Riemannian submanifold

of a Riemannian manifold

equipped with the Riemannian metric inherited from

is a Riemannian manifold (with or without boundary) and

is an immersed submanifold or an embedded submanifold (with or without boundary), the pullback

is a Riemannian metric on

already has a Riemannian metric

, then the immersion (or embedding)

is called an isometric immersion (or isometric embedding) if

Hence isometric immersions and isometric embeddings are Riemannian submanifolds.

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.

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