Riemannian submanifold

Not to be confused with Sub-Riemannian manifold.

The sphere $S^{n}$ with the round metric is a Riemannian submanifold of $\mathbb {R} ^{n+1}$.

A Riemannian submanifold $N$ of a Riemannian manifold $M$ is a submanifold $N$ of $M$ equipped with the Riemannian metric inherited from $M$.

Specifically, if $(M,g)$ is a Riemannian manifold (with or without boundary) and $i:N\to M$ is an immersed submanifold or an embedded submanifold (with or without boundary), the pullback $i^{*}g$ of $g$ is a Riemannian metric on $N$, and $(N,i^{*}g)$ is said to be a Riemannian submanifold of $(M,g)$. On the other hand, if $N$ already has a Riemannian metric ${\tilde {g}}$, then the immersion (or embedding) $i:N\to M$ is called an isometric immersion (or isometric embedding) if ${\tilde {g}}=i^{*}g$. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.

For example, the n-sphere $S^{n}=\{x\in \mathbb {R} ^{n+1}:\lVert x\rVert =1\}$ is an embedded Riemannian submanifold of $\mathbb {R} ^{n+1}$ via the inclusion map $S^{n}\hookrightarrow \mathbb {R} ^{n+1}$ that takes a point in $S^{n}$ to the corresponding point in the superset $\mathbb {R} ^{n+1}$. The induced metric on $S^{n}$ is called the round metric.