Lorentz surfaces in pseudo-Riemannian
space forms of neutral signature
with horizontal reflector lifts
Kazuyuki HASEGAWA (Kanazawa University)
Joint work with Dr. K. Miura (Ochanomizu Univ.)
0. Introduction.
f e
(M , g ) : oriented n-dimensional pseudo-Riemannian manifold
(M, g) : oriented pseudo-Riemannian submanifold
f
f : M → M : isometric immersion
H : mean curvature vector field
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Problem. f
: Classify f : M → M with H = 0
• f
Riemannian case : When M = S n(c), M = S 2(K) of constant
curvatures c > 0 and K > 0, a full immersion f with H = 0 is congru-
ent to the standard immersion (E. Calabi, J. Diff. Geom., 1967). The
f
twistor space of M = S n(c) and the twistor lift play an important
role.
f
• In this talk, we consider the problem above in the case of M = Qn(c) =
s
pseudo-Riemannian space form of constant curvature c and index s, and
M =Lorentz surfaces , in particular, n = 2m and s = m.
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• This talk is consists of
1. Examples of Lorentz surfaces in pseudo-Riemannian space forms
with H = 0.
2. Reflector spaces and reflector lifts.
3. A Rigidity theorem.
4. Applications.
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1. Examples of Lorentz surfaces with H = 0.
• In “K. Miura, Tsukuba J. math., 2007”, pseudo-Riemannian subman-
ifolds of constant curvature in pseudo-Riemannian space forms with
H = 0 are constructed from the Riemannian standard immersion us-
ing “ Wick rotations ”.
K[x] := K[x1, . . . , xn+1] : polynomial algebra in n + 1 variables
x1, . . . , xn+1 over K (K = R or C).
Kd[x] ⊂ K[x] : space of d-homogeneous polynomials
Rn+1 : pseudo-Euclidean space of dim=n + 1 and index=t
t
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⎧
n+1
X ⎪
⎨ −1
∂ (1 ≤ i ≤ t)
4Rn+1 := − εi , where εi =
t ∂x2
i
⎪
⎩ 1 (t + 1 ≤ i ≤ n + 1)
i=1
H(Rn+1) := Ker4Rn+1
t t
Hd(Rn+1) := H(Rn+1) ∩ Kd[x]
t t
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We define ρt : C[x] → C[x] by
⎧
⎪√
⎨ −1xi (1 ≤ i ≤ t)
ρt(xi) =
⎪
⎩ xi (t + 1 ≤ i ≤ n + 1)
• ρt is called “Wick rotation”.
Set σt := ρt ◦ ρt and Pt± := Ker(σt|R[x] ∓ idR[x]).
We define H± (Rn+1) := Pt± ∩ Hd(Rn+1).
d,t 0 0
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• Hd(Rn+1) = H+ (Rn+1) ⊕ H− (Rn+1)
0 d,t 0 d,t 0
{ui}i :orthonormal basis Hd(Rn+1) s.t.
0
u1, . . . , ul ∈ H− (Rn+1),
d,t 0
ul+1, . . . um+1 ∈ H+ (Rn+1),
d,t 0
m+1
X
(ui)2 = (x2 + · · · + x2 )2,
1 n+1
i=1
where l = l(d, t) = dim H− (Rn+1) and m+1 = m(d, t)+1 = dim Hd(Rn+1).
d,t 0 0
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Hereafter, we assume n = 2 and t = 1 (for simplicity). Then we see
that m = 2d + 1 and l = d. We define
Ui := Imρt(ui) (i = 1, . . . , d)
and
Ui := Reρt(ui) (i = d + 1, . . . , 2d + 1)
d(d+1)
Using these function, we define an immersion from φ : Q2(1) → Q2d(
1 d 2
)
by φ = (U1, . . . , U2d+1). Note that φ satisfies H = 0.
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For example, when d = 2 (n = 1, t = 1), φ : Q2(1) → Q4(3) is given by
1 2
√
3 1 2
U1 = ( +2x + y + z ), U2 = (y − z 2)
2 2 2
6 2
U3 = xy, U4 = zx, U5 = yz,
d(d+1)
• Composing homotheties and anti-isometries of Q2(1) and Q2d(
1 d 2
),
we can obtain immersions from Q2(2c/d(d + 1)) to Q2d(c) with H = 0
1 d
6
(c = 0). We denote this immersion by φd,c .
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2. Reflector spaces and reflector lifts.
(See “G. Jensen and M. Rigoli, Matematiche (Catania) 45 (1990), 407-
443. ”)
f e
(M , g ) : oriented 2m-dim. pseudo-Riemannian manifold of
neutral signature (index= 1 dim M )
f
2
f
Jx ∈ End(TxM ) s.t.
2
◦ Jx = I,
◦ (Jx)∗gx = −ex,
e g
◦ dim Ker(Jx − I) = dim Ker(Jx + I) = m,
◦ Jx preserves the orientation.
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f
Zx : the set of such Jx ∈ End(TxM )
[
f
Z(M ) := Zx
f
x∈M
(the bundle whose fibers are consist of para-complex structures)
f
• Z(M ) is called the f
reflector space of M , which is one of corre-
sponding objects to the twistor space in Riemannian geometry.
(M, g) : oriented Lorentz surface with para-complex structure
J ∈ Z(M ).
f
f : M → M : isometric immersion
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e f e
Def. : The map J : M → Z(M ) satisfying J (x)|TxM = f∗ ◦ Jx is
reflector lift of f .
e e e
Def. : The reflector lift is called horizontal if ∇J = 0, where ∇ is
f
the Levi-Civita connection of M .
e e
• ∇J = 0 ⇒ H = 0
• Surfaces with horizontal reflector lifts are corresponding to superminimal
surfaces in Riemannian cases.
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Prop. The immersion φm,c : Q2(2c/m(m + 1)) → Q2m(c) in §1
1 m
admits horizontal reflector lift.
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3. A Rigidity theorem.
At first, we recall the k-th fundamental forms αk .
f e
(M , g ) : oriented n-dim pseudo-Riemannian manifold
(M, g) : oriented pseudo-Riemannian submanifold
f
f : M → M : isometric immersion
e f
∇ : Levi-Civita connection of M .
e e e
We define ∇X1 := X1, ∇(X1, X2) = ∇X1 X2 and inductively for k ≥ 3
e e e
∇(X1, . . . , Xk ) = ∇X1 ∇(X2, . . . , Xk ),
where Xi ∈ Γ(T M ).
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We define
e
Osck (f ) := {(∇(X1, . . . , Xk ))x | Xi ∈ Γ(T M ), 1 ≤ i ≤ k}
x
and
[
k
Osc (f ) := Osck (f ).
x
x∈M
f
• T M = Osc1(f ) ⊂ Osc2(f ) ⊂ · · · ⊂ Osck (f ) ⊂ · · · ⊂ T M .
• We define deg(f ) =the minimum integer d such that Oscd−1(f ) (
Oscd(f ) and Oscd(f ) = Oscd+1(f ).
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• There exists the maximum number n (1 ≤ n ≤ deg(f )) such that
f
Osc1(f ), . . . , Oscn(f ) are subbundles on T M . Then we call f to be
nicely curved up to order n.
• When f is nicely curved up to order n, there exists the maximum
number m (1 ≤ m ≤ n) such that all induced metrics of subbundles
Osc1(f ), . . . , Oscm(f ) are nondegenerate. Then f is called
nondegenerately nicely curved
up to m.
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• 1 ≤ m ≤ n ≤ deg(f ).
• When f is called nondegenerately nicely curved up to m, we can
define the (k −1)-th normal space N k−1 which is defined the orthogonal
complement subspace of Osck−1(f ) in Osck (f ).
Def. : When f is nondegenerately nicely curved up to m, we
define (k + 1)-th fundamental form αk+1 as follows :
k
e
αk+1(X1, . . . , Xk+1) := (∇(X1, . . . , Xk+1)N ,
where Xi ∈ Γ(T M ) (1 ≤ i ≤ k + 1) and 1 ≤ k ≤ m − 1
f
• If M = Qn(c), αk+1 is symmetric.
s
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Thm. ¯
: Let f , f : M → Q2m(c) be an isometric immersions from
m
a connected Lorentz surface M with horizontal reflector lifts. If
¯
both immersions f and f are nondegenerately nicely curved up to
¯
m, then there exist an isometry Φ of Q2m(c) such that f = Φ ◦ f .
m
Remark. The immersion φm,c : Q2(2c/m(m + 1)) → Q2m(c) in §1
1 m
is nondegenerately nicely curved up to m.
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Cor. : Let f : M → Q2m(c) be an isometric immersions from a
m
connected Lorentz surface M with horizontal reflector lifts. If f is
nondegenerately nicely curved up to m and the Gaussian curvature
K is constant, then K = 2c/m(m + 1) and f is locally congruent
to φm,c.
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4. Applications.
V , W : vector spaces with inner products h , i
α : V × V · · · × V → W : symmetric k-multilinear map to W .
For the sake of simplification, we set
α(X k ) := α(X, .{z , X )
| .. }
k
for X ∈ V .
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Def. : We say that α is spacelike (resp. timelike) isotropic if
hα(uk ), α(uk )i is independent of the choice of all spacelike (resp.
timelike) unit vectors u. The number hα(uk ), α(uk )i is called space-
like (resp. timelike) isotropic constant of α.
• α is spacelike isotropic with spacelike isotropic constant λ ⇐⇒ α is
timelike isotropic with timelike isotropic constant (−1)k λ.
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Def. : We say that the k-th fundamental form αk is spacelike
(resp. timelike) isotropic if αk is spacelike (reps.
p timelike)
isotropic at each point p ∈ M . The function λk : M → R de-
fined by λk (p) := spacelike isotropic constant of αk
p is called the
spacelike (resp. timelike) isotoropic function. If the spacelike (resp.
timelike) isotoropic function λk is constant, then k-th fundamental
form αk is called constant spacelike (resp. timelike) isotropic .
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Prop. : Let f : M → Qn(c) be an isometric immersion with H =
s
0. If the higher fundamental forms αk are spacelike isotropic for
2 ≤ k ≤ m and their spacelike isotropic functions are everywhere
nonzero on M . Then we have n ≥ 2m and s ≥ m. In particular, if
n = 2m, then s = m and f admits horizontal reflector lift.
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Cor. : If f : M → Q2m(c) be an isometric immersion with H = 0
s
such that αk are spacelike isotropic for 2 ≤ k ≤ m whose isotropic
functions are everywhere nonzero and M has a constant Gaussian
curvature K, then we have s = m, K = 2c/m(m + 1) and f is
locally congruent to the immersion φm,c
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Consider the following condition :
(†) An isometric immersion f maps each null geodesic of M into
f
a totally isotropic and totally geodesic submanifold L of M .
e
• L is totally isotropic : ⇐⇒ g |L = 0.
e
• L is totally geodesic : ⇐⇒ ∇X Y ∈ Γ(T L) for all X, Y ∈ Γ(T L).
Remark. The immersion φm,c satisfies the condition (†).
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Cor. : Let f : M → Q2m(c) be an isometric immerison with
s
H = 0. If K = 2c/m(m + 1) and the condition (†) holds, then
s = m and f is locally congruent to the immersion φm,c.
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