Editing Riemannian manifold (section)

=== 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}}