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# Conformal geometry of curves(Differential Geometry and Submanifold Theory)

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## curves

### O’Hara

*

Department of Mathematics, Tokyo Metropolitan University

### Remi Langevin

Institut de Math\’ematiques de Bourgogne, Universite de Bourgogne

By

knot

### an

embedding from $S^{1}$ into $S^{3}$

### or

$\mathbb{R}^{3}$,

its image.

By

### we mean a

disjoint union ofknots. We only study 2-component

Let $K$ be a knot, and let $x,y$ be

### a

pair of distinct points

### on

$K$. Let

$\Sigma(x, x+dx, y, y+dy)$ be a sphere through the fourpoints $x$,$x+dx,y$,$y+$ $dy$. It is uniquely

### determined

unless these four points

### are

concircular. By

identifying $\Sigma(x, x+dx, y,y+dy)$ with the complex sphere $\mathbb{C}\cup\{\infty\}$, we

can consider four points $x,x+dx_{?}y$, and $y+dy$

### as

four complex numbers. Let $\Omega$ denote the

### cross

ratioofthese four complex numbers, and call it the

ratio of

### a

knot $K$

### .

Although the four complex numbers

### are

not uniquely determined,their

ratio

be uniquely

The

infinitesimal

ratio

be considered

### as a

complex valued 2-form

### on

$K\mathrm{x}$ $K\backslash \triangle$. It is, by definition, invariant under Mobiustransformations.

We show that the energy

### of

knots $E_{\mathrm{o}}^{(2)}$

### can

be expressed in terms ofthe

infinitesimal

ratio. An

ofknots is

### a

functional onthe space of

knots which blows up

### as a

knot degeneratesto a singular knot with double

points. It

### was

introducedto produce

### a

“canonicalembedding” foreach knot type

### as an

embeddingwhich gives the minimum value of the energywithin

its isotopy class. $E_{\mathrm{o}}^{(2)}$

is defined by

$E_{\mathrm{o}}^{(?)}.(K)=-4+ \iint_{K\mathrm{x}K}(\frac{1}{|x-y|^{2}}-\frac{1}{d_{K}(x,y)^{2}})$dxdy,

where $d_{K}(x, y)$ denotes the arc-length between$x$ and $y$.

We then give interpretations ofthe real and imaginary parts ofthe

in-finitesimal

ratiofrom

### a

conformal geometric viewpoints, i.e. by using

’supported byJSPS duringhisstay at Dijon,France from September 2004to March

2005

(2)

### 135

what is invariant under M\"obiustransformations.

The first interpretation of the real part of the

ration

is given

### as

follows. A pair of ordered distinct points

be considered

### as

an oriented 0-sphere $S^{0}$. Let $\mathrm{S}(3, 0)$ be

### a

space of oriented $S^{0}$ in $S^{3}$

### .

It

is homeomorphic to the two points configuration space $S^{3}\mathrm{x}$ $S^{3}\backslash$ L. It

### can

also be identified with the total space of the cotangent bundle $T^{*}S^{3}$

### .

Then the real part of the infinitesimal

### cross

ratio is equal to the pull-back ofthe standardsymplectic formof$T^{*}g^{3}\underline{\simeq}s^{3}$)$\langle$$S^{3}\backslash$A by

### an

inclusion map

$K\mathrm{x}$ $K\backslash$IIS $\mathrm{C}arrow tS^{3}\mathrm{x}$ $S^{3}\backslash$ L.

The secondinterpretationofthe realpart of the infinitesimal

ration is given

follows. Let

### us

first consider $S^{3}$

### as

the set of the points at

infinity ofthe upper half light

### cone

in the Minkowski space $\mathbb{R}^{4,1}$. Then

### can

consider $\mathrm{S}(3,0)$

### as

the oriented Grassmannian manifold of the set of

2-dimensional vector subspace of mixed-type in $\mathbb{R}^{4,1}$

It allows

### us

to endow

semi-Riemannian structure with signature $(3, 3)$ to $\mathrm{S}(3,0)$

(This fact

### can

be proved in several ways, for example, by using PKicker coordinates, or

by considering$\mathrm{S}(3_{7}0)$

### as

a homogeneous space $SO(4_{?}1)/SO(3)\mathrm{x}$SO$(\mathit{1}, 1)$.

But it is convenient to

### use

of pencils of codimension 1 spheres in $S^{1}$ for

proofs.) As points $x$ and $y$

### move

along $K$, the set ofpairs $(x, y)$ form

### a

surfacein $\mathrm{S}(3,0)$

### .

Thenthereal part of the infinitesimal

### cross

ratio is equal

to the “imaginary”

### area

element of this surface.

Unlike the real part, the imaginary part of the infinitesimal

ratio

have

### a

global interpretation. If

### we

consider $S^{3}$

### as

the boundary of

thhe hyperbolic 4-space, the imaginary part of the

### infinitesimal cross

ratio

is locally equal to the “transversal

### area

form” of geodesies in $\mathbb{H}^{4}$ joining

$x$

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