Riemannian submanifold explained
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] Sn=\{x\inRn+1:\lVertx\rVert=1\}
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.
Notes and References
- Book: Lee, John. Introduction to Riemannian Manifolds. 2018. 2nd.
- Book: Chen, Bang-Yen. Geometry of Submanifolds. 1973. Mercel Dekker. New York. 0-8247-6075-1. 298.
