hyperbolic plane in quadratic spaces

hyperbolic plane in quadratic spaces

A

non-singular

(

http://planetmath.org/NonDegenerateQuadraticForm

)

isotropic quadratic space

ℋ

of

dimension

2 (over a field) is called a

hyperbolic plane

. In other
words,

ℋ

is a 2-dimensional

vector space

over a field
equipped with a

quadratic form

Q

such that there exists a non-zero
vector

v

with

Q

⁢

(

v

)

=

0

.

Examples

. Fix the ground field to be

ℝ

, and

ℝ

2

be the two-dimensional vector space over

ℝ

with the

standard basis

(

0

,

1

)

and

(

1

,

0

)

.

1.

Let

Q

1

⁢

(

x

,

y

)

=

x

⁢

y

. Then

Q

1

⁢

(

a

,

0

)

=

Q

1

⁢

(

0

,

b

)

=

0

for all

a

,

b

∈

ℝ

.

(

ℝ

2

,

Q

1

)

is a hyperbolic
plane. When

Q

1

is written in matrix form, we have

Q

1

⁢

(

x

,

y

)

=

(

x

y

)

⁢

(

0

1

2

1

2

0

)

⁢

(

x

y

)

=

(

x

y

)

⁢

M

⁢

(

Q

1

)

⁢

(

x

y

)

.

2.

Let

Q

2

⁢

(

r

,

s

)

=

r

2

-

s

2

. Then

Q

2

⁢

(

a

,

a

)

=

0

for all

a

∈

ℝ

.

(

ℝ

2

,

Q

2

)

is a hyperbolic
plane. As above,

Q

2

can be written in matrix form:

Q

1

⁢

(

x

,

y

)

=

(

x

y

)

⁢

(

1

0

0

-

1

)

⁢

(

x

y

)

=

(

x

y

)

⁢

M

⁢

(

Q

2

)

⁢

(

x

y

)

.

From the above examples, we see that the name “hyperbolic plane”
comes from the fact that the associated quadratic form resembles the
equation of a

hyperbola

in a two-dimensional

Euclidean plane

.

It’s not hard to see that the two examples above are equivalent
quadratic forms. To transform from the first form to the second,
for

instance

, follow the linear substitutions

x

=

r

-

s

and

y

=

r

+

s

,
or in matrix form:

(

1

1

-

1

1

)

⁢

M

⁢

(

Q

1

)

⁢

(

1

-

1

1

1

)

=

(

1

1

-

1

1

)

⁢

(

0

1

2

1

2

0

)

⁢

(

1

-

1

1

1

)

=

(

1

0

0

-

1

)

=

M

⁢

(

Q

2

)

.

In fact, we have the following

Proposition

. Any two hyperbolic planes over a field

k

of
characteristic not 2 are

isometric quadratic spaces

.

Proof.

From the first example above, we see that the quadratic space with the quadratic form

x

⁢

y

is a hyperbolic plane.

Conversely

, if we can show that any hyperbolic plane

ℋ

is

isometric

the example (with the ground field switched from

ℝ

to

k

), we are done.

Pick a non-zero vector

u

∈

ℋ

and suppose it is
isotropic:

Q

⁢

(

u

)

=

0

. Pick another vector

v

∈

ℋ

so

{

u

,

v

}

forms a basis for

ℋ

. Let

B

be
the

symmetric bilinear form

associated with

Q

. If

B

⁢

(

u

,

v

)

=

0

,
then for any

w

∈

ℋ

with

w

=

α

⁢

u

+

β

⁢

v

,

B

⁢

(

u

,

w

)

=

α

⁢

B

⁢

(

u

,

u

)

+

β

⁢

B

⁢

(

u

,

v

)

=

0

, contradicting the fact that

ℋ

is non-singular. So

B

⁢

(

u

,

v

)

≠

0

. By dividing

v

by

B

⁢

(

u

,

v

)

, we may assume that

B

⁢

(

u

,

v

)

=

1

.

Suppose

α

=

B

⁢

(

v

,

v

)

. Then the matrix associated with the quadratic form

Q

corresponding to the basis

𝔟

=

{

u

,

v

}

is

M

𝔟

⁢

(

Q

)

=

(

0

1

1

α

)

.

If

α

=

0

then we are done, since

M

𝔟

⁢

(

Q

)

is

equivalent

to

M

𝔟

⁢

(

Q

1

)

via the

isometry

T

:

ℋ

→

ℋ

given by

T

=

(

1

2

0

0

1

)

⁢

, so that

⁢

T

t

⁢

(

0

1

1

0

)

⁢

T

=

(

0

1

2

1

2

0

)

.

If

α

≠

0

, then the trick is to replace

v

with an isotropic vector

w

so that the bottom right cell is also 0. Let

w

=

-

α

2

⁢

u

+

v

. It’s easy to verify that

Q

⁢

(

w

)

=

0

. As a result, the isometry

S

required has the matrix form

S

=

(

1

-

α

2

0

1

)

⁢

, so that

⁢

S

t

⁢

(

0

1

1

α

)

⁢

S

=

(

0

1

1

0

)

.

∎

Thus we may speak of the hyperbolic plane over a field without any ambiguity, and we may identify the hyperbolic plane with either of the two quadratic forms

x

⁢

y

or

x

2

-

y

2

. Its notation, corresponding to the second of the forms, is

⟨

1

⟩

⊥

⟨

-

1

⟩

, or simply

⟨

1

,

-

1

⟩

.

A

hyperbolic space

is a finite dimensional

orthogonal direct sum

of hyperbolic planes. It is always even dimensional and has the notation

⟨

1

,

-

1

,

1

,

-

1

,

…

,

1

,

-

1

⟩

or simply

n

⁢

⟨

1

⟩

⊥

n

⁢

⟨

-

1

⟩

, where

2

⁢

n

is the dimensional of the hyperbolic space.

Remarks.

•

The notion of the hyperbolic plane encountered in the theory of quadratic forms is different from the “hyperbolic plane”, a 2-dimensional space of

constant

negative

curvature

(

Euclidean

signature

) that is commonly used in

differential geometry

, and in

non-Euclidean geometry

.

•

Instead of being associated with a quadratic form, a hyperbolic plane is sometimes defined in terms of an

alternating form

. In any case, the two definitions of a hyperbolic plane coincide if the ground field has characteristic 2.

Title

hyperbolic plane in quadratic spaces

Canonical name

HyperbolicPlaneInQuadraticSpaces

Date of creation

2013-03-22 15:41:47

Last modified on

2013-03-22 15:41:47

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 11E88

Classification

msc 15A63

Defines

hyperbolic plane

Defines

hyperbolic space

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