Technische
Universitat
Berlin
Mathematische
Fachbdibliothek
Inv.-Nr:
HA
2,
)
IG
TECHNISCHE
UNIVERSITAT
BERLIN
Examples
of
Weyl
Geometries
in
Affine
Differential
Geometry
Martin
Peikert
Preprint
No.
579/1998
PREPRINT
REIHE
MATHEMATIK
|
Fachbereich
03
|
_
Examples
of
Weyl
Geometries
in
Affine
Differential
Geometry
Martin
Peikert*
'
Abstract
We
study
relations
between
Weyl
geometries
and
Codazzi
structures
(see
[1])
and
investigate
examples
of
Weyl
geometries
on
affine
hypersurfaces.
Keywords:
Weyl]
connections,
Codazzi
structures,
affine
hypersurfaces.
MS
Classification:
53A15,
53B15
0
Introduction
In
section
1
we
summarize
basic
facts
on
Weyl
structures.
In
the
literature
there
are
differences
in
the
definition
of
a
Weyl
structure
depending
on
the
choice
of
a
constant;
for
this
reason
we
define
a
Weyl
structure
using
an
arbitrary
constant
to:
A
Weyl
structure
on
a
C™-manifold
M
is
given
by
a
quadruple
W
=
{V,
C,~,7}
(where
V
is
a
torsion-free
connection,
C
a
conformal
class
of
semi-Riemannian
metrics,
me
R\
{0}
and
T
a
class
of
one-forins)
satisfying
the
compatibility
condition
(1.4)
below.
In
[1]
the
authors
investigate
the
relation
between
Codazzi
structures
C
=
{P,C}
(where
P
is
a
projective
class
of
connections
and
C
a
conformal
class
of
semi-Riemannian
metrics)
and
Weyl
structures
W
on
a
manifold
M.
They
show
how
a
Wey]
structure
can
be
constructed
from
a
Codazzi.structure
and
vice
versa.
We
give
a
short
introduction
to
these
constructions
(with
an
arbitrary
non-zero
real
number
to)
in
section
2
and
prove
a
necessary
and
sufficient
condition
for
a
bijective
relation
between
Weyl
and
Codazzi
structures.
In
section
3
we
investigate
two
naturally
arising
one-forins
in
the
theory
of
affine
hypersurfaces,
i.e.
the
connection
form
7
and
the
T
chebychev
form
T.
We
show
that
the
vanishing
of
the
exterior
derivative
of
one
of
them
is
equivalent
to
the
vanishing
of
the
the
exterior
derivative
of
the
other,
so
the
construction
of
a
non
trivial
Weyl
structure
(see
definition
1.3.1
below)
is
either
possible
with
both
of
them
or
none
of
them.
For
this
Weyl
geometry,
we
prove
the
following
two
results:
(i)
Only
on
hyperquadrics,
the
induced
connection
can
be
realised
as
either
a
Weyl
connection
or
the
Levi-Civita
connection
of
the
affine
metric
h.
(ii)
The
induced
connection
of
the
centroaffine
hypersurface
geometry
is
invariant
under
gauge
transfor-
mations
of
the
Weyl
geometry.
We
refer
the
reader
to
[11]
or
[6]
for
definitions
and
facts
on
affine
differential
geometry,
for
more
detailed
proofs
see
[9].
In
this
work
we
always
assume
that
the
C'-
manifolds
are
simply
connected
and
real.
*Department
of
Mathematics,
Technische
Universitat
Berlin,
e-mail:
bho
1
Weyl
structures
Let
Af
be
a
connected
C’*-manifold
of
dimension
n
>
2,
let
¥(M)
the
set
of
tangent
vector
fields
and
let
u,v,w,...
denote
elements
from
¥(M).
1.1.
Projective
structure
and
conjugate
connections
Definition
1.1.1
Let
V,
V*
be
torsion-free
connections
on
a
manifold
M.
(i)
V
and
V™
are
called-projectively
equivalent
iff
there
exists
a
one-form
&
such
that,
for
allu,v
€
¥
(M),
Vuv
=
Vivt
&(u)u
+
a(v)u,
(1.1)
we
will
write
Vi
~
V*,
|
(ii)
The
set
P(V)
:-=
{V*|V*
~
V}
is
called
the
projective
class
of
V.
(il)
A
connection
V
is
called
flat
iff
its
curvature
tensor
is
identical
zero.
(iv)
V
is
called
projectively
flat
if
it
is
projectively
equivalent
to
a
flat
connection.
(v)
The
projective
curvature
tensor
W
with
respect
to
V
is
defined
by
Wu,v)w
=
R(u,v)w
—
P(u,w)v
+
P(v,w)ut+
Plu, v)w
—
Pv,
w)w
where
P(u,v)
=
Ky
{nRie(u,
v)
—
Rie(v,
u)}.
—
2-1]
The
projective
flatness
of
a
connection
is
related
to
the
projective
curvature
tensor
W
and
to
P
in
the
following
way
(for
the
proof
see
[2]):
Theorem
1.1.2
(Weyl)
A
connection
V
on
a
manifold
M
is
projectively
flat
if
and
only
if
(i)
W
is
equal
to
zero
on
M
and
(ii)
(VuP)(o,w)
=
(VyP)(u,w)
for
all
u,v,w
€
¥(M).
Definition
1.1.3
Let
h
be
a
semi-Riemannian
metric
on
a
manifold
M
and
u,v,w
€
X(M).
(i)
Two
sonnendions
V
and
V*
are
called
conjugate
w.r.t.
h
iff
they
satisfy
wh(v,w)
=
h(Vyv,w)
+
h(v,
View).
(1.2)
The
triple
{V,h,
V*}
is
called
a
conjugate
triple.
(ii)
Let
V
be
a
connection
and
Tt
a
one-form
such
that
Vh+7h
is
totally
symmetric.
Define
a
torsion-free
connection
V*
by
uh(v,w)
=
h(VFv,w)
+
h(v,
Vuw))
+
F(v)h(u,
w).
(1.3)
Then
V*
is
said
to
be
semi-conjugate
to
V
relative
to
h
by
7.
Remark
1.1.4
Let
V*
be
semi-conjugate
to
V
relative
to
h
by
7.
Define
V
via
V,v
=
Vu
+
A(u,
v)T,
then
it
is
easy
to
see
that
{V,h,
V}
is
a
conjugate
triple.
Proposition
1.1.5
([7])
Let
{V,h,
V*}
be
a
conjugate
triple
and
R,
R*
the
curvature
tensors
of
V
and
V*
respectively.
For
all
u,v,w,z€
X(M),
we
have
the
relation
h(R(u,
v)w,
z)
+
h(w,
R*(u,v)z)
=
0.
1.2
Weyl
geometry
Definition
1.2.1
Letn
>
2
and
M
be
a
n-dimensional
manifold
endowed
with
a
torsion-free
connection
V.
Consider
a
conformal
class
of
semi-Riemannian
metrics
C
:=
{Bh|0
<
BE
C™(M)},
a
constant
m
€
R\
{0}
and
a
sel
of
one-forms
T
:=
{On|
he
C}.
(i)
The
quadruple
{VC
,t0,
7}
is
called
a
Weyl
structure
W
on
M
iff
for
the
connection
V
and
all
semi-
Riemannian
metrics
h
€C
the
following
condition
is
satisfied:
Vi
=
wb,
Oh
=:
WOWh.
(1.4)
The
condition
(1.4)
is
called
compatibility
condition
and
(M,W)
is
called
a
Weyl
manifold.
(il)
If
for
the
connection
V,
an
arbitrary
semi-Riemannian
metric
h
and
wm
€
R
\
{0}
given
there
exists
a
one-form
0
such
that
the
compatibility
condition
is
satisfied,
then
V
is
called
Weyl
connection
and
we
write
V
=:
V(h,
to,
8).
If
we
inake
a
conforinal
change
of
the
metric
there
exists
a
transformation
of
the
one-form
that
preserves
the
compatibility
condition
for
all
metrics
in
C.
Definition
1.2.2
Let
0
<
BE
C™(M)
and
V
=
V(h,
iu,0).
The
snapping
h—
Bh,
6+
6++dlnp.
(1.5)
is
called
gauge
transformation.
Remark
1.2.3
From
definition
1.2.2
one
easily
verifies
that
the
compatibility
condition
is
invariant
under
gauge
transformations.
a
From
a
given
metric
h
€
C,-a
real
non-zero
number
tv
and
a
one-form
6
one
can
construct
a
torsion-free
connection
V
satisfying
(1.4);
thus
(h,
10,9)
induce
a
Weyl
geometry.
Lemma
1.2.4
Leth
eC,
mw
€
R\
{0},
6
be
a
one-form
and
V
the
Levi-Civita
connection
of
h.
Then
(i)
the
Weyl
connection
V
=
V(h, 0,0)
can
be
expressed
in
terms
of
h,vo
and
6
by
Vu
=
Vyv
—
=
{A(u)u
+
O(v)u
+
h(a,
vo}
for
allu,v
€
¥(M),
(1.6)
where
@
is
defined
by
A(u)
=:
h(u,9);
this
construction
is
invariant
under
gauge
transformations;
(ii)
the
connection
V
defined
in
(i)
satisfies
the
compatibility
condition.
Proof.
For
the
proof
see
[12].
|
Remark
1.2.5
For
a
Weyl
connection
V
it
is
obvious
that
its
curvature
tensor
R
defined
by
R(u,v)w
:=
{VuVu-—VuVu-
Viu,v)
}w
and
its
Ricci
tensor
Ric
defined
by
Ric(v,
w)
:=
trace{u
>
R(u,v)w}
are
invariant
under
gauge
transformations.
1.3
Weyl
curvature
Definition
1.3.1
Let
V
=
V(h,
10,0)
be
a
Weyl
connection
and
u,v,
w
€
£(M).
(i)
The
length
curvature
F:
X(M)
x
X(M)
—
C™(M)
is
defined
by
F(u,v)
=
—w
d6(u,
v)
|
(1.7)
where
dO
denotes
the
exterior
derivative
of
6.
We
call
a
Weyl
structure
trivial
iff
F
=
0.
(ii)
The
directional
curvature
is
the
mapping
K:
X(M)
x
X(M)
x
¥(M)
>
X(M)
defined
by
K(u,
v)w
=
R(u,v)w
—
F(u,
v)w.
-
(1.8)
Remark
1.3.2
F
is
obviously
gauge
invariant
and
so
is
K,
too.
Lemma
1.3.3
The
following
relations
for
R,
F
and
K
(i)
n(R(u,v)w,w)
=
F(u,v)h(w,w),
(ii)
A(K(u,v)w,z)
=
—h(w,
K(u,
v)z),
(ii)
(a)
A(u,v)wl,»w
and
(b)
F(u,v)w|lw,
(iv)
2F(u,v)h(w,z)
=
h(R(u,v)w,
z)
+
h(w,
R(u,
v)z)
hold
for
allh
€C
and
u,v,w,z
€
¥(M).
Proof.
(i)
—
(iii)
are
well
known,
see
[12].
(iv)
follows
directly
from
(i)
and
(iii).
LJ
The
relation
1.3.3
(iv)
is
similar
to
a
relation
of
curvature
tensors
of
conjugate
connections.
As
a
consequence
of
proposition
1.1.5
and
lemma
1.3.3
(iv)
we
get
for
an
arbitrary
choice
of
a
semi-Riemannian
metric
h €
C:
Corollary
1.3.4
Let
V
=
V(h,
10,
8)
be
a
Weyl
connection,
{V,h, V*}
the
conjugate
triple
and
R,
R*
the
curvature
tensors
of
V
respectively
V*.
The
equation
R*(u,v)w
a
R(u,v)w
—
2F(u,v)w
is
valid
for
all
u.v,w
€
X(M).
This
relation
is
gauge
invariant.
Remark
1.3.5
The
conjugate
connection
V*
of
{V,h}
is
torsion-free
if
9
=
0;
then
we
have
V*
=
V
=
V.
Theorem
1.3.6
([3])
If
and
only
if
F
=
0
then
the
Weyl
connection
V
=
V(h,
10,0)
is
the
Levi-Civita
connection.
of
an
appropriate
metric
h®
€
C.
Furthermore
the
symmetry
of
the
Ricci
tensor
Ric
is
closely
related
to
the
vanishing
of
the
length
curvature,
a
straightforward
computation
shows
Ric(u,v)
—
Rie(v,u)
=n
F(u,v).
This
proves:
Proposition
1.3.7
Let
V
=
V(h,
tv,
6)
be
a
Weyl
connection,
Ric
the
Ricci
tensor
and
F
the
length
curva-
ture
wrt.
V.
Then
we
have
that
F
=
0
if
and
only
if
Ric
is
symmetric.
The
projective
flatness
of
a
Weyl
connection
is
also
related
to
the
vanishing
of
the
length
curvature.
Theorem
1.3.8
Let
V
=
V(h,
tv,
0)
be
a
Weyl
connection
and
W
the
projective
curvature
tensor
w.r.t.
V.
If
in
>
3
then
we
get
that
W=0
implies
F
=
0.
Proof.
Let
n
>
3
and
u,v,w
€
X(M)
be
h-orthogonal
by
pairs.
Then
Gx
Wu,
v)w
=
K(u,
v)w
—
Pu,
w)v
+
P(v,
w)u
—
F(u,v)w
+
P(u,
v)w
—
P(v,
u)w.
It
is
sufficient
to
show
F(u,v)
#
P(u,
v)
—
P(v,u)
because
the
other
terms
are
h-orthogonal
to
w.
Using
proposition
1.3.7
we
have
Plu,
v)
—
Pv,
u)
=
SF
(u,v).
L]
Remark
1.3.9
If
the
length
curvature
does
not
vanish
the
Weyl
connection
cannot
be
projectively
flat
in
dim(A)
>
3.
In
dimension
two
an
analogon
to
this
is
not
known.
2
Codazzi
structures
2.1
Introduction
Definition
2.1.1
Let
P*
:=
P(V*)
be
a
projective
class
of
torsion-
jree
connections!
,
C
a
conformal
class
of
semi-Ricmannian
metrics
and
u,v,w
€
X(M).
(i)
If
for
nn,
phones
hEC
there
exists
a
V"
€
P*
such
that
the
equation
(Vi)
(v,
w)
=
(Veh)
(u,
w)
|
(2.1)
holds,
then
the
pair
{V*,h}
will
be
called
a
Codazzi
pair.
(ii)
Equation
(2.1)
is
called
Codazzi
equation
for
V*
and
h.
(ili)
Jf
there
exists
a
Codazzi
pair
{V*,h}
in
{P*,C},
then
iP",
C}
ts
called
a
Codazzi
structure
on
M.
(iv)
A
Codazzi
transformation
is
the
mapping
he
Bhai
h®,
Viuwn
Vivtd
InB(u)u
+
dlnB(u)u
=:
V**.
(2.2)
Remark
2.1.2
If
{P*,C}
is
a
Codazzi
structure
then
for
every
h €
C
there
exists
a
V
€
P*
such
that
{h,
V}
is
a
Codazzi
pair;
the
Inapping
A
+>
V
is
injective,
not
surjective.
2.2
The
constructions
Tn
[1],
the
authors
considered
two
constructions:
(i)
Given
a
Weyl]
structure?
one
constructs
a
Codazzi
structure
aud
(ii)
vice
versa,
construct
a
Weyl
structure
from
a
given
Codazzi
structure.
(i)
Construct
a
Codazzi
structure
from
a
given
Weyl
structure
{V
=
V(h,to,4),C,
10
7}
on
M:
Ving
lemma
1.2.4
(i)
we
have
V,,v
=
Vu
—
2
{6(
(u)v
+
6(v)u
—
h(u,v)0}.
Define
Viv=Vuvt
.
{
(uw)
+
6(v)u
+
h(u,vjo
(2.3)
and
a
connection
V
that
is
conjugate
to
V*
with
respect
to
h:
Vuv:=
Vyv-
2
&
{8(u)
+
6(v)u
+
A(u,
vo}.
The
result
is
that
{V*,h}
and
{V,h}
are
Codazzi
pairs
and
V*
is
projectively
equivalent
to
V.
‘that
do
not
necessarily
have
a
symmetric
Ricci
tensor
2
with
t
=
2,
which
is
not
a
necessary
condition.
One
can
choose
an
arbitrary
tw
€
R
\
{0}.
6
Remark
2.2.1
For
the
curvature
tensors
R*
of
V*
and
R
of
V
of
conjugate
connections
we
have
proposition
1.1.5
(Le.
h(R(u,
v)w,z)+h(w,
R*(u,v)z)
=
0);
given
a
Weyl
structure
one
can
construct
a
Codazzi
structure,
in
this
situation
proposition
1.1.5
is
equivalent
to
lemma
1.3.3
(iv)
(i.e.
h(R(u,v)w,
z)
+
h(w;,
R(u,v)z)
=
2F(u,v)h(w,
z)).
Lemina
1.1.5
is
not
conformally
invariant
but
lemma
(1.3.3)
(iv)
obviously
is
invariant
uuder
gauge
transformations
that
include
a
conformal
change
of
the
metric.
(ii)
Again,
following
[1],
we
can
construct
a
Weyl
structure
from
a
Codazzi
structure:
let
{P,C}
bea
Codazzi
structure.
For
a
fixed
Codazzi
pair
{V,h}
€
{P,C},
we
define
the
(1.2)-tensor
C:
C:=V-V..
(2.4)
V
and
V
are
torsion-free,
therefore
C
is
a
symmetric
(1.2)-tensor.
For
C'
define
the
associated
one-form
nT(v)
:=
trace
{ur
C(u,v)}.
(2.5)
A
Codazzi
transformation
induces
the
following
transformation
formulas
for
C
and
T
(see
[11],
propo-
sition
5.1.3.)
C¥(u,v)
=
C(u,v)-
5
{dlnB(u)v
+
dlnB(v)u
+
h(u,v)grad,
np},
(2.6)
Lom
rd
‘
wl
®
=
ST
-
dnp.
(2.7)
Here
we
see
that
the
one-form
2
T
transforins
like
the
one-form
6
that
appears
in
the
gauge
trans-
formation
(1.5)
with
t=
—2.
Therefore
this
one-form
is
eligible
to
construct
a
Weyl
connection:
Vuvi=
Vyut
“is
{T(u)
+T(v)u—
R(t,
vr}
(2.8)
where
T
is
defined
by
T(u)
=:
A(u,
7).
Remark
2.2.2
The
connection
V
is
not
necessarily
projectively
equivalent
to
the
given
V;
the
invariant
formulation
of
lemma
1.1.5
for
the
curvature
tensors
is
only
possible
if
V
and
V
are
in
the
same
projective
class.
Lemma
2.2.3
({1])
Consider
two
Codazzi
structures
{P,C}
and
{P*,C}
and
define
the
symmetric
(1.2)-
hensor
field
y
for
any
two
Codazzi
pairs
{V,h}
and
{V*,h}
by
y(v,
w)
:=
Vyw-Viw.
The
Codazzi
structures
define
the
same
Weyl
structure
if
and
only
if
y
is
apolar,
which
means
trace
{ur
y(u,v)}
=
0.
Following
[1]
prescribe
a
Weyl
structure
and
construct
a
Codazzi
structure.
From
this
Codazzi
structure
again
a
Weyl
structure
can
be
constructed.
This
latter
Weyl
structure
coincides
with
the
given
one.
On
the
other
hand,
if
we
start
with
a
given
Codazzi
structure
and
first
construct
a
Weyl
structure
and
then
from
this
a
Codazzi
structure
again,
the
latter
Codazzi
structure
need
not
to
coincide
with
the
given
one.
Theorem
2.2.4
Let
{P,C}
be
a
Codazzi
structure
and,
for
a
fived
h
€
C,
construct
a
Weyl
connection
V=V(h,
—
ee
T)
following
(2.4)
-
(2.8).
From
that
Weyl
connection
construct
a
projective
class
P*
as
in
(2.3);
this
implies
that
{P*,C}
is
a
Codazzi
structure.
Then
~P=P*
if
and
only
if,
in
P,
there
exists
a
Weyl
connection
compatible
with
hi:
Proof.
Let
V
€
P
be
a
Weyl
connection
and
show
that
V
is
equal
to
V(h,
ait
T)
€
P*,
using
the
projective
equivalence
of
V
and
a
V
€
P
and
lemma
1.2.4
(i).
—
|
LJ
Remark
2.2.5
Let
C
be
a
given
conformal
class;
then
there
are
two
types
of
projective
classes:
projective
classes
that
contain
a
Weyl
connection
compatible
with
C
and
others
that
do
not
contain
such
a
connection.
3
Affine
differential
geometry
of
hypersurfaces
3.1
Introduction
Let
M
be
orientable
and
A
a
real
affine
space
of
dimension
n+
1
equipped
with
the
canonical
flat
connection
V;
let
V
be
the
real
vector
space
associated
to
A
and
V*
its
dual
space.
Let
2:
M
>
A
be
an
immersion
with
an
arbitrary
transversal
field
y.
Then
we
have
the
structure
equations
Vuda(v)
=
da(Vuv)+h(u,v)y,
(3.1)
dy(v)
=
da(—Sv)+7(v)y.
(3.2)
Here,
h
is
a
symmetric
(0.2)-tensor
field,
V
a
torsion-free
connection,
called
the
induced
connection,
S
a
(1.1)-tensor
field,
called
the
shape
operator
and
7
a
one-form
called
the
connection
form.
Choose
a
conormal
field
Y:
M
—
V*
as
the
unique
solution
of
Y(y)=1
and
Y(d2x(v))
=0
(v
€
X(M)).
(3.3)
If
x
is
a
regular
hypersurface
—
i.e.
h
is
nondegenerate
—
then
h
is
called
the
affine
metric.
Remark
3.1.1
The
regularity
of
z
is
independent
of
the
choice
of
y
and
equivalent
to
rank(dY,Y)
=n+1.
Let
x
be
regular,
then
we
can
consider
Y
as
a
hypersurface
Y:
M
—3
V*
with
transversal
field
(—Y’)
and
structure
equation
VudY
(uv)
=
dY¥
(Viv)
+
$(u,v)(-Y).
(3.4)
V"
is
a
torsion-free
connection,
called
the
conormal
connection,
and
S
is
a
symmetric
(0.2)-tensor
field.
Iu
this
section
we
assuine
all
hypersurfaces
to
be
regular;
then
the
pair
{Y,y}
satisfying
(3.3)
is
called
a
normalisation.
Lemma
3.1.2
Some
basic
relations
of
the
coefficients
of
the
structure
equations
are:
(i)
{V*,h}
is
a
Codazzi
pair,
(ii)
V
and
V*
are
semi-conjugate
relative
to
h
by
T
and
(iii)
S(u,v)
=
A(Su,v)
+
(VEF)(v)
—
F(u)F(v).
Proof.
The
proof
of
(i)
follows
[7];
use
that
Vv,
as
defined
in
remark
1.1.4,
and
V*
are
torsion-free
and
conjugate
with
respect
to
h.
For
(ii)
show
that
Y(dy(v))
=:
<
Y,dy(v)
>
=
f(v),
use
(i)
and
the
fact
that
for
a
conjugate
triple
{V*,
h,
Vv}
we
have:
V*h
is
totally
symmetric
iff
VA
is
totally
syinmetric,
see
[11],
4.4.1;
(iii)
follows
from
(i),
(ii)
and
the
structure
equations
(3.1),
(3.2)
and
(3.4).
LJ
Corollary
3.1.3
The
Levi-Civita
connection
V
of
h
in
terms
of
V,
V*
and
@
és
given,
using
the
notation
of
remark
1.1.4,
by
Vou
=
$(Veut+
Vyut
A(u,v)r)
=
3(Viu+
Vy).
The
integrability
conditions
for
the
hypersurfaces
2
and
Y
in
terms
of
V
and
V*
read
h(v,Su)—h(u,
Sv)
=
2dF(v,u),
(3.5)
(VyS)u—(VuS)v
=
F(v)Su—7(u)Sv,
(3.6)
R(u,v)w
=
hiv,
w)Su—h(u,w)Sv
(3.7)
(Vuh)(u,w)
+7(v)h(u,w)
=
(Vyh)(v,w)
+
F(u)a(v,
w),
(3.8)
R*(u,v)w
=
S(v,w)u—S(u,w)v,
(3.9)
(ViS)(u,w)
=
(VES)(v,w);
(3.10)
the
proof
is
analogous
to
[11],
4.8.1
and
4.8.2,
compare
also
[8]
and
[4].
Remark
3.1.4
By
the
equations
(3.9)
and
(3.10)
it
can
be
seen
that,
like
in
the
case
of
relative?
norma-
lisatious,
V*
is
projectively
flat
because
the
integrability
conditions
of
V*
are
the
same
as
in
the
case
of
relative
normalisations
(for
the
proof
see
[11],
4.10.3.2.).
Moreover
the
Ricci
tensor
Ric*
of
V*
is
symmetric
(see
[11],
4.8.1.).
3
3.2
The
vanishing
of
the
derivative
of
the
connection
form
A
natural
question
that
arises
is:
under which
conditions
for
the
connection
form
is
it
possible
to
construct.
uou-trivial
Weyl
structures.
The
existence
of
a
connection
form
with
non-vanishing
exterior
derivative
is
proved
by
Opozda:
Theorem
3.2.1
([8])
Let
M
be
a
simply
connected
n-dimensional
manifold
endowed
with
a
connection
V,
asymmetric
bilinear
form
h,
a
(1,1)-tensor
field
S
and
a
one-form
7
such
that
equations
(3.5)
—
(3.8)
are
satisfied.
Then
there
are
an
nondegenerate
immersion
x:
M
>
A
and
a
vector
field
y
transversal
to
x
such
thal
V,h,S
and
+
are
the
objects
induced
by
{x,y}
via
(3.1)
and
(3.2).
Here
we
can
see
that
there
are
no
further
restrictions
to
7,
so
we
can
assume
that
the
exterior
derivative
of
the
connection
form
dves
not
vanish.
In
this
case
we
can
construct
a
non-trivial
Weyl
connection
using
T.
The
following
lemma
gives
conditions
to
V,h
and
S,
resp.,
which
imply
d7
=
0.
Lemma
3.2.2
Let
x
be
a
hypersurface
with
transversal
field
y
and
conormal
field
Y.
The
following
properties
are
equivalent:
(i)
dt
=
0;
(ii)
S
as
selfadjoint
w.r.t.
h;
(ui)
V
has
a
symmetric
Ricci
tensor;
(iv)
dT
=0
where
T
(u)
:=
4trace
{vu
H
(Vyu
—
Veu)};
T
is
called
the
Tchebychev
form.
Proof.
For
(i)
©
(ii)
use
(3.5);
(ii)
©
(iii)
follows
from
(3.7)
and
(iii)
=
(iv)
is
shown
in
[7],
proposition
4.1
and
4.4,
.
C]
Remark
3.2.3
Because
of
the
equivalence
of
(i)
and
(iv)
there
are
either
two
one-forms
to
construct
a
non-trivial
Weyl
connection,
or
none.
2
‘s
ee
“where
y
is
chosen
such
that
7
is
equal
to
zero
Lemma
3.2.4
Let
x:
M
—
A
be
a
hypersurface
with
transversal
field
y
and
conormal
field
Y.
Let
C
be
the
conformal
class
of
metrics
such
that
h €
C,
V
the
Levi-Civita
connection
of
h
and
u,v
€
X(M).
If
any
one
of
the
conditions
(i)
—
(iv)
is
satisfied,
then
d7
=
0:
(i)
the
induced
connection
is
a
Weyl
connection,
(ii)
there
exists
h®
€C
such
that
{V,h}
is
a
Codazzi
pair,
(iii)
Vu
=
Vyu
—
Alu,
v)T
with
T(u)
=:
h(u,r)
for
all
(wu
€
X(M)),
(iv)
V
as
projectively
equivalent
to
V.
Proof.
(i)
Let
V
be
a
Weyl
connection.
From
lemma
1.2.4
(i)
we
know
that
there
exists
a
one-form
@
such
that
Vyv
=
Vy
—
F1{A(u)u
+
a(v)u
—
h(u,v)a},
where
a
is
defined
by
A(w)
=:
h(u,@).
Using
corollary
3.1.3
we
get
Viv
=
Vyv
+
3
{a(u)v
+
&(v)u
—
h(u,v)a}
—
h(u,v)?.
Additionally
we
have
(Vih)(v,w)
=
—w
a(u)h(v,w)
+
F(v)h(u,
w)
+
7(w)h(u,v).
Lemma
3.1.2
(i)
implies
a
=
—i7;
we
get
Viv
=Vyv-
4
{7(u)u
+
F(v)u
+
A(u,v)T}.
(3.11)
This
aud
remark
3.1.4
imply
d7
=
0.
(ii)
A
straightforward
computation
shows
d7
=
0.
(iii)
Using
corollary
3.1.3
we
have
Vu
=
Vuv—2
h(w,v)r.
Lemma
1.2.4
(i)
shows
7
=
0,
this
implies
d7 =
0.
(iv)
Let
@
be
a
one-form
such
that
Vv
=
Vyv
—
&(u)u
—
A(v)u.
A
similar
calculation
as
in
the
proof
of
(i)
shows
Viv
=
Vue
+
&(u)u
+
&(v)u
—
h(u,v)r.
The
Codazzi
property
of
{V*,h}
leads
us
to
Viv
=
Vayu
-
T(u)u
—
T(v)u
—
A(w,
v)r.
(3.12)
With
the
same
arguinentation
as
in
(i)
we
get
d7
=
0.
LI
Remark
3.2.5
In
parts
(ii)
and
(iv)
of
the
above
proof,
it
can
be
seen
from
the
equations
(3.11)
and
(3.12)
that
the
cubic
form
C(u,v,w)
:=
h(Vyu
—
Viv,w)
has
the
form
C(u,v,w)
=
&(u)h(v,
w)
+
&(v)h(w,
uw)
+
a(w)h(u,v).
This
is
a
uecessary
and
sufficient
condition
for
2(M/)
to
be
a
quadric
—
as
shown
in
[5],
theorem
8.
Therefore,
only
on
quadrics,
the
induced
connection
can
be
realised
as
a
connection
that
is
projectively
equivalent
to
the
Levi-Civita
connection,
or
as
a
Weyl
connection.
3.3
Transformations
of
the
transversal
field
Lemma
3.3.1
(Transformation
Lemma)
Consider
a
hypersurface
7:
M
3
A
with
two
normalisations
{Y,y}
and
{
Y*
,y*}
with
the
same
orientation.
There
are
a
function
0
<
¢
€
C™(M)
and
a
vector
field
1)
©
X(M)
such
that
y#
=
b'{y
+
da(n)}.
Then
we
have
(i)
Y#
=4Y,
(ii)
Vitu
=
Viv
+
dlnd(u)u
+
dlng(v)u,
(iii)
h#
=
gh,
(iv)
V#u
=
Vyv
—
A(u,v)n
and
(v)
F#(v)
=
F(v)
+
H(v)
—
dlnd(v),
where
7
is
given
by
(wu)
=
h(u,)
for
all
u
€
%(M).
10
Proof.
Straightforward
calculations.
|
CI
If
we
choose
¢
=
3
and
7)
=
(1+
+
grad,
InZ
then
we
have
a
gauge
transformation
(h,7)
4
(Gh,
F+id
Inf),
which
is
induced
by
a
transformation
of
the
transversal
field.
Moreover,
if
we
set
a
=
—1-it
is
easy
to
see
that
V
is
invariant
under
transformations
of
the
transversal
field
with
¢
and
7
chosen
as
above.
If
we
assume
that
V
is
a
Weyl
connection
then
lemma
3.2.4
shows
that
it
is
a
trivial
Weyl
connection.
3.4
The
centroaffine
connection
as
a
gauge
invariant
connection
Definition
3.4.1
Let
w:
M
—
V
be
a
hypersurface
with
0
¢
2(M),
and
x
be
transversal
to
a(M).
Let
{Yvy}be
a
normalisation
and
define
the
associated
support
function
p=
<
Y,-2>.
As
«
and
y
are
transversal
we
can
express
y
in
terms
of
a
and
dx;
by
straightforward
computations
we
get
y=
—p
‘a
+
da(grad,
np
+
7)
and
the
decomposition
Vida(v)
=
da(Vuv
+
h(u,
v)
{grad
Inp
+
T})
—
pu!
h(u,
v)
a.
Definition
3.4.2
Let:
M
—
V
be
a
hypersurface
with
normalisation
{Y,y}
and
associated
support
func-
tion
p
#0.
Let
V
be
the
induced
connection
and
7
the
connection
form.
Define
the
connection
°
Vu
=
Vuv
+
h(u,
v)
{grad,
Inp
+
r})
where
T
is
given
by
T(u)
=
h(u,T)
for
all
wu
€
X(M).
Remark
3.4.3
V
again
is
a
torsion-free
connection
(V:
torsion-free,
h
symmetric).
The
geometric
inter-
pretation
of
V
is
that the
pregeodesics
of
this
connection
are
intersections
of
the
hypersurface
with
planes
that
coutain
the
point
ag.
This
connection
is
studied
in
[10]
for
the
case
of
relative
normalisations.
Por
a
transformation
yr
Bo'y
+
(1+
+
)da(grad,
Inf)
=:
y*
we
have
p>
Bp
:=
p*;
the
other
quantities
change
as
in
the
transformation
lemma.
We
get
<2
II
a!
Vitv
+
h®
(u,v)
{erad),»#
Inp*
+
7*
\
=
Vyv—-(1++)h(u,
v)grad,
Ing
+
Bh(u,v)
37!
(grad,
InBp
+
7
+
+
grad,
Inf)
=
Vu.
We
have
proved:
Proposition
3.4.4
Let
«:M
—
V
be
a
hypersurface
as
in
definition
3.4.1.
Then
V
is
gauge
invariant.
The
connection
induced
by
a
transversal
field
that
is
a
multiple
of
its
position
vector
field
2
is
well-known
in
affine
differential
geometry
of
hypersurfaces:
it
is
known
as
centroaffine
connection.
Therefore
we
have
Corollary
3.4.5
The
centroaffine
connection
is
gauge
invariant.
‘Lemma
3.4.6
If
V
is
a
Weyl
connection,
then
it
is
trivial.
11
Proof.
From
the
definition
of
V
we
can
see
that
V
is
induced
by
the
transversal
field
—p~!a.
Lemma
3.2.4
gives
that
the
induced
connection
is
a
trivial
Weyl
connection.
LI
Acknowledgements.
I
would
like
to
thank
Professor
U.
Simon,
Professor
B.
Opozda
and
Professor
S.
Ivanov,
M.
Wiehe
and
the
other
members
of
the
group
“Affine
Differentialgeometrie”
at
the
TU
Berlin
for
helpful
discussions.
References
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U.
Simon,
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H.
Viesel,
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Raum
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—
