t
TECHNISCHE
UNIVERSITAT
BERLIN
S
S220
ISOPARAMETRIC
SURFACES
IN
3-DIMENSIONAL
DE
SITTER
SPACE
AND
ANTI-DE
SITTER
SPACE
Huili
LIU
Guosong
ZHAO
Preprint
No.
657/1999
PREPRINT
REIHE
MATHEMATIK
FACHBEREICH
3
ISOPARAMETRIC
SURFACES
IN
3-DIMENSIONAL
DE
SITTER
SPACE
AND
ANTI-DE
SITTER
SPACE
Hui
Liu))?)3)4),
Guosona
ZHAO!)3)
ABSTRACT.
A
spacelike
surface
M
in
3-dimensional
de
Sitter
space
5?
or
3-dimensional
anti-
de
Sitter
space
HE
is
called
isoparametric,
if
M
has
constant
principal
curvatures.
A
timelike
surface
is
called
isoparametric,
if
its
minimal
polynomial
of
the
shape
operator
is
constant.
In
this
paper,
We
determine
the
spacelike
isoparametric
surfaces
and
the
timelike
isoparametric
surfaces
in
S?
and
H8.
$1.
Introduction.
A
hypersurface
M
of
a
complete
simply-connected
Riemannian
manifold
R"t1(c)
of
constant
curvature
c
is
isoparametric
if
M
has
constant
principal
curvatures.
There
are
many
results
about
the
isoparametric
hypersurfaces
in
Riemannian
space
forms
(cf.
[CE],
[CHEN],
[N-R]).
For
the
isoparametric
hypersurfaces
in
the
indefinite
space
forms,
Nomizu
[NO]
derived
the
Cartan
formula
for
spacelike
isoparametric
hypersurfaces
in
Lorentzian
space
forms.
Hahn
[HA]
considered
the
general
case
of
indefinite
space
forms
of
curvature
c
and
obtained
the
Cartan-type
formula.
Magid
[MA]
studied
Lorentzian
hypersurfaces
in
the
Minkowski
space
E?.
He
obtained
a
complete
classification
of
isoparametric
hyper-
surfaces
in
E?.
In
this
paper,
we
consider
the
problem
in
3-dimensional
de
Sitter
space
S?
and
3-dimensional
anti-de
Sitter
space
H?.
A
spacelike
surface
M
in
3-dimensional
de
Sitter
space
S?
or
3-dimensional
anti-de
Sitter
space
H}
is
called
isoparametric,
if
M
has
constant
principal
curvatures.
A
timelike
surface
M
in
3-dimensional
de
Sitter
space
S?
or
3-dimensional
anti-de
Sitter
space
H?
is
called
isoparametric,
if
its
minimal
polynomial
of
the
shape
operator
is
constant.
we
will
prove
the
following
theorems.
Theorem
1.1.
Let
x:
M
—
S?
be
a
spacelike
isoparametric
surface,
then,
by
a
transfor-
mation
in
E;,
it
can
be
written
as
the
one
of
the
following
surfaces:
(i)
the
totally
umbilical
surface;
(ii)
z(u,v)
=
(asin(u),
acos(u),
bsinh(v),bcosh(v)),
a?
—b?
=
1.
1991
Mathematics
Subject
Classification.
53050,
52021,
53C40.
Key
words
and
phrases.
isoparametric
surface,
de
Sitter
and
anti-de
Sitter
space,
principal
curvature.
1)
Partially
supported
by
DFG466-CHV-II3
/127/0.
2)
Partially
supported
by
Technische
Universitat
Berlin.
3)
Partially
supported
by
NSFC.
4)
Partially
supported
by
SRF
for
ROCS,
SEM;
the
SRF
of
Liaoning
and
the
Northeastern
University.
Typeset
by
A,yS-TRxX
Z
HUILI
LIU,
GUOSONG
ZHAO
Theorem
1.2.
Let
c
:
M
—
Hi
be
a
spacelike
isoparametric
surface,
then,
by
a
transfor-
mation
in
Es,
it
can
be
written
as
the
one
of
the
following
surfaces:
(i)
the
totally
umbilical
surface;
(ii)
z(u,v)
=
(asinh(u),
bsinh(v),
acosh(u),bcosh(v)),
a?
+
6?
=
1.
Theorem
1.3.
Let
«
:
M
—
S}
be
a
timelike
isoparametric
surface such
that
the
mean
curvature
H
and
the
Gauss
curvature
«
satisfy
H?
—-K+1#40,
then,
by
a
transformation
in
Et,
tt
can
be
written
as
the
following
surface:
1 1
1
1
Fa
(u+v),acos
Fiat
v),
Bosh
Te
(u
—
v),
bsinh
Jan
—
v)),
z(u,v)
=
(asin
where
a?
+
0?
=
1.
Theorem
1.4.
Let:
M
-
H®
be
a
timelike
isoparametric
surface such
that
the
mean
curvature
H
and
the
Gauss
curvature
«
satisfy
H*
—«K—1>
0,
then,
by
a
transformation
in
ES,
it
can
be
written
as
the
one
of
the
following
surfaces:
1
1
1 1
i
u,v)
=
(asinh
—=(u-+t
v),
bcosh
——(u
—
v),a
cosh
——(u+v),
bsinh
——(u—
v)),
()
#(u2)
=
(asinh
T-(u
+
v),
boosh
T
(u—
9),
acosh
=
(u-+
v),bsinh
(uv)
where
a?
—
b?
=
1.
1
1
1 1
u+v),acos
—=-(u-+
v),
bsin
——(u
—
v),
bcos
——-(u
—
v)),
Jaq”
+?)
0098
Fee
(4+
0),
Bsn
Tae
(u—
v)obeos
Fe
(u—
0)
(ii)
a(u,v)
=
(asin
where
a*
—
6?
=
—1.
§2.
Preliminaries.
Let
Ky” be
the
m-dimensional
pseudo-Euclidean
space
with
the
natural
basis
€1,...,
€
its
metric
<
,
>
is
given
by
m3)
m—q
.
m
(2.1)
<2,y
>=
Liyi
—
S-
Ljyj,
tT,
y
EE,
i=1
j=m—qt+l
where
©
=
(11,
22,-..,Lm),
Y
=
(Y1,Y2)
++)
Ym).
The
n-dimensional
de
Sitter
space
S?
and
n-dimensional
anti-de
Sitter
space
Hf’
are
defined
by
(2.2)
t=
(2
CET!
:<
2,2
>=1),
(2.3)
HY
=
(2
€
EST?
:<.a4,2
>=
—1).
It
is
well
known
that
S?
and
Hi}
are
the
complete
connected
pseudo-Riemannian
hypersur-
faces
with
constant
sectional
curvature
1
and
-1
in
Eft!
and
E%*"',
respectively
({O]).
Let
N
be
a
pseudo-Riemannian
manifold
with
the
pseudo-Riemannian
metric
g
and
M
be
a
submanifold
of
N.
If
the
pseudo-Riemannian
metric
9
of
N
induces
a
Riemannian
metric
ISOPARAMETRIC
SURFACES
3
g
(respectively,
a
pseudo-Riemannian
metric,
a
degenerate
quadric
form)
on
M,
then
M
is
called
a
spacelike
(respectively,
timelike,
degenerate)
submanifold.
We
denote
by
V
the
covariant
differentiation
with
respect
to
the
indefinite
Riemannian
metric
of
E{
(or
E3)
and
by
V
and
V
the
covariant
differentiations
with
respect
to
the
induced
metric
of
S?
(or
H?)
and
M,
respectively.
We
denote
by
n(x)
=
—ez,
(x
€
S3,
e=1,
€
Hj,
«
=
-1),
the
normal
vector
field
of
S?
(or
H?)
in
E}
(or
E$);
€,
the
normal
vector
field
of
M
in
S}
(or
Hf).
Then,
considering
that
M
is
locally
embedded
in
S?
(or
H}),
we
have
the
following
Gauss’s
and
Weingarten’s
formulas.
VxY
=VxY+<X,Y>n
(2.4)
VxY
=VxY
+A(X,
VE
Vx
=
—A(X),
where
X
and
Y
are
tangent
vector
fields
on
M,
and
A
is
a
field
of
type
(1,1)
tensor
(Weingarten
operator)
on
M
corresponding
to
€,
i.e.,
(2.5)
<
A(X),
Y
>=
A(X,Y)
<€,E>.
Proposition
2.1.
Let
x
:
M
—
S#
(or
H})
be
a
timelike
surface
in
S?
(or
H?).
Then
the
Weingarten
operator
A
of
x
has
real
eigenvalues
if
and
only
if
the
mean
curvature
H
and
the
Gauss
curvature
k
of
x
satisfying
H*
—k
>
0.
Proof.
Let
«
:
M
—
S}
(or
H})
be
a
timelike
surface
and
{e1,e2}
be
a
local
pseudo-
orthonormal
basis
of
TM
such
that
the
metric
of
x
is
given
by
ds*
=e”
(du?
—
dv’).
From
(2.4)
we
have
A(é1)
=
hyye1
—
hizee
A(e2)
=
haie1
—
hazea,
where
h;;
=
h(e;,e;).
Thus
A
has
real
eigenvalues
if
and
only
if
(hit
—_
hoo)?
—_
4(h?,
_—
hizh22)
=
4(H?
—_
kK)
=
0.
O
It
is
easy
to
see
by
Theorem
1.3
and
Proposition
2.1:
Corollary
2.1.
Let
x:
M
—
S?
be
a
timelike
isoparametric
surface
such
that
its
Wein-
garten
operator
has
real
eigenvalues,
then,
by
a
transformation
in
E7,
it
can
be
written
as
the
following
surface:
1
iat,
=
Ta
—
v),
bsinh
——(u
—v)),
/
2b
1
z(u,v)
=
(asin
ks
+
v),acos
—=—(u+
v),
bcosh
/
2a
/2a
where
a*
+
b?
=
1.
4
HUILI
LIU,
GUOSONG
ZHAO
83.
Spacelike
isoparematric
surfaces
in
S?
and
Hi.
In
this
section,
we
prove
the
Theorem
1.1
and
Theorem
1.2
given
in
section
1.
Let
zt:
M
—
S}
(or
H})
C
Ef
(or
E$)
be
a
spacelike
surface
of
3-dimensional
de
Sitter
space
S¥
(or
anti-de
Sitte
space
H?)
with
the
metric
given
by
(3.1)
g
=
2e”
(du?
+
du”)
=
2e”|dz|?
=
e”
(dz
@dzZ+
dz
@
dz),
where
z=
u+
iv,
dz
=
du+idv.
Then
from
<
2,5
>=
e
(for
S?3,
¢
=
1;
for
HS,
e
=
—1)
and
g
=<
da,
dx
>=
—
<
2,d*z
>=
e’
(dz
@
dz+
dz
@
dz)
we
have
<2z,0
>
=<
03,0
>H=<
Lz,0z
>=<
z,Lz
>=<
L,2zz
>=
0
(3.2)
<2,
Lez
>
SS
Uz,
0
zz
P<
Lz,
Lgz
DH
<
Lz,
023
>H<
Lz,L
zz
>=
O
<%z,03
>=
—-—
<2,20.3
>=e”.
We
use
O
L
O
_O
8 O
L
O
4
i
O
as
Sees
met
eal
ames
Fm
g=
=
-(—+i—).
"Oz
2'du
Av”
~*
7
2
O0u
dv
Let
A
be
the
Laplacian
of
g,
then
A
=
2e°
"0205,
k=
—e
"Wes,
where
«
is
the
Gauss
curvature
of
g.
We
choose
€
€
S}
(or
H#)
such
that
<€,%z
>=<
£,%2
>=<f,2>=0,
<&,€
>=
—1.
Then
we
have
(3.3)
Lez
=W2zlzt+
pl,
p=—<Bzz,€>.
The
mean
curvture
H
of
z
is
given
by
(3.4)
HE
=e
"225
+
EL.
If
H
#0,
we
have
(3.5)
Oo
p=
—H'e™”
<
Gez,02z
>.
Let
®
=
ydz?.
Then
@
is
global
defined
and
&?
is
independent
of
the
choise
of
€.
For
the
surface
x
we
have
the
following
structure
equations:
Leg
=
WzLz
+
v&
Lez
=
—ce"
n+
Hervé
(3.6)
Lez
=
Welz
+
YE
€,
=
Hx,+
ye
“xz
&z
=
ge
“xr,
+
Hz;.
ISOPARAMETRIC
SURFACES
or
From
@zzz
=
£zzz
we
obtain
the
integrability
conditions
for
the
structure
equations
of
x:
(3.7)
wes
+e
|p|?
=
ce”
+
H?e”
.
z=
He”
that
is
(3
8)
Pz
=
He”
—~kK+e
*"
|p|?
=
H?
-e.
By
(2.4)
and
(2.5)
we
have
1,
(3.9)
p=
~ge
o(hu
an
hoe
+
2ihy2),
where
h,;
is
the
second
fundamental
form
of
z.
If
¢
:
M
—
Sj
(or
H})
is
a
isoparametric
surface,
then
by the
definition
we
know
that
the
Gauss
curvature
«
and
the
mean
curvature
H
of
x
are
constant.
From
(3.8)
we
know
that
yz
=
0
and
e~?”|y|?
is
constant.
If
y
=
0,
the
surface
is
totally
umbilical.
If
y
4
0,
from
(3.8)
we
have
0
=
A(loge~*”
|p|?)
=
A(—2w
+
log
y
+
log)
=
—2Aw.
Therefore
we
get
1
K=-—e
"wiz
=
5
Au
=
(),
The
surface
is
flat
and
we
can
choose
the
coordinate
z
such
that
w
=
0.
In
this
case,
(3.8)
becomes
(3.10)
=
Be
=
0
lel?
=
H?
-«.
But
yz
=
0
and
|y|?
=
constant
yield
that
~
is
constant.
Then
we
have
the
following
structure
equations
for
the
surface
x:
Lez
=
€
Leg
=
—Ex
+
HE
(3.11)
Lez
=
PF
Es
=
He,
+
PLZ
€z
=
pt,
+
Herz,
where
|y|?
=
H?
—
e.
By
a
transformation
of
z
we can
assume
that
y=
p>
0.
We
solve
the
equations
(3.11)
under
the
conditions
y?
=
H?
—e
and
y
>
0.
From
py
=
422
=
“ez
we
get
Fy,
=
0.
Then
the
surface
x
can
be
written
as
(3.12)
x=
f(u)+g(v),
f(u),g(v)
€
FE}
(or
Ep).
6
HUILI
LIU,
GUOSONG
ZHAO
From
(3.11)
and
(3.12)
we
have
(H
—
9)fuu
—
(H+
9)
toy
=
Sepa;
then
(H
—
vp)
f"(u)
—
dey
f(u)
=a
(H+
y)g"'(v)
+
4depg(v)
=
a,
where
a
is
constant
vector
in
E{
(or
E$).
By
a
translation
in
E}
(or
Ej)
we
may
assume
that
a
=
0.
Then
we
obtain
elt
_
Ae
f"(u)
=
Ag)
(3.13)
7,
—
AEP
g
(v)
=
Hag”
yp”?
=
H?
~e.
(a)
When
¢
=
1,
(3.13)
is
f"(u)
=
2
p(w)
HI
—
(3.14)
W(,)
—
749
9
U)=
5
re
(H
+
y)(H
—
y)
=1.
Therefore
4
4
f(u)
=
cg
sinh
(
=
u)
+
c4
cosh
(
r
u)
H—p
H-yp
(3.15)
for
H-w>Oor
_
H
—
(3.16)
;
g(v)
=
¢3
sinh
(
7)
+
c4
cosh
(
|
;
for
H
—
»
<
0.
The
surface
is
congruent
to
the
surface
(ii)
of
Theorem
1.1.
(b)
When
¢
=
—1,
(8.13)
is
f"(u)
=
2
fF
(u)
yp-—fH
4
(3.17)
Y
g(v)
yp?
=
B41,
WW
_
ISOPARAMETRIC
SURFACES
7
Therefore
me
al
4p
4ip
f(u)
=
cy
sinh
(\/-
=
7
+
c3
cosh
(
D
=)
+
h
4p
C4
COS
ot
a
y
—
H
>
0.
The
surface
is
congruent
to
the
surface
(ii)
of
Theorem
1.2.
This
completes
the
proof
of
the
Theorem
1.1
and
Theorem
1.2.
(3.18)
a(v)
=
casinh
(f=?
§4.
Timelike
isoparematric
surfaces
in
S?
and
H3.
In
this
section,
we
prove
the
Theorem
1.3
and
Theorem
1.4
given
in
section
1.
Let
a:
M
-
S?
(or
H?)
Cc
Ef
(or
E$)
be
a
timelike
surface
of
3-dimensional
de
Sitter
space
S3
(or
anti-de
Sitte
space H?)
with
the
metric
given
by
(4.1)
g
=e"
(du
®
duv+
dv
@
du).
Then
from
<
2,2
>=e
(for
S?,
«
=
1;
for
H3,
«
=
—1)
and
g
=<
dz,
dr
>=
—
<
z,d*z
>=
e“”
(du
@
du
+
dv
@
du)
we
have
<
fy,
>
=<
Ly,
UL
>H<
Ly,
Ly
>
=<
Ly,
Ly
>=<
L,0yy
>=
0
(4.2)
<2,
2y0
>
==
Cy
Ban
><
Lay
Soy
P<
My,
Day
>
HK
Fy.
Fay
>=
0
<
Ly,
Ly
>
=
—
<2L,Lyy
>=e”™.
We
use
3 3
Oon=
—,
AW==—.
Ou
Oy
Let
A
be
the
Laplacian
of
g,
then
A
—
2e°
Oy»;
K
=
Ee
Way,
where
«
is
the
Gauss
curvature
of
g.
We
choose
€
€
S?
(or
H)
such
that
<
€,t,
>=<
fa,
>=<
fe
>=0,
<é,f>=—1.
Then
we
have
(4.3)
‘
=
WyTy
+
v6,
p=<
Cans
€
>
Lyy =
Wyly
+
WE,
pa<
Lyv,§
>.
The
mean
curvture
H
of
x
is
given
by
(4.4)
HE
=e
“tay
tex.
8
HUILI
LIU,
GUOSONG
ZHAO
If
H
£0,
we
have
-1—
(4.5)
p=
He
OS
Begs
Be
.
1
=
.
w
=H
ev
<
Lyy,
luv
>
-
Let
®
=
ydu?,
©
=
Wdv2.
Then
©
and
W
are
global
defined
and
®?
and
W?
are
independent
of
the
choise
of
€.
For
the
surface
x
we
have
the
following
structure
equations:
Luu
=
Wuly
+
ve
Lyy
=
—Ee"n
+
Hevé
(4.6)
Lyy
=
WyLy
+
We
Ey
=
—H2ry,—-
pe
"Ly
Ey
=
—We
“a,
—
Hx,.
FYOM
Lyyu
=
Luuy
aNd
Lyyy
=
Lyyy
we
obtain
the
integrability
conditions
for
the
structure
equations
of
z:
Wuy
—€
“yy
=
—ce”
—
He
(4.7)
Yy
=
Ae”
w,
=
Hye”
that
is
_)
=
He”
(4.8)
Wy
=
Hye”
K+e
yy
=
H*
+e.
If
«
:
M
->
S?
(or
H§)
is
a
isoparametric
surface,
then
by the
definition
we
know
that
the
Gauss
curvature
«
and
the
mean
curvature
H
of
x
are
constant.
From
(4.8)
we
know
that
yy
=
0,
vy
=
0
and
e~?”
yy
is
constant.
If
H?
-K
+e
4
0,
by
(4.8)
we
know
that
yy
#0.
Then
from
(4.8)
we
get
0
=
A(loge~*”
py)
=
A(—2w
+
logy
+
logy)
=
—2Aw.
Therefore
we
get
1
(4.9)
K=—-e
"Wy
=
~
Aw
=
QJ.
The
surface
is
flat
and
we
can
choose
the
coordinate
(u,v)
such
that
w
=
0.
In
this
case,
(4.8)
becomes
Yy
=
A,
=0
(4.10)
vy,
=
H,
=0
yy
=
H*
+e.
ISOPARAMETRIC
SURFACES
9
But
gy
=
0,
dy
=
0
and
yy
=
constant
yield
that
y
and
7
are
constant.
Then
we
have
the
following
structure
equations
for
the
surface
z:
Luu
=
pt
Luy
=
—EX+
HE
(4.11)
Loy
=
WE
Cu
=
—
Hay
—
PLy
oa
=
Lay
—
Hz,
where
pp
=
H*
+.
We
solve
the
equations
(4.11)
under
the
conditions
yy
>
0.
From
(4.11)
we
have
Lov
+
WH
ty,
—
Epa
=
0.
By
a
parametric
transformation
(u,v)
>
(4/|plu,
\/|w|v)
we
obtain
Cuuu
+
2H
(signy)ryy,
—
ex
=
0
Dvn
+
2H
(sign)
yy
—
ex
=
0,
where
signy
=
1,
when
y
>
0;
signy
=
—1,
when
y
<
0.
If
pw
>
0,
then
(4.13)
becomes:
Cuuuy
+
2H
(signy)ryy,
—
ex
=
0
Lyvoy
+
2A
(signy)fyy
—
ex
=
0.
(4.13)
(4.14)
(a)
When
¢
=
1,
(4.15)
(u,v)
=
cy
sinA(u+
v)
+
cg
cos
A(ut
v)
+
€3
cosh
p(u
—
v)
+
cg
sinh
pp(u
—
v),
where
€1,
C2,
C3,
C4
€
Ef
are
constant
vectors
and
?
=
H(signy)
+
VH?
+1,
p=
—H(signy)
+
/
H2
+1.
The
surface
is
congruent
to
the
surface
given
by
Theorem
1.3.
(b)
When
e
=
—1,
(4.16)
x(u,v)
=
c,
sinh
A(u
+
v)
+c
cosh
A(u
—
v)
+.€3
cosh
p(u
+
v)
+
casinh
p(u
—
v),
where
Cy,
C2,
C3,
Ca
€
ES
are
constant
vectors
and
\*
=
—H(signy)+
VH?-1,
p=
—H(signy)
—
/
H2
—
1,
—H(signy)
>
0.
The
surface
is
congruent
to
the
surface
(i)
given
by
Theorem
1.4.
(4.17)
z(u,v)
=c,sinA(u
+
v)
+
c2cosA(ut+
v)
+
¢3
sin
p(u
—
v)
+
c4
cos
u(u
—
v),
where
cj,
C2,
¢3,
c4
€
Ey
are
constant
vectors
and
?
=
H(signy)
—
/H?2-1,
p=
H(signy)+/H?2-1,
H(signy)
>
0.
The
surface
is
congruent
to
the
surface
(ii)
given
by
Theorem
1.4.
This
completes
the
proof
of
the
Theorem
1.3
and
Theorem
1.4.
Acknowledgements.
We
would
like
to
thank
Professor
U.
Simon
for
his
hospitality
during
our
research
stay
at
the
TU
Berlin.
10
HUILI
LIU,
GUOSONG
ZHAO
REFERENCES
[CE]
T.
Cecil,,
Lie
Sphere
Geometry:
with
Applications
to
Submanifolds,
New
York,
Springer,
1992.
[CHEN]
B.
Y.
Chen,,
Riemannian
Submanifolds,
Handbook
of
Differential
Geometry,
North
Holland,
Vol-
ume
1,
1999.
[HA]
J.
Hahn,
Isoparametric
hypersurfaces
in
the
Pseudo-Riemannian
space
forms,
Math.
Z.
187
(1984),
195-208.
[MA]
M.
A.
Magid,
Lorentzian
isoparametric
hypersurfaces,
Pacific
J.
Math.
118
(1985),
165-197.
[N-R]
R.
Niebergall
and
P.
J.
Ryan,
Isoparametric
hypersurfaces—The
affine
case,
Geometry
and
Topol-
ogy
of
Submanifolds
V,
World
Scientific
(1993),
201-214.
[NO]
K.
Nomizu,
On
isoparametric
hypersurfaces
in
the
Lorentzian
space
forms,
Japan
J.
Math.
7
(1981),
217-226.
[O]
B.
O’Niell,
Semi-Riemannian
Geometry,
Academic
Press,
Orland,
1983.
Hult
Liu:
DEPARTMENT
OF
MATHEMATICS,
NORTHEASTERN
UNIVERSITY,SHENYANG
110006,
P.
R.
CHIna
E-mail
address:
GUOSONG
ZHAO:
DEPARTMENT
OF
MATHEMATICS,
SICHUAN
UNIVERSITY,
CHENGDU
610064,
P.
R.
CHINA
E-mail
address:
