CR submanifold

CR submanifold

Suppose that

M

⊂

ℂ

N

is a

real submanifold

of real dimension

n

.

Take

p

∈

M

,

then let

T

p

⁢

(

ℂ

N

)

be the

tangent vectors

of

ℂ

N

at the point

p

.

If we identify

ℂ

N

with

ℝ

2

⁢

N

by

z

j

=

x

j

+

i

⁢

y

j

,

we can take the following vectors as our basis

∂

∂

⁡

x

1

|

p

,

∂

∂

⁡

y

1

|

p

,

…

,

∂

∂

⁡

x

N

|

p

,

∂

∂

⁡

y

N

|

p

.

We define a real

linear mapping

J

:

T

p

⁢

(

ℂ

N

)

→

T

p

⁢

(

ℂ

N

)

such that for any

1

≤

j

≤

N

we have

J

(

∂

∂

⁡

x

1

|

p

)

=

∂

∂

⁡

y

1

|

p

and

J

(

∂

∂

⁡

y

1

|

p

)

=

-

∂

∂

⁡

x

1

|

p

.

Where

J

is referred to as the

complex structure

on

T

p

⁢

(

ℂ

N

)

.

Note that

J

2

=

-

I

,

that is applying

J

twice we just negate the vector.

Let

T

p

⁢

(

M

)

be the tangent space of

M

at the point

p

(that is, those vectors of

T

p

⁢

(

ℂ

N

)

which are tangent to

M

).

Definition

.

The

subspace

T

p

c

⁢

(

M

)

defined as

T

p

c

⁢

(

M

)

:=

{

X

∈

T

p

⁢

(

M

)

∣

J

⁢

(

X

)

∈

T

p

⁢

(

M

)

}

is called the

complex tangent space

of

M

at the point

p

,

and if the

dimension

of

T

p

⁢

(

M

)

is constant for all

p

∈

M

then the
corresponding

vector bundle

T

c

⁢

(

M

)

:=

⋃

p

∈

M

T

p

c

⁢

(

M

)

is called the

complex bundle

of

M

.

Do note that the complex tangent space is a real (not complex)

vector space

, despite its rather unfortunate name.

Let

ℂ

⁢

T

p

⁢

(

M

)

and

ℂ

⁢

T

p

⁢

(

ℂ

N

)

be the complexified vector spaces, by just allowing the coefficents of the vectors to
be

complex numbers

. That is for

X

=

∑

a

j

∂

∂

⁡

x

1

|

p

+

b

j

∂

∂

⁡

x

1

|

p

we allow

a

j

and

b

j

to be complex numbers. Next we can extend the mapping

J

to be

ℂ

-linear on these new vector spaces and still get that

J

2

=

-

I

as before. We notice
that the operator

J

has two

eigenvalues

,

i

and

-

i

.

Definition

.

Let

𝒱

p

be the

eigenspace

of

ℂ

⁢

T

p

⁢

(

M

)

corresponding to the eigenvalue

-

i

.

That is

𝒱

p

:=

{

X

∈

ℂ

⁢

T

p

⁢

(

M

)

∣

J

⁢

(

X

)

=

-

i

⁢

X

}

.

If the dimension of

𝒱

p

is constant for all

p

∈

M

,

then
we get a corresponding vector bundle

𝒱

which we call the

CR bundle

of

M

.

A

smooth section

of the CR bundle is then called
a

CR vector field

.

Definition

.

The submanifold

M

is called a

CR submanifold

(or just

CR manifold

) if the dimension of

𝒱

p

is constant for all

p

∈

M

.

The complex dimension of

𝒱

p

will then be called the

CR dimension

of

M

.

An example of a CR submanifold is for example a

hyperplane

defined by

Im

⁡

z

N

=

0

where the CR dimension is

N

-

1

.

Another less trivial example is the

Lewy hypersurface

.

Note that sometimes

𝒱

p

is written as

T

p

0

,

1

⁢

(

M

)

and referred to as the

space of antiholomorphic vectors

, where an

antiholomorphic vector

is a tangent vector which can be written in terms of the basis

∂

∂

⁡

z

¯

j

|

p

:=

1

2

(

∂

∂

⁡

x

j

|

p

+

i

∂

∂

⁡

y

j

|

p

)

.

The CR in the name refers to Cauchy-Riemann and that is because the vector space

𝒱

p

corresponds to differentiating with respect to

z

¯

j

.

References

1

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

2

Albert Boggess.
,
CRC, 1991.

Title

CR submanifold

Canonical name

CRSubmanifold

Date of creation

2013-03-22 14:49:04

Last modified on

2013-03-22 14:49:04

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Definition

Classification

msc 32V05

Synonym

CR manifold

Synonym

Cauchy-Riemann submanifold

Related topic

GenericManifold

Related topic

TotallyRealSubmanifold

Related topic

TangentialCauchyRiemannComplexOfCinftySmoothForms

Related topic

ACRcomplex

Defines

CR bundle

Defines

CR vector field

Defines

complex tangent space

Defines

complex bundle

Defines

space of antiholomorphic vectors

Defines

antiholomorphic vector

Defines

CR dimension

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