Riemannian submanifold

A Riemannian submanifold of a Riemannian manifold is a submanifold of equipped with the Riemannian metric inherited from.
Specifically, if is a Riemannian manifold and is an immersed submanifold or an embedded submanifold, the pullback of is a Riemannian metric on, and is said to be a Riemannian submanifold of. On the other hand, if already has a Riemannian metric, then the immersion is called an isometric immersion if. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.
For example, the n-sphere is an embedded Riemannian submanifold of via the inclusion map that takes a point in to the corresponding point in the superset. The induced metric on is called the round metric.