construction of Riemann surface using paths

construction of Riemann surface using paths

Note: All arcs and curves are assumed to be smooth in this entry.

Let

f

be a

complex function

defined in a disk

D

about a point

z

0

∈

ℂ

. In this entry, we shall show how to construct a

Riemann surface

such that

f

may be analytically continued to a

function

on this surface by considering paths in the

complex plane

.

Let

𝒫

denote the class of paths on the complex plane having

z

0

as an

endpoint

along which

f

may be analytically continued. We may define an

equivalence relation

∼

on this set —

C

1

∼

C

2

if

C

1

and

C

2

have the same endpoint and there exists a one-parameter family of paths along which

f

can be analytically continued which includes

C

1

and

C

2

.

Define

𝒮

as the

quotient

of

𝒫

modulo

∼

. It is possible to extend

f

to a function on

𝒮

. If

C

∈

𝒫

, let

f

⁢

(

C

)

be the value of the

analytic continuation

of

f

at the endpoint of

C

(not

z

0

, of course, but the other endpoint). By the

monodromy theorem

, if

C

1

∼

C

2

, then

f

⁢

(

C

1

)

=

f

⁢

(

C

2

)

. Hence,

f

is well defined on the quotient

𝒮

.

Also, note that there is a

natural projection

map

π

:

𝒮

→

ℂ

. If

C

is an

equivalence class

of paths in

𝒮

, define

π

⁢

(

C

)

to be the common endpoint of those paths (not

z

0

, of course, but the other endpoint).

Next, we shall define a class of subsets of

𝒮

. If

f

can be analytically continued from along a path

C

from

z

0

to

z

1

then there must exist an

open disk

D

′

centered about

z

1

in which the continuation of

f

is

analytic

. Given any

z

∈

D

′

, let

C

⁢

(

z

)

be the

concatenation

of the path

C

from

z

0

to

z

1

and the straight line segment from

z

1

to

z

(which lies inside

D

′

). Let

N

⁢

(

C

,

D

′

)

⊂

𝒮

be the set of all such paths.

We will define a topology of

𝒮

by taking all these sets

N

⁢

(

C

,

D

′

)

as a basis. For this to be legitimate, it must be the case that, if

C

3

lies in the

intersection

of two such sets,

N

⁢

(

C

1

,

D

1

)

and

N

⁢

(

C

2

,

D

2

)

there exists a basis element

N

⁢

(

C

3

,

D

3

)

contained

in the intersection of

N

⁢

(

C

1

,

D

1

)

and

N

⁢

(

C

2

,

D

2

)

. Since the endpoint of

C

3

lies in the intersection of

D

1

and

D

2

, there must exist a disk

D

3

centered about this point which lies in the intersection of

D

1

and

D

2

. It is easy to see that

N

⁢

(

C

3

,

D

3

)

⊂

N

⁢

(

C

1

,

D

1

)

∩

N

⁢

(

C

2

,

D

2

)

.

Note that this topology has the Hausdorff property. Suppose that

C

1

and

C

2

are distinct elements of

𝒮

. On the one hand, if

π

⁢

(

C

1

)

≠

π

⁢

(

C

2

)

, then one can find

disjoint

open disks

D

1

and

D

2

centered about

C

1

and

C

2

. Then

N

⁢

(

C

1

,

D

1

)

∩

N

⁢

(

C

2

,

D

2

)

=

∅

because

π

⁢

(

N

⁢

(

C

1

,

D

1

)

)

∩

π

⁢

(

N

⁢

(

C

2

,

D

2

)

)

=

D

1

∩

D

2

=

∅

. On the other hand, if

π

⁢

(

C

1

)

=

π

⁢

(

C

2

)

, then let

D

3

be the smaller of the disks

D

1

and

D

2

. Then

N

⁢

(

C

1

,

D

3

)

∩

N

⁢

(

C

2

,

D

3

)

=

∅

.

To

complete

the proof that

𝒮

is a Riemann surface, we must exhibit coordinate

neighborhoods

and

homomorphisms

. As coordinate neighborhoods, we shall take the neighborhoods

N

⁢

(

C

,

D

)

introduced above and as homomorphisms we shall take the

restrictions

of

π

to these neighborhoods. By the way that these neighborhoods have been defined, every element of

𝒮

lies in at least one such neighborhood. When the domains of two of these homomorphisms overlap, the

composition

of one homomorphism with the

inverse

of the restriction of the other homomorphism to the overlap region is simply the

identity map

in the overlap region, which is analytic. Hence,

𝒮

is a Riemann surface.

Title

construction of Riemann surface using paths

Canonical name

ConstructionOfRiemannSurfaceUsingPaths

Date of creation

2013-03-22 14:44:23

Last modified on

2013-03-22 14:44:23

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

21

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 30F99

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