Chapter 7 Metric and Affine Conformal Geometry

7 Metric and Affine Conformal

We consider the metric extrinsic geometry of quantities associated to

For

brevity

we

write
instead of
R-

N, R denote the left and right normal

*df

M

:

HPI.

M

L:=

f

Geometry

df (Y)

=

is the normal space,

so we

(7.2)

need to compute

1

II (X, But differentiation of

Y)

=

(X df (Y) -

2

-

NX

-

df (Y) R).

(7.2) yields

dN(X)df (Y)R +

NX

-

df (Y)R

+

Ndf (Y)dR(X)

=

X

-

df (Y),

or

X

-

df (Y)

-

NX

-

df (Y)R

=

dN(X)df (Y)R

=

-dN(X)

F. E. Burstall et al.: LNM 1772, pp. 39 - 46, 2002 © Springer-Verlag Berlin Heidelberg 2002

*

+

df (Y)

Ndf (Y)dR(X) +

*df (Y)dR(X).

40

7 Metric and Affine Conformal

Proposition

The

8.

'Rdf

mean

Geometry

curvature vector 'H

dfR

(*dR + RdR),

2

2

2

trace II is

given by

(*dN

+

dR +

*dNdf

(7.4)

NdN)df,

(7.5)

NdN).

(7-3)

Proof. By definition of the trace, 4'H jdfJ2

dN

=

*dfdR

=

-df (*dR

-

*

df

-

RdR)

+

df

*

(*dN

+

+

but

(*dN + NdN)df

=

*dNdf

=

-df

A

-

dN

dR

=

*

df

-dN A

=

df

=

-d(Ndf)

-df (*dR + RdR).

If follows that

27ildfI2

=

-df (*dR

RdR),

+

and

2 ldfTf Similarly for

dR +

dRR)Tf

(*dR + RdR)Tf

N.

Proposition

9. Let K denote the Gaussian curvature

and let K' denote the normal curvature

Kjwhere X E

=

and

TpM,

:=
R)

unit vectors. Then

1

KIdf 12 K

1

ldf 12

=

2 =

(< *dR, RdR

-1 (< *dR, RdR 2

> +
)

(7.6)

>

*dN, NdN >)

(7.7)

-


).

>,

The

1

N)

-

< *dN

>

NdN, NdN

-

-




NdN)df, dfRdR

(7.7).

have

T7r =

>

RdR), NdNdf

pull-back of the 2-sphere

Integrating this for compact

3-space (R

dfdR, N

-ldf 12

II(X, JX), II(X, X)

R*dA

In

*




>

dNdf, N

dR,R*dR

*dR, RdR




*dNdf


+





>

>

NdNdf

+

>

dNdf

dNdf)

+

dR + N

-

RdR >

dNdf, Ndf

+ < +

dR,dfRdR