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Riemannian submanifold

From Wikipedia, the free encyclopedia
The sphere with the round metric is a Riemannian submanifold of .

A Riemannian submanifold of a Riemannian manifold is a submanifold of equipped with the Riemannian metric inherited from .

Specifically, if is a Riemannian manifold (with or without boundary) and is an immersed submanifold or an embedded submanifold (with or without boundary), 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 (or embedding) is called an isometric immersion (or isometric embedding) if . Hence isometric immersions and isometric embeddings are Riemannian submanifolds.[1][2]

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.

References

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  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.