- Source: Riemannian submanifold
A Riemannian submanifold
N
{\displaystyle N}
of a Riemannian manifold
M
{\displaystyle M}
is a submanifold
N
{\displaystyle N}
of
M
{\displaystyle M}
equipped with the Riemannian metric inherited from
M
{\displaystyle M}
.
Specifically, if
(
M
,
g
)
{\displaystyle (M,g)}
is a Riemannian manifold (with or without boundary) and
i
:
N
→
M
{\displaystyle i:N\to M}
is an immersed submanifold or an embedded submanifold (with or without boundary), the pullback
i
∗
g
{\displaystyle i^{*}g}
of
g
{\displaystyle g}
is a Riemannian metric on
N
{\displaystyle N}
, and
(
N
,
i
∗
g
)
{\displaystyle (N,i^{*}g)}
is said to be a Riemannian submanifold of
(
M
,
g
)
{\displaystyle (M,g)}
. On the other hand, if
N
{\displaystyle N}
already has a Riemannian metric
g
~
{\displaystyle {\tilde {g}}}
, then the immersion (or embedding)
i
:
N
→
M
{\displaystyle i:N\to M}
is called an isometric immersion (or isometric embedding) if
g
~
=
i
∗
g
{\displaystyle {\tilde {g}}=i^{*}g}
. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.
For example, the n-sphere
S
n
=
{
x
∈
R
n
+
1
:
‖
x
‖
=
1
}
{\displaystyle S^{n}=\{x\in \mathbb {R} ^{n+1}:\lVert x\rVert =1\}}
is an embedded Riemannian submanifold of
R
n
+
1
{\displaystyle \mathbb {R} ^{n+1}}
via the inclusion map
S
n
↪
R
n
+
1
{\displaystyle S^{n}\hookrightarrow \mathbb {R} ^{n+1}}
that takes a point in
S
n
{\displaystyle S^{n}}
to the corresponding point in the superset
R
n
+
1
{\displaystyle \mathbb {R} ^{n+1}}
. The induced metric on
S
n
{\displaystyle S^{n}}
is called the round metric.
References
Kata Kunci Pencarian:
- Geodesik
- Geometri kompleks
- Lipatan (matematika)
- Riemannian submanifold
- Riemannian manifold
- Pseudo-Riemannian manifold
- Riemannian
- List of differential geometry topics
- Glossary of Riemannian and metric geometry
- Riemannian geometry
- List of things named after Bernhard Riemann
- Simons' formula
- Curvature of Riemannian manifolds
