>N</math>, and <math>(N, i^*g)</math> is said to be a ''Riemannian submanifold'' of <math>(M,g)</math>.{{sfn|Lee|2018|p=15}}

In the case where <math>N \subseteq M</math>, the map <math>i : N \to M</math> is given by <math>i(x) = x</math> and the metric <math>i^*g</math> is just the restriction of <math>g</math> to vectors tangent along <math>N</math>. In general, the formula for <math>i^*g</math> is
: <math>i^*g_p(v,w) = g_{i(p)} \big( di_p(v), di_p(w) \big), </math>
where <math>di_p(v)</math> is the [[pushforward (differential)|pushforward]] of <math>v</math> by <math>i.</math>

Examples:
* The [[N-sphere|<math>n</math>-sphere]]
*: <math>S^n=\{x\in\mathbb{R}^{n+1}:(x^1)^2+\cdots+(x^{n+1})^2=1\}</math>
:is a smooth embedded submanifold of Euclidean space <math>\mathbb R^{n+1}</math>.{{sfn|Lee|2018|p=16}} The Riemannian metric this induces on <math>S^n</math> is called the ''round metric'' or ''standard metric''.
* Fix real numbers <math>a,b,c</math>. The [[ellipsoid]]
*:<math>\left\{(x,y,z) \in \mathbb R^3 : \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \right\}</math>
:is a smooth embedded submanifold of Euclidean space <math>\mathbb R^3</math>.
* The [[Graph_of_a_function|graph]] of a smooth function <math>f:\mathbb{R}^n\to\mathbb{R}</math> is a smooth embedded submanifold of <math>\mathbb{R}^{n+1}</math> with its standard metric.
* If <math>(M,g)</math> is not simply connected, there is a covering map <math>\widetilde{M}\to M</math>, where <math>\widetilde M</math> is the [[universal cover]] of <math>M</math>. This is an immersion (since it is locally a diffeomorphism), so <math>\widetilde M</math> automatically inherits a Riemannian metric. By the same principle, any [[smooth covering space]] of a Riemannian manifold inherits a Riemannian metric.

On the other hand, if <math>N</math> already has a Riemannian metric <math>\tilde g</math>, then the immersion (or embedding) <math>i : N \to M</math> is called an ''[[isometric immersion]]'' (or ''[[isometric embedding]]'') if <math>\tilde g = i^* g</math>. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.{{sfn|Lee|2018|p=15}}

=== 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.

=== Positive combinations of metrics ===
Let <math>g_1, \ldots, g_k</math> be Riemannian metrics on <math>M.</math> If <math>f_1, \ldots, f_k</math> are any positive smooth functions on <math>M</math>, then <math>f_1 g_1 + \ldots + f_k g_k</math> is another Riemannian metric on <math>M.</math>