Fundamental theorems of Lagrangian surfaces in S
2× S
2Makoto Kimura
Department of Mathematics, Faculty of Science and Engineering, Shimane University, Matsue, Shimane 690-8504, Japan
e-mail : mkimura@riko.shimane-u.ac.jp Kaoru Suizu
Seikyo Gakuen High School
Kawachi-nagano, Osaka, 586-8585, Japan e-mail : suizu@seikyo.ed.jp
(2000 Mathematics Subject Classification : 53C40.)
Abstract. Existence andSO(3)×SO(3)-congruence of Lagrangian immersion from oriented 2-dimensional Riemannian manifold to the Riemannian product of 2-spheres are studied.
In particular, we will show that two minimal Lagrangian immersions areSO(3)×SO(3)- congruent if and only if the corresponding angle functions are coincide.
1 Introduction
In this note we will discuss about existence andSO(3)×SO(3)-congruence of Lagrangian isometric immersions form 2-dimensional Riemannian manifold to the Riemannian product of unit 2-spheres, according to [3]. The author would like to thank Professor Young Jin Suh for inviting me to this international workshop.
2 Fundamental Theorems of Submanifolds
At first, we recall Fundamental Theorem of surfaces in 3-dimensional space forms. Let (M2, g) be a 2-dimensional Riemannian manifold, which is isometrically immersed in 3-dimensional real space formMf3(c) of constant sectional curvaturec.
We denote∇, R, hthe Levi-Civita connection, the curvature tensor and the second fundamental tensor ofM2, respectively. Then Gauss equation and Codazzi equation are written as follows:
g(R(X, Y)Z, W) =c(g(Y, Z)g(X, W)−g(X, Z)g(Y, W)) (2.1)
+h(Y, Z)h(X, W)−h(X, Z)h(Y, W), (∇Xh)(Y, Z) = (∇Yh)(X, Z).
(2.2)
119
Theorem 2.1. Let(M2, g)be a2-dimensional simply connected Riemannian man- ifold and let h be a symmetric (0,2)-tensor field on M. Suppose M satisfies both Gauss equation (2.1)and Codazzi equations (2.2). Then there exists an isometric immersionf : (M2, g)→Mf3(c), andf is unique up to isometries ofMf3(c).
Next we recall Fundamental Theorem of Lagrangian submanifolds in complex space forms. Let Mfn be a complex n-dimensional K¨ahler manifold with K¨ahler metric h , i and complex structure J. Let Mn be a Lagrangian submanifold in Mfwith second fundamental formσ: T M×T M →T⊥M, where T M and T⊥M denote the tangent bundle and the normal bundle ofM (inMf), respectively. Then for tangent vector fields X, Y, Z ∈ T M, tensor field T of type (0,3), defined by T(X, Y, Z) = hσ(X, Y), Zi is symmetric. Hence the second fundamental form of Lagrangian submanifolds in K¨ahler manifolds is essentially given byintrinsicinfor- mation.
LetMfn(4c) be a complex space form of constant holomorphic sectional curvature 4c. Then Gauss equation and Codazzi equation of Lagrangian submanifoldsM in Mfn(4c) are written as:
g(R(X, Y)Z, W) =c(g(Y, Z)g(X, W)−g(X, Z)g(Y, W)) (2.3)
+X
i
(T(Y, Z, ei)T(X, W, ei)−T(X, Z, ei)T(Y, W, ei)), (∇WT)(X, Y, Z) = (∇XT)(W, X, Z),
(2.4)
wheree1, e2, . . . , en is an orthonormal basis of tangent space of M.
Theorem 2.2. [1] Let(Mn, g)be ann-dimensional simply connected Riemannian manifold and letT be a symmetric(0,3)-tensor field onM. SupposeM satisfies both Gauss equation (2.3)and Codazzi equations (2.4). Then there exists a Lagrangian isometric immersion f : (Mn, g)→Mfn(4c), and f is unique up to (holomorphic) isometries ofMfn(4c).
With respect to general K¨ahler manifold Mfn and its Lagrangian submanifold Mn, Gauss and Codazzi equations on M do not guarantee the existence of La- grangian isometric immersion intoMf.
3 Lagrangian surfaces inS2×S2
In this section, we will discuss about existence and congruence of Lagrangian isometric immersion from 2-dimensional Riemannian manifold to the Riemannian product of 2-spheres. LetS2 be a unit sphere in R3. For any p∈S2, we define a linear transformationJ of the tangent spaceTpS2 ofS2at pas
(3.1) Jv=p×v
by the vector product × of R3, so J is a complex structure on S2. Then the special orthogonal groupSO(3) acts naturally forS2and is the isometry group for the Riemannian metric on S2 which is induced by the standard inner product of R3. Moreover SO(3) preserves J. Standard symplectic form θ onS2 is given by θp(u, v) = (p×u)·v, whereu, v ∈TpS2 and·is the induced Riemannian metric on S2 by the inclusionS2⊂R3.
We define a complex structure (J, J) onS2×S2by (3.2) (J, J)(X1, X2) = (JX1, JX2)
for all tangent vectors (X1, X2) to S2×S2. Let h , i be the product metric on S2×S2 defined by
h(X1, X2),(Y1, Y2)i=X1·Y1+X2·Y2.
Thenh, iis a Hermitian metric andS2×S2is a K¨ahler manifold with respect to the complex structure (J, J). S2×S2 is considered as a symplectic manifold with symplectic formθe= (pr1)∗θ+ (pr2)∗θ, where pr1,pr2 :S2×S2 →S2 are projec- tion maps into first factor and second factor, respectively, and θ is the standard symplectic form onS2.
Example 1. Let xi : Ii →S2 (i= 1,2) be curves in a 2-sphere with arc-length parameter si, and let x : I1×I2 → S2 be the product immersion defined by x(s1, s2) = (x1(s1), x2(s2)). Then clearlyxis a Lagrangian immersion. If κi (i = 1,2) are curvatures of spherical curvesxi, then we getx00i(si) =κi(si)Jx0i(si)−xi(si).
So we have
σ¡
∂/∂s1, ∂/∂s1
¢=¡
κ1(s1)Jx01(s1), 0¢ , (3.3)
σ¡
∂/∂s2, ∂/∂s2
¢=¡
0 , κ2(s2)Jx02(s2)¢ ,
and
2H = σ¡
∂/∂s1, ∂/∂s1
¢+σ¡
∂/∂s2, ∂/∂s2
¢
= ¡
κ1(s1)Jx01(s1), κ2(s2)Jx02(s2)¢ .
Consequently the product immersion xis minimal if and only if κ1≡κ2≡0, that is, eachxi is a great circle ofS2. Hence, we have
Proposition 3.1. Let x be a product immersion : M1×M2 → S2×S2. If x is a minimal immersion, then xis totally geodesic and each Mi (i = 1,2) is a great circle of S2.
LetPe be the tensor field of type (1,1) onS2×S2, defined by (3.4) Pe(X1, X2) = (X1,−X2).
Then we have (cf. [4])
Pe2= 1, (3.5)
hP X, Ye i=hX,P Ye i (3.6)
tracePe= 0, (3.7)
∇¯Pe= 0 (3.8)
where X,Y are any tangent vectors of S2 ×S2 and ¯∇ denotes the Levi-Civita connection of S2×S2. Pe is called the almost product structure of S2×S2. (3.2) and (3.4) imply
(3.9) PeJe=JeP .e
LetM2 be an oriented Riemannian manifold of dimension 2 and let (3.10) x:M2→S2×S2, x(p) = (x1(p), x2(p))
be a Lagrangian immersion, i.e.,x∗θe= 0. If{e1, e2}is an orthonormal basis for the tangent space TpM2 at p∈ M2, which is compatible with the orientation of M2, then {Jee1,Jee2} is an orthonormal basis for an orthogonal complementTp⊥M2 of TpM2. Thus{e1, e2,Jee 1,Jee 2}is an (oriented) orthonormal basis forTx(p)(S2×S2).
So we put
(3.11) P xe ∗X =x∗P X+Jxe ∗QX
forX∈TpM2 whereP andQare linear endomorphisms inTpM2. Then it follows from (3.5)∼(3.9) that
traceP = 0, P2−Q2= 1, P Q+QP = 0, (3.12)
hP X, Yi=hX, P Yi, hQX, Yi+hX, QYi= 0.
(3.13)
Then, from (3.12), there exists an orthonormal basis {e1, e2} of TpM2 compatible with the orientation ofM2 andϕ∈[−π/4, π/4] such that
(3.14)
(P e1= cos 2ϕe1, P e2=−cos 2ϕe2, Qe1=−sin 2ϕe2, Qe2= sin 2ϕe1.
Here we note that such{e1, e2} is uniquely determined up toei 7→ −ei (i= 1,2).
Clearly ϕis continuous and when ϕ∈(−π/4, π/4), ϕis differentiable. We call ϕ the angle function for a Lagrangian immersion x from an oriented 2-dimensional Riemannian manifoldM2 toS2×S2.
Note that the angle functionϕis essentially same as the K¨ahler angle (=π/2− 2ϕ) ofM2 inS2×S2 with respect to the complex structure (J,−J).
Example 2. Letφ± :S2→S2×S2be a Lagrangian immersion given by (x, y, z)7→
((x, y,∓z),(x, y,±z)), where (x, y, z) is an orthogonal coordinate system on R3. Then we can see that the angle functionϕofφ± is identically equal to±π/4.
4 Results
SO(3)×SO(3)-congruence Theorem for Lagrangian isometric immersionM2→ S2×S2 is described as:
Theorem 4.1. Let M2 be a connected and oriented 2-dimensional Riemannian manifold and let x1, x2 : M2 → S2 ×S2 be Lagrangian isometric immersions with which the angle functions ϕ1, ϕ2 take the values in (−π/4, π/4). We denote σi(i = 1,2) the second fundamental forms of xi, respectively, and Ti(X, Y, Z) = hσi(X, Y),JZi, the corresponding symmetric tensor fields one M2. Then there is an isometry g ∈ SO(3)×SO(3) such that x2 = g◦x1 if and only if ϕ1 = ϕ2, and T1=T2 hold.
With respect tominimal Lagrangian surfaces inS2×S2, we have:
Theorem 4.2. Let M2 be an oriented 2-dimensional Riemannian surface and x1, x2 : M2 → S2×S2 be minimal Lagrangian immersions. Let ϕi : M2 → (−π/4, π/4) be the angle function of xi(i = 1,2). Then there is an isometry g∈SO(3)×SO(3) such thatx2=g◦x1 if and only ifϕ1=ϕ2.
Theorem 4.3. Let M2 be a simply connected oriented Riemannian manifold of dimension2and letKbe the Gauss curvature ofM2. Suppose there exists a function ϕ:M2→(−π/4, π/4) such that
K= (sin22ϕ)/2−2k gradϕk2, 4ϕ+ 2kgrad ϕk2tan 2ϕ=−sin 4ϕ/4.
Then there exists a Lagrangian isometric minimal immersion x: M2 →S2×S2 such that ϕis the angle function ofx.
Essential point of the proof is the following Cartan-Griffith’s theorem [2]:
Theorem 4.4. Let G be a Lie group with Lie algebrag and Maurer-Cartan form Ω. (i) Let M be a manifold on which there exists a g-valued 1-formΦsatisfying
(4.1) dΦ =−1
2[Φ∧Φ].
Then for any pointp∈M there exists a neighborhood U ofpand a mapf :U→G such that f∗Ω = Φ. (ii) Given mapsf1, f2:M →G, then f1∗Ω =f2∗Ωif and only if f1=La◦f2 for some fixeda∈G, whereL is the left translation onG.
References
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[2] P. Griffiths,On Cartan’s Method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775–814.
[3] M. Kimura and K. Suizu,Fundamental theorems of Lagrangian surfaces inS2× S2, Osaka J. Math.,44(2007), 829–850.
[4] G. D. Ludden and M. Okumura,Some integral formulas and their applications to hypersurfaces of S2×S2, J. Differential Geometry 9(1974), 617–631.
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