Riemannian submanifold
A Riemannian submanifold <math>N</math> of a Riemannian manifold <math>M</math> is a submanifold <math>N</math> of <math>M</math> equipped with the Riemannian metric inherited from <math>M</math>.
Specifically, if <math>(M,g)</math> is a Riemannian manifold (with or without boundary) and <math>i : N \to M</math> is an immersed submanifold or an embedded submanifold (with or without boundary), the 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>. 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.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>
For example, the n-sphere <math>S^n = \{ x \in \mathbb R^{n+1} : \lVert x \rVert = 1 \}</math> is an embedded Riemannian submanifold of <math>\mathbb R^{n+1}</math> via the inclusion map <math>S^n \hookrightarrow \mathbb R^{n+1}</math> that takes a point in <math>S^n</math> to the corresponding point in the superset <math>\mathbb R^{n+1}</math>. The induced metric on <math>S^n</math> is called the round metric.