Riemannian submanifold

Not to be confused with Sub-Riemannian manifold.

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

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

Specifically, if ${\displaystyle (M,g)}$ is a Riemannian manifold (with or without boundary) and ${\displaystyle i:N\to M}$ is an immersed submanifold or an embedded submanifold (with or without boundary), the pullback ${\displaystyle i^{*}g}$ of ${\displaystyle g}$ is a Riemannian metric on ${\displaystyle N}$, and ${\displaystyle (N,i^{*}g)}$ is said to be a Riemannian submanifold of ${\displaystyle (M,g)}$. On the other hand, if ${\displaystyle N}$ already has a Riemannian metric ${\displaystyle {\tilde {g}}}$, then the immersion (or embedding) ${\displaystyle i:N\to M}$ is called an isometric immersion (or isometric embedding) if ${\displaystyle {\tilde {g}}=i^{*}g}$. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.^[1][2]

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

References

1. Lee, John (2018). Introduction to Riemannian Manifolds (2nd ed.).
2. Chen, Bang-Yen (1973). Geometry of Submanifolds. New York: Mercel Dekker. p. 298. ISBN 0-8247-6075-1.