submanifold

submanifold

There are several conflicting definitions of what a submanifold is, depending on which author you are reading. All that agrees is that a submanifold is a subset of a manifold which is itself a manifold, however how

structure

is inherited from the

ambient space

is not generally agreed upon.
So let’s start with

differentiable

submanifolds of

ℝ

n

as that’s the most useful case.

Definition

.

Let

M

be a subset of

ℝ

n

such that for every point

p

∈

M

there exists a

neighbourhood

U

p

of

p

in

ℝ

n

and

m

continuously differentiable functions

ρ

k

:

U

→

ℝ

where the

differentials

of

ρ

k

are

linearly independent

,
such that

M

∩

U

=

{

x

∈

U

∣

ρ

k

⁢

(

x

)

=

0

,

1

≤

k

≤

m

}

.

Then

M

is called a

submanifold

of

ℝ

n

of

dimension

m

and of codimension

n

-

m

.

If

ρ

k

are in fact smooth then

M

is a

smooth submanifold

and
similarly if

ρ

is

real analytic

then

M

is a

real analytic
submanifold

. If
we identify

ℝ

2

⁢

n

with

ℂ

n

and we have a
submanifold there it is called a

real submanifold

in

ℂ

n

.

ρ

k

are usually called the

local defining functions

.

Let’s now look at a more general definition. Let

M

be a manifold of dimension

m

. A subset

N

⊂

M

is said to have the

submanifold property

if there exists an integer

n

≤

m

, such that for
each

p

∈

N

there is a

coordinate

neighbourhood

U

and a coordinate

function

φ

:

U

→

ℝ

m

of

M

such that

φ

⁢

(

p

)

=

(

0

,

0

,

0

,

…

,

0

)

,

φ

⁢

(

U

∩

N

)

=

{

x

∈

φ

⁢

(

U

)

∣

x

n

+

1

=

x

n

+

2

=

…

=

x

m

=

0

}

if

n

<

m

or

N

∩

U

=

U

if

n

=

m

.

Definition

.

Let

M

be a manifold of dimension

m

.
A subset

N

⊂

M

with the submanifold property
for some

n

≤

m

is called a

submanifold

of

M

of dimension

n

and of codimension

m

-

n

.

The ambiguity arises about what

topology

we require

N

to have. Some authors require

N

to have the

relative topology

inherited from

M

, others don’t.

One could also mean that a subset is a submanifold if it is a

disjoint
union

of submanifolds of different dimensions. It is not hard to see that
if

N

is

connected

this is not an issue (whatever the topology on

N

is).

In case of differentiable manifolds,
if we take

N

to be a

subspace

of

M

(the topology on

N

is the relative topology inherited from

M

) and the differentiable structure of

N

to
be the one determined by
the coordinate neighbourhoods above then we call

N

a

regular submanifold

.

If

N

is a submanifold and the

inclusion map

i

:

N

→

M

is an

imbedding

, then we
say that

N

is an

imbedded

(or

embedded

)

submanifold

of

M

.

Definition

.

Let

p

∈

M

where

M

is a manifold. Then the

equivalence class

of all
submanifolds

N

⊂

M

such that

p

∈

N

where we say

N

1

is

equivalent

to

N

2

if there is some open neighbourhood

U

of

p

such
that

N

1

∩

U

=

N

2

∩

U

is called the

germ of a submanifold

through the point

p

.

If

N

⊂

M

is an

open subset

of

M

, then

N

is called the

open submanifold

of

M

. This is the easiest class of examples of submanifolds.

Example of a submanifold (a in fact) is the unit sphere in

ℝ

n

. This is in fact a hypersurface
as it is of codimension 1.

References

1

William M. Boothby.
,
Academic Press, San Diego, California, 2003.

2

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

Title

submanifold

Canonical name

Submanifold

Date of creation

2013-03-22 14:47:20

Last modified on

2013-03-22 14:47:20

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 32V40

Classification

msc 53C40

Classification

msc 53B25

Classification

msc 57N99

Related topic

Manifold

Related topic

Hypersurface

Defines

real submanifold

Defines

codimension of a manifold

Defines

local defining functions

Defines

real submanifold

Defines

smooth submanifold

Defines

real analytic submanifold

Defines

regular submanifold

Defines

imbedded submanifold

Defines

embedded submanifold

Defines

germ of a submanifold

Defines

open submanifold

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