Spinorial Characterizations of Surfaces into 3-dimensional pseudo-Riemannian Space Forms

0.13 MB·English

by  M. -A. Lawn

Spinorial Characterizations of Surfaces into
3-dimensional pseudo-Riemannian Space Forms
Marie-Am´elie Lawn∗ and Julien Roth†
0
1
0
2 Abstract

n
Wegiveaspinorialcharacterizationofisometricallyimmersedsurfaces
a
ofarbitrarysignatureinto3-dimensionalpseudo-Riemannianspaceforms.
J
For Lorentzian surfaces, this generalizes a recent work of the first author
5 inR2,1 tootherLorentzian spaceforms. Wealsocharacterizeimmersions
2
of Riemannian surfaces in these spaces. From this we can deduce analo-

gousresultsfortimelikeimmersions ofLorentziansurfacesinspaceforms
]
G of corresponding signature, as well as for spacelike and timelike immer-
D sions of surfaces of signature (0,2), hence achieving a complete spinorial
description for this class of pseudo-Riemannian immersions.
.
h
keywords: Dirac Operator, Killing Spinors, Isometric Immersions, Gauss and
t
a Codazzi Equations.
m
[ subjclass: DifferentialGeometry,GlobalAnalysis,53C27,53C40,53C80,58C40.

1
v
1 Introduction
8
9
1 A fundamental question in the theory of submanifolds is to know whether a
4 (pseudo-)Riemannian manifold (Mp,q,g) can be isometrically immersed into a
. r,s
1 fixed ambient manifold (M ,g). In this paper, we focus on the case of hyper-
0 surfaces,thatis,codimension1. Whentheambientspaceisaspaceform,asthe
0 pseudo-Euclidean space Rp,q and the pseudo-spheres Sp,q of positive constant
1 curvature,orthe pseudo-hyperbolicspaces Hp,q ofnegative constantcurvature,
:
v the answer is given by the well-known fundamental theorem of hypersurfaces:
i
X
Theorem. [9](Mp,q,g)beapseudo-Riemannianmanifoldwithsignature(p,q),
r p+q =n. Let A be a symmetric Codazzi tensor , that is, d∇A=0, satisying
a
R(X,Y)Z =δ hA(Y),ZiA(X)−hA(X),ZiA(Y) +κ hY,ZiX−hX,ZiY
h i h i
with κ∈R for all x∈M and X,Y,Z ∈T M.
x
Then, if δ = 1 (resp. δ = −1), there exists locally a spacelike (resp. timelike)
isometric immersion of M in Mp+1,q(κ) (resp. Mp,q+1(κ)).
IntheRiemanniancaseandforsmalldimensions(n=2or3),anothernec-
essary and sufficient condition is now well-known. This condition is expressed
∗[e-mail]
†[e-mail]
1
in spinorial terms, namely, by the existence of a special spinor field. This work
initiated by Friedrich [4] in the late 90’s for surfaces of R3 was generalized for
surfaces of S3 and H3 [8] and other 3-dimensional homogeneous manifolds [10].
The first author [5] uses this approach to give a spinorial characterization
of space-likeimmersionsof Lorentziansurfacesin the Minkowskispace R2,1. In
this paper, we give a generalization of this result to Lorentzian or Riemannian
surfaces into one of the three Lorentzian space forms, R2,1, S2,1 or H2,1. These
finally allows us to give a complete spinorial characterization for spacelike as
e
well as for timelike immersions of surfaces of arbitrary signature into pseudo-
Riemannian space forms.
We will begin by a section of recalls about extrinsic pseudo-Riemannian
spin geometry. For further details, one refers to [1, 2] for basic facts about spin
geometry and [1, 3, 7] for the extrinsic aspect.
2 Preliminaries
2.1 Pseudo-Riemannian spin geometry
Let(Mp,q,g), p+q=2,be anorientedpseudo-Riemanniansurfaceofarbitrary
signature isometrically immersed into a three-dimensional pseudo-Riemannian
spin manifold (Nr,s,g). We introduce the parameter ε as follows: ε = i if the
immersion is timelike and ε = 1 if the immersion is spacelike. Let ν be a unit
vector normal to M. The fact that M is oriented implies that M carries a
spin structure induced from the spin structure of N and we have the following
identification of the spinor bundles and Clifford multiplications:
ΣN ≡ΣM.
|M
(cid:26) X ·ϕ|M = εν•X •ϕ)|M,
(cid:0)
where · and • arethe Cliffordmultiplications, respectivelyonM and N. More-
over, we have the following well-known spinorial Gauss formula
ε
(1) ∇ ϕ=∇ ϕ− A(X)·ϕ,
X X
2
where A is the shape operator of the immersion. Finally, we recall the Ricci
identity on M
1
(2) R(e ,e )ϕ= ε ε R e ·e ·ϕ,
1 2 1 2 1221 1 2
2
where e ,e is a local orthonormal frame of M and ε =g(e ,e ).
1 2 j j j
The complex volume element on the surface depends on the signature and is
defined by
ωC =iq+1e ·e .
p,q 1 2
Obviously ωC 2 =1 independently of the signature and the action of ωC splits
p,q
ΣM into two eigenspaces Σ±M of real dimension 2. Therefore, a spinor field
ϕ can be written as ϕ = ϕ+ +ϕ− with ωC ·ϕ± = ±ϕ±. Finally, we denote
ϕ=ωC·ϕ=ϕ+−ϕ−.
2
2.2 Restricted Killing spinors
Let (Mp,q,g), p + q = 2 be a surface of the pseudo-Riemannian space form
Mr,s(κ), r+s = 3, p 6 r, q 6 s. This space form carries a Killing spinor ϕ,
that is satisfying ∇ ϕ = λX •ϕ, with κ = 4λ2. From the Gauss formula (1),
X
the restriction of ϕ on M satisfies the equation
ε
(3) ∇ ϕ= A(X)·ϕ+λX •ϕ.
X
2
But we have
X •ϕ=ε2ν•ν•X •ϕ=−ε2ν•X •ν•ϕ=−εX·(ν•ϕ).
Moreover,the complex volume element ωC =−ise •e •ν of Mr,s(κ) over M
r,s 1 2
acts as the identity on ΣMr,s(κ) ≡ΣM. Thus, we have
|M
ν•ϕ = ωC •ϕ=−isν•e •e •ν•ϕ
r,s 1 2
= isν•e •ν•e •ϕ
1 2
= isε2(εν•e )•(ενe •ϕ)
1 2
= isε2e ·e ·ϕ.
1 2
Hence a simple case by case computation shows that we have
X •ϕ=−isε3X·e ·e ·ϕ=iX·ωC ·ϕ=iX·ϕ.
1 2 p,q
in the six possible cases (for the three possible signatures (2,0), (1,1), (0,2) of
the surface with respectively ε=1 or i).
Wewillcallaspinorsolutionofequation(3)arealspecialKillingspinor(RSK)-
spinor if ε∈R, and an imaginary special Killing spinor (ISK)-spinor if ε∈iR.
2.3 Norm assumptions
Inthis section,we precisethe normassumptionsuseful forthe statementofthe
main result. Let (Mp,q,g) be a pseudo-Riemannian surface and ϕ spinor field
on M. Let ε = 1 or i and λ ∈ R or iR. We say that ϕ satisfies the norm
assumption N (p,q,λ,ε) if the following holds:
±
1. For p=2,q =0 or p=0,q=2:
• If ε=1, then X|ϕ |2 =±2ℜehiηX·ϕ,ϕi.
1
• If ε=i, then Xhϕ,ϕi=±2ℜehiηX·ϕ ,ϕ i.
1 1
2. For p=1,q =1: ϕ is non-isotropic
3 The main result
We now state the main result of the present paper.
Theorem 1. Let (Mp,q,g), p+q =2 be an oriented pseudo-Riemannian man-
ifold. Let H be a real-valued function. Then, the three following statements are
equivalent:
3
1. There exist two nowhere vanishing spinor fields ϕ and ϕ satisfying the
1 2
norm assumptions N (p,q,λ,ε) and N (p,q,λ,ε) respectively and
+ −
Dϕ =2εHϕ +2iλϕ and Dϕ =−2εHϕ −2iλϕ .
1 1 1 2 2 2
2. There exist two spinor fields ϕ and ϕ satisfying
1 2
ε ε
∇ ϕ = A(X)·ϕ −iλX·ϕ , and ∇ ϕ =− A(X)·ϕ +iλX·ϕ ,
X 1 2 1 1 X 2 2 1 2
where A is a g-symmetric endomorphism and H =−1tr(A).
2
3. ThereexistsalocalisometricimmersionfromM intothe(pseudo)-Riemannian
space form Mp+1,q(4λ2) (resp. Mp,q+1(4λ2)) if ε = 1 (resp. ε = i) with
mean curvature H and shape operator A.
Remark 1. Note that, in this result, two spinor fields are needed to get an
isometric immersion. Nevertheless, for the case of Riemannian surfaces in Rie-
mannian space forms (Friedrich [4] and Morel [8]) only one spinor solution
of one of the two equations is sufficient. This is also the case for surfaces of
signature (0,2) in space forms of signature (0,3).
In order to prove this theorem, we give two technical lemmas.
Lemma 3.1. Let (Mp,q,g) be an oriented (pseudo)-Riemannian surface and
η,λ two complex numbers. If M carries a spinor field satisfying
∇ ϕ=ηA(X)·ϕ+iλX ·ϕ,
X
then, we have
ε ε R +4η2det(A)−λ2 e ·e ·ϕ=2ηd∇A(e ,e )·ϕ.
1 2 1212 1 2 1 2
(cid:0) (cid:1)
Proof : An easy computation yields
∇ ∇ ϕ = η∇ A(Y)·ϕ+η2A(Y)·A(X)·ϕ+iηλA(Y)·X ·ωC·ϕ
X Y X
+iλ∇ Y ·ωC·ϕ+iηλY ·ωC·A(X)·ϕ−λ2Y ·ωC·X·ωCϕ.
X
Hence (the other terms vanish by symmetry)
R(e ,e )ϕ = ∇ ∇ ϕ−∇ ∇ ϕ−∇ ϕ
1 2 e1 e2 e2 e1 [e1,e2]
= η ∇ A(e )−∇ A(e )−A([e ,e ]) ϕ+η2(A(e )A(e )−A(e )A(e ))ϕ
e1 2 e2 1 1 2 2 1 1 2
−(cid:0)λ2(e ·ωC·e ωC−e ·ωC·e ωC)ϕ.(cid:1)
2 1 1 2
Since we have
A(e )A(e )−A(e )A(e )=−2det(A)
2 1 1 2
and
C C C C C 2 C 2
e ·ω ·e ω −e ·ω ·e ·ω =e ·e · ω −e ·e · ω =2e ·e ,
2 1 1 2 1 2 2 1 1 2
(cid:0) (cid:1) (cid:0) (cid:1)
by the Ricci identity (2), we get
1
ε ε R e e ·ϕ=ηd∇A(e ,e )−2η2det(A)e ·e ϕ−2λ2e ·e ·ϕ,
1 2 1221 1 2 1 2 1 2 1 2
2
4
and finally
(4) (−ε ε R +4η2det(A)+4λ2)e ·e ·ϕ=2ηd∇A(e ,e )·ϕ.
1 2 1212 1 2 1 2
(cid:3)
Now, we give this second lemma
Lemma 3.2. Let (Mp,q,g) be an oriented (pseudo)-Riemannian surface and λ
a complex number. If M carries a spinor field solution of the equation
(5) Dϕ=±(εHϕ+2iλϕ)
and satisfying the norm assumption N (p,q,λ,ε), then this spinor satisfies
±
ε
∇ ϕ=± A(X)·ϕ−iλX ·ϕ .
X
2
(cid:16) (cid:17)
Proof : We give the proof for the sign +. The other case is strictly the same .
Case of signature (1,1): We define the endomorphism B by
ϕ
(B )i =g(B (e ),e )=β (e , e ):=hε∇ ϕ, e ·ϕi.
ϕ j ϕ i j ϕ i j ei j
Obviously Using ei·ϕ± as a normalized dual frame of Σ∓M and the same
hϕ+,ϕ−i
proof as in [5] we can show that
1
h∇ ϕ,e ·ϕ±i=hε∇ ϕ,εe ·ϕ±i=− hB (X)·ϕ,e ·ϕ∓i.
X i X i 2εhϕ+,ϕ−i ϕ i
and hence ∇ ϕ=− 1 B (X)·ϕ. Moreover
X 2εhϕ+,ϕ−i ϕ
β (e , e ) = h∇ ϕ, e ·ϕi=−hε∇ ϕ, e2·e ·ϕi
ϕ 1 2 e1 2 e1 1 2
= −hεe ·∇ ϕ, e ·e ·ϕi=−hεDϕ+εe ·∇ ϕ, e ·e ·ϕi
1 e1 1 2 2 e2 1 2
= −ε2Hhϕ, e ·e ·ϕi−h2iελωC·ϕ, e ·e ·ϕi+β (e ,e )
1 2 1 2 ϕ 2 1
C
= −h2iελω ·ϕ, e ·e ·ϕi+β (e ,e ),
1 2 ϕ 2 1
since for any ϕ,ψ ∈Γ(ΣM)
hϕ, e ·e ·ψi=he ·e ·ϕ, ψi=−he ·e ·ϕ, ψi=−hϕ, e ·e ·ψi=0.
1 2 2 1 1 2 1 2
Let now consider the decomposition β (X,Y) = S (X,Y)+T (X,Y) in the
ϕ ϕ ϕ
symmetric part S and antisymmetric part T . Hence, we see easily that if
ϕ ϕ
λ/ε ∈ iR, then β is symmetric, i.e., T = 0. and if λ/ε ∈ R, then T (X) =
ϕ ϕ ϕ
2iλ/εωC·X. In the two cases, we have
ε
∇ ϕ= A(X)·ϕ−iλX·ϕ,
X
2
by setting A=2S . We verify easily that tr(A)=2tr(S )=2tr(B )=−2H.
ϕ ϕ ϕ
Case of signature (2,0) or (0,2): The proof is fairly standard following
the technique used in [4], [8] and [10]. We consider the tensors Q± defined by
ϕ
Q±(X,Y)=ℜe ε∇ ϕ±,Y ·ϕ∓ .
ϕ X
(cid:10) (cid:11)
5
Then, we have
tr(Q±)=−ℜe εDϕ±,ϕ∓ =−ℜe ε(εH ±2iλϕ∓,ϕ∓ =−ε2 H ±2ℜe(λ) |ϕ∓|2.
ϕ
(cid:10) (cid:11) (cid:10) (cid:11) (cid:0) (cid:1)
Moreover,we have the following defect of symmetry of Q±,
ϕ
Q±(e ,e ) = ℜe ε∇ ϕ±,e ·ϕ∓ =ℜe εe ·∇ ϕ±,e ·e ·ϕ∓
ϕ 1 2 e1 2 1 e1 1 2
= ℜe(cid:10)εDϕ±,e ·e ·ϕ(cid:11)∓ −ℜ(cid:10)e ε∇ ϕ±,e ·e ·ϕ∓ (cid:11)
1 2 e2 1 2
= ℜe(cid:10)(ε2H ±2iελ)ϕ∓,e(cid:11) ·e ·(cid:10)ϕ∓ +ℜe ε∇ ϕ±,e(cid:11) ·ϕ∓
1 2 e2 1
= 2ℜe(cid:10)(ελ)|ϕ∓|2+Q±(e ,e ). (cid:11) (cid:10) (cid:11)
ϕ 2 1
ϕ± ϕ±
Then, using the fact that εe · and εe · form a local orthonormal
1 |ϕ±|2 2 |ϕ±|2
frame of Σ∓M for the real scalar product ℜeh·,·i, we see easily that
Q+(X) Q−(X)
∇ ϕ+ =ε ϕ ·ϕ− and ∇ ϕ− =ε ϕ ·ϕ+.
X |ϕ−|2 X |ϕ|+|2
Q+ Q−
We setW = ϕ − ϕ . Fromthe abovecomputations,wehaveimmediately
|ϕ−|2 |ϕ+|2
that W + ℜe(iλ/ε)Id is symmetric and trace-free. Now, we will show that
W +ℜe(iλ/ε)Id is of rank at most 1. First, we have
X|ϕ+|2+ε2X|ϕ−|2 =2ℜe εW(X)·ϕ−,ϕ+ .
(cid:10) (cid:11)
Moreover,from the norm assuption N(p,q,λ,ε), we have
X|ϕ+|2+ε2X|ϕ−|2 =2ℜehiλX ·ϕ,ϕi=4ℜe iλX ·ϕ−,ϕ+ .
(cid:10) (cid:11)
We deduce immediately that W +2ℜe(iλ/ε)Id is of rank at most 1 and hence
vanishes identically since it is symmetric and trace-free. Thus, we have the
following relation
|ϕ+|2Q+−|ϕ−|2Q− =−2ℜe(iλ/ε)|ϕ+|2|ϕ−|2g.
ϕ ϕ
From now on, we will distinguish two cases.
• Case 1: iλ/ε∈R.
Then we are in one of these two possible situations: ε = i and λ ∈ R or ε = 1
and λ∈iR. The second situation was studied by Morel [8].
So we define the following tensor F := Q+ −Q− +2iελ(|ϕ+|2−|ϕ−|2)g. We
ϕ ϕ
have then
Q+(X) Q−(X)
∇ ϕ = ∇ ϕ++∇ ϕ− =ε ϕ ·ϕ++ε ϕ ·ϕ−
X X X |ϕ−|2 |ϕ−|2
F(X)
= ε ·(ϕ++ϕ−)−iλX ·ϕ−−iλX ·ϕ+
|ϕ|2
ε
= A(X)·ϕ−iλX ·ϕ,
2
where we have set A = 2F . We conclude by noticing that A is a symmetric
|ϕ2|
tensor with tr(A)=−2H.
• Case 2: iλ/ε∈iR.
6
Then we are in one of these two possible situations: ε= i and λ∈iR or ε =1
and λ∈R. The second situation was studied by Morel [8].
In this case, we have from the previous computations that W vanishes identi-
cally. So we set
Q+ Q−
ϕ ϕ
F = =
|ϕ−|2 |ϕ+|2
and then we have ∇ ϕ = F(X)·ϕ, where F(X) is defined by g(F(X),Y) =
X
F(X,Y). Nevertheless, F is not symmetric. We define the following symmetric
tensor A(X,Y)= 1 (F(X,Y)+F(Y,X)). We compute immediately
|ϕ|2
A(e ,e )=2F(e ,e )/|ϕ|2 , A(e ,e )=2F(e ,e )/|ϕ|2,
1 1 1 1 2 2 2 2
A(e ,e )=2F(e ,e )/|ϕ|2−2λ/ε and A(e ,e )=2F(e ,e )/|ϕ|2+2λ/ε.
1 2 1 1 2 2 2 2
Finally, we conclude that
ε ε
∇ ϕ= A(X)·ϕ+λX ·ω·ϕ= A(X)·ϕ−iλX ·ϕ.
X
2 2
(cid:3)
Now, we can give the proof of Theorem 1. We have already proven that 3.
implies2. whichimplies1. Moreover,Lemma3.2showsthat1. implies2. Now,
we will prove that 2. implies 3. For this, we use Lemma 3.1, but we need to
distinguish the three cases for the different signatures. Let ϕ=ϕ++ϕ−.
Case of signature (2,0): Here, ωC =ie e ,hencee ·e ·ϕ=−iωC·ϕ=−iϕ¯.
1 2 1 2
Hence formula (4) becomes
−i(−R +ε2det(A)+4λ2)ϕ¯=εd∇A(e ,e )·ϕ.
1212 1 2
G2,0 C2,0
| {z } | {z }
or equivalently εC ·ϕ± = ±iG ϕ∓. Applying two times this relation we
2,0 2,0
have finally
ε2||C ||2ϕ± =−G2 ϕ±.
2,0 2,0
Again we have two cases.
• Spacelike immersion: ε=1, M2,0 ֒→M3,0.
We refer to [4] for the immersion in R3,0 and to [8] for S3 and H3. Only one
(RSK)-spinor is needed.
• Timelike immersion: ε=i, M2,0 ֒→M2,1.
Two(ISK)-spinorsareneeded. Wededucefromtheaboverelationsbetweenϕ±
1
and ϕ± that hC ·ϕ ,ϕ i = 0. Moreover, in this case we have hϕ ,ϕ i = 0.
2 2,0 1 2 1 2
Thus,sincethespinorbundleΣM isofcomplexrank2,wehaveC ·ϕ =fϕ
2,0 1 1
where f is a complex-valued function over M. By taking the inner product by
ϕ , we see immediately that f only takes imaginary values, that is f =ih with
1
h real-valued. Thus, we have ±G ϕ± =ihϕ±. Since ϕ± do not vanish simul-
2,0 1 1 1
taneously, we deduce that h and G vanish identically. Thus C vanishes too
2,0
and the Gauss and Codazzi equation are satisfied. Then, we get the conclusion
by the fundamental theorem of hypersurfaces given above.
7
Case of signature (1,1): ωC =−e e , hence e ·e ·ϕ=−ωC·ϕ=−ϕ¯.
1 2 1 2
Hence formula (4) becomes
−(R +ε2det(A)+4λ2)ϕ¯=εd∇A(e ,e )·ϕ.
1212 1 2
G1,1 C1,1
| {z } | {z }
or equivalently εC ·ϕ± = G ϕ∓. Applying two times this relation we have
1,1 1,1
finally
ε2||C ||2ϕ± =G2 ϕ±.
1,1 1,1
• Spacelike immersion: ε=1, M1,1 ֒→M2,1.
Wereferto[5]fortheimmersioninR2,1. Letusconsidertheotherspaceforms.
Here again, we need two (RSK)-spinors. Since ϕ± do not vanish at the same
1
point, we have clearly that ||C ||=G2 >0. Moreover,we have
1,1 1,1
−||C ||2hϕ ,ϕ i = hC ·ϕ ,C ·ϕ i
1,1 1 2 1,1 1 1,1 2
= −G2 he ·e ϕ ,e ·e ·ϕ i
1,1 1 2 1 1 2 2
= G2 hϕ ,ϕ i.
1,1 1 2
Since hϕ ,ϕ i never vanishes, we deduce that ||C || = −G2 6 0. Conse-
1 2 1,1 1,1
quently, ||C || = G = 0. Moreover, C is not isotropic. Indeed, since
1,1 1,1 1,1
G =0, we have C ·ϕ =0 and thus C automatically vanishes as proved
1,1 1,1 1 1,1
in [5].
•Timelike immersion: ε=i,M1,1 ֒→M1,2. Itiseasytoseethatcomputations
similar to the one for the previous case give the result.
Two (ISK)-spinors are needed.
Case of signature (0,2) ωC =−ie e , hence e ·e ·ϕ=iωC·ϕ=iϕ¯.
1 2 1 2
Hence formula (4) becomes
i(−R +ε2det(A)+4λ2)ϕ¯=εd∇A(e ,e )·ϕ.
1212 1 2
G0,2 C0,2
| {z } | {z }
or equivalently εC ·ϕ± = ±iG ϕ∓. Applying two times this relation we
0,2 0,2
have finally
ε2||C ||2ϕ± =−G2 ϕ±.
0,2 0,2
• Spacelike immersion: ε=1, M0,2 ֒→M1,2.
Similar computations to the case M2,0 ֒→ M2,1 give the result. Two (ISK)-
spinors are needed.
• Timelike immersion: ε=i, M0,2 ֒→M0,3.
We get ||C ||2ϕ± = G2 ϕ±, hence C = G2 = 0 as the norm of C is
0,2 0,2 0,2 0,2 0,2
negativedefinite. ThisisasimilarcomputationtothecaseM2,0 ֒→M3,0. Only
one (ISK)-spinor is needed.
Let us summarize these results. In the tabular below we give the number of
(RSK)-(resp. (ISK)-)spinorsonthe surfaceMp,q solutionsofthespecialKilling
equation (3), or equivalently of the Dirac equation (5), which is sufficient for
the surface to be immersed, depending on the signature (p,q) and on the type
ε of the immersion.
8
ε 1 i
(p,q)
(0,2) 2 RSK-spinor 1 ISK-spinor
(1,1) 2 RSK-spinors 2 ISK-spinors
(2,0) 1 RSK-spinor 2 ISK-spinors
Table 1: Number of spinors needed
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