Riemannian submanifold

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

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

References

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