Riemannian submanifold



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.