A K¨ahler manifold is a nearly K¨ahler manifold

Nova S´erie

ON TOTALLY REAL SUBMANIFOLDS IN A NEARLY K ¨AHLER MANIFOLD *

Zhong Hua Hou

Abstract: LetM^mbe a totally real submanifold of a nearly K¨ahler manifold ¯M^2m. We prove an important relationship between the covariant differential of the second fundamental form ofM^m and that of the almost complex structure of ¯M^2m. And we show an application to the pinching problem on the square of the length of the second fundamental form ofM^m.

0 – Introduction

Let ( ¯M^2m, g, J) be an almost Hermitian manifold with Riemannian metric g and almost complex structure J. ¯M^2m is called anearly K¨ahler manifold if the almost complex structure J satisfies g(JX, JY) =g(X, Y) and ( ¯∇_XJ)(X) = 0, for any tangent vector fields X and Y on ¯M^2m. A K¨ahler manifold is a nearly K¨ahler manifold.

The canonical example of non-K¨ahler nearly K¨ahler manifold is the six dimen- sional standard unit sphereS⁶. There are many other non-K¨ahler nearly K¨ahler manifolds such as RP⁷×RP⁷, F₄/A₂×A₂ and U(4)/U(2)×U(1)×U(1) etc.

A. Gray in [5] stated that a 3-symmetric space with a naturally reductive G-invariant pseudo-Riemannian metric is a nearly K¨ahler manifold. Many inter- esting theorems about the topology and the geometry of nearly K¨ahler manifolds have been proved by many authors (cf. e.g. [4], [6], [8], [10], [11] and [12] etc.).

Received: June 29, 2000.

1980Mathematics Subject Classification: Primary53C42; Secondary32v40.

Keywords and Phrases: Nearly K¨ahler manifold; Totally real submanifold.

* Work done under partial support by SRF for ROCS, SEM.

* Work done under support by Funda¸c˜ao da Universidade de Lisboa through CMAF, FCT and Programa PRAXIS XXI.

The geometry of submanifolds in an almost Hermitian manifold is a very active topic in the theory of submanifolds. There have been many results on geometry of submanifolds in a K¨ahler manifold.

The theory of submanifolds in a nearly K¨ahler manifold say ¯M^2m was studied by many authors. LetM^m be a totally real submanifold of ¯M^2m. Denote byA_ξ the shape operator on the tangent bundle T M of M in the direction of a unit normal vector field ξ in the normal bundle N M. When ¯M is a complex space form, Chen–Ogiue [1] proved that

AJ X(Y) = AJ Y(X) ,

for any two vector fieldsX and Y tangent to M. Ejiri [3] showed that the same property holds for a 3-dimensional totally real submanifold inS⁶. In section 1, we shall prove that the same property also holds for a totally real submanifold in a nearly K¨ahler manifold.

We define a skew-symmetric tensor field Gof type (1,2) by G(X, Y) = ( ¯∇XJ)Y ,

where X and Y are vector fields on ¯M^2m and ¯∇ is the Levi–Civita connection on ¯M^2m. Ejiri [3] proved that, for a 3-dimensional totally real submanifoldM³ in S⁶, G(X, Y) is orthogonal to M³ for any vector fields X and Y tangent to M³. By applying this fact, he proved that such M³ is orientable and minimal.

In section 2, we shall prove that, for any totally real submanifoldM^m in ¯M^2m, G(X, Y) is orthogonal toMⁿ for any vector fields X and Y tangent to M^m.

Denote by σ the second fundamental form of M^m. We proceed to show an important relationship betweenσ and G. Precisely, we shall prove

Proposition 1.4. LetM^m be a Lagrangian submanifold of a nearly K¨ahler manifold ( ¯M^2m, g, J). Let σ be the second fundamental form of M^m. Define a skew-symmetric tensor field G of type (1,2) by G(X, Y) = ( ¯∇XJ)Y for any vector fields X and Y tangent to M^m, where ∇¯ is the Levi–Civita connection with respect tog. Then

g^³∇σ(X, Y, Z), JW^´ = g^³∇σ(X, Y, W), JZ^´ + g^³σ(Y, Z), G(W, X)^´

− g^³σ(Y, W), G(Z, X)^´ for any vector fieldsX,Y,Z and W tangent toM^m.

As an application, we shall prove a pinching theorem on the square of the length of the second fundamental form of a 3-dimensional totally real submanifold M³ inS⁶. Precisely, we shall prove the following

Proposition 2.1. LetM³be a closed3-dimensional totally real submanifold of S⁶. Denote by S the square of the length of the second fundamental form of M³. If S <5/2, then M³ is totally geodesic.

Remark 0.1. Simons [9] and Chern–do Carmo–Kobayashi [2] proved that, for a closedn-dimensional minimal submanifoldMⁿin an (n+p)-dimensional unit sphereS^n+p,Mⁿis totally geodesic ifS < n/(2−1/p). MoreoverS =n/(2−1/p) when and only when p= 1 or n=p= 2. A.M. Li and J.M. Li in [7] improved this result. They proved that ifp≥2 and S <2n/3, thenMⁿis totally geodesic.

Moreover, S = 2n/3 when and only when n = p = 2. Therefore we wonder whether or not this pinching constant could be improved whenn≥3 andp≥2.

Proposition 2.1 give an affirmative answer to this expectation because 5/2>2 = 3×(2/3).

1 – Geometry of totally real submanifolds in a nearly K¨ahler manifold Let ( ¯M^2m, g, J) be an almost Hermitian manifold with Riemannian metric g and almost complex structure J. Let {e1, e2, ..., e2m} be a local orthonormal frame field on the tangent bundle TM¯^2m. Let {ω1, ω2, ..., ω2m} be its coframe field and (ω_ab) be the Levi–Civita connection form associated with this coframe field. Then we have the Cartan structure equations:

(1.1)











∇e¯ a=ωabeb ;

ωab+ωba = 0 ; dωa=ωab∧ωb ;

Ω¯ab+ ¯Ωba= 0 ; dω_ab=ω_ac∧ω_cb+ ¯Ω_ab ,

where (Ω_ab) is the curvature form. The almost complex structure J of ¯M satis- fies:

(1) J: TxM¯ →TxM¯ is linear;

(2) J(JX) =−X;

(3) g(JX, JY) =g(X, Y);

for any X, Y in TxM¯. Under the above orthonormal frame field and associated coframe field,J can be expressed asJ =Jabωaeb. In this case,J(X) =JabXaeb

for anyX =Xaea∈TxM. Moreover conditions (2) and (3) can be represented¯ by

JacJbc=δab , (1.2)

JacJcb=−δab , (1.3)

where (δ_ab) is the Kronecker symbol. From (1.2) and (1.3) we have

(1.4) Jab+Jba = 0 .

Put ˜J = ¹₂J_ab ωa∧ω_b. From (1.4) it follows that ˜J is a 2-form which is called thefundamental form associated the almost complex structure J.

The covariant differential of (Jab), say (Jab,c), is defined to be (1.5) ∇J¯ ab := Jab,cωc = dJab+Jcbωca+Jacωcb . It follows from (1.3) that

(1.6) J_ae,cJ_eb+J_aeJ_eb,c = 0 .

Definition 1.1. ( ¯M^2m, g, J) is called a nearly K¨ahler (orTachibana)mani- foldif the covariant differential ¯∇J of J satisfies ( ¯∇_XJ)(X) = 0 for any tangent vectorX.

Remark 1.1. Equation ( ¯∇XJ)(X) = 0 for any tangent vectorXis equivalent to

(1.7) J_ab,c+J_ac,b = 0 ,

for alla,band c.

Let ( ¯M^2m, g, J) be a nearly K¨ahler manifold. LetM^m be a totally real sub- manifold, or a Lagrangian submanifold, of ¯M^2m. Then it follows that the image of the tangent spaceTxM^m under the mapping of the almost complex structure is the normal space N_xM^m, at every pointx ∈Mⁿ. From now on, we agree on the following index ranges:

1≤a, b, c, ...≤2m , 1≤i, j, k, ...≤m, and i^∗=m+i for 1≤i≤m . Choose{e1, e2, ..., em;e1^∗, e2^∗, ..., em^∗}to be a local orthonormal frame field of the tangent bundleTM¯^2m such thatei lies inT M^m and ei^∗ =Jei lies inN M^m, for all 1≤i≤m. We call such a kind of frame anadapted frame field on ¯M^2m. Let {ω1, ω2, ..., ωm;ω1^∗, ω2^∗, ..., ωm^∗} be the associated coframe field. Denote (ωab) to be the associated Levi–Civita connection form. Then (Jab) can be ex- pressed as

(1.8) Jab =







δij, a=i, b=j^∗ ;

−δij, a=i^∗, b=j ; 0, otherwise .

From (1.5) and (1.8), we infer

(1.9) −∇J¯ i^∗j^∗ = ¯∇Jij = ωi^∗j+ωij^∗, ∇J¯ ij^∗ = ¯∇Ji^∗j = ωi^∗j^∗−ωij . Restricting (1.1) toMⁿ, we getω_k^∗ = 0 for all k. The structure equations of Mⁿ are

(1.10)











dωi =ωij ∧ωj ;

Ωij =ωik^∗∧ωk^∗j+ ¯Ωij ; dωij =ωik∧ωkj+ Ωij ;

Ωi^∗j^∗ =ωi^∗k∧ωkj^∗+ ¯Ωi^∗j^∗ ; dωi^∗j^∗=ωi^∗k^∗∧ωk^∗j^∗+ Ωi^∗j^∗ .

From (1.1) we have the following relations:

ωij^∗∧ωi = 0, dωij^∗ = ωik∧ωkj^∗+ωil^∗∧ωl^∗j^∗+ ¯Ωij^∗ . Denote the curvature form ( ¯Ωab) by

(1.11) Ω¯ab = 1

2 R¯abcd ωc∧ωd . By Cartan’s lemma, we have

(1.12) ω_ij^∗=h^j_ik^∗ω_k, h^j_ik^∗=h^j_ki^∗, h^l_ijk^∗ =h^l_ikj^∗ + ¯R_l^∗_ijk , where (h^l_ijk^∗ ) is the covariant differential of (h^l_ij^∗) defined by

(1.13) ∇h^l_ij^∗ = h^l_ijk^∗ ωk = dh^l_ij^∗+h^l_kj^∗ ωki+h^l_ik^∗ωkj+h^s_ij^∗ωs^∗l^∗ .

The second fundamental formσ and its covariant differential ∇σ are defined by (1.14) σ=h^k_ij^∗ωiωjek^∗, ∇σ= (∇h^k_ij^∗)ωiωjek^∗ .

From the first equation of (1.9) and (1.12) we derive (1.15) Jij,k= h^j_ik^∗−hⁱ_jk^∗ . It follows from (1.7) and (1.15) that

0 = Jij,k+Jik,j = h^j_ik^∗+h^k_ij^∗−2hⁱ_kj^∗ , (1.16)

0 = Jjk,i+Jji,k = h^k_ji^∗+hⁱ_jk^∗ −2h^j_ik^∗ . (1.17)

From (1.16) and (1.17), we derive

(1.18) h^j_ik^∗=hⁱ_jk^∗ or Jij,k =h^j_ik^∗−hⁱ_jk^∗ = 0 .

It is known that the shape operator Ae_j∗ of M^m with respect to ej^∗ can be expressed asAe_j∗(ei) =h^j_ik^∗ek. From (1.18) we get the following

Proposition 1.1. LetM^m be a totally real submanifold of a nearly K¨ahler manifoldM¯^2m. Denote byA_ξthe shape operator ofM^mwith respect to a normal vector fieldξ of M^m. Then

(1.19) A_{J X}(Y) =A_{J Y}(X),

for any two vector fieldsX andY tangent toM^m.

Remark 1.2. When ¯M^2m is chosen to be the 6-dimensional unit sphere S⁶ andMⁿ is a 3-dimensional totally submanifold of S⁶, Ejiri [3] has proved (1.19) in different way.

As an direct application of Proposition 1.1, we have the following

Proposition 1.2. LetM^m be a totally real submanifold of a nearly K¨ahler manifold M¯^2m. Let {e₁, e₂, ..., e_m} be a local orthonormal frame field on M^m. DenoteAi^∗ = (hⁱ_jk^∗)to be the coefficient matrix of Ae_i∗ for everyi. Then

(1.20) Tr

µ X

i

A²_i∗

¶2

= ^X

i,j

(TrAi^∗Aj^∗)² .

Proof:

Tr µ

X

i

A²_i^∗

¶2

= ^X

i,j

hⁱ_sk^∗hⁱ_kl^∗h^j_lr^∗h^j_rs^∗ = ^X

l,s

h^s_ik^∗h^l_ki^∗h^l_jr^∗ h^s_rj^∗ = ^X

l,s

(TrAl^∗As^∗)² .

We define a skew-symmetric tensor field Gof type (1,2) by (1.21) G(X, Y) = ( ¯∇XJ)Y ,

for any vector fields X, Y on ¯M^2m. Then G(ea, eb) = Jab,cec for any a and b.

Moreover it follows from (1.18) that

(1.22) G(ei, ej) =Jij,k^∗ek^∗ ∈ N M^m , for anyiand j, which implies the following

Proposition 1.3. LetM^m be a totally real submanifold of a nearly K¨ahler manifoldM¯^2m. Define a skew-symmetric tensor fieldGof type(1,2)byG(X, Y) = ( ¯∇XJ)Y. Then G(X, Y) is normal to M for any two vector fieldsX and Y tan- gent toM^m.

Let us consider the behavior of (h^l_ijk^∗ ). From (1.9), (1.13) and (1.20) we infer

∇h^l_ij^∗ = h^l_ijk^∗ ω_k

= dh^l_ij^∗+h^l_kj^∗ ωki+h^l_ik^∗ωkj+h^k_ij^∗ωk^∗l^∗

= dhⁱ_lj^∗+h^k_lj^∗ω_ki+hⁱ_lk^∗ω_kj+hⁱ_kj^∗ ω_k^∗_l^∗

= dhⁱ_lj^∗+hⁱ_lk^∗ωkj+h^k_lj^∗(ωk^∗i^∗−∇J¯ k^∗i) +hⁱ_kj^∗( ¯∇Jk^∗l+ωkl)

= (dhⁱ_lj^∗+hⁱ_kj^∗ ω_kl+hⁱ_lk^∗ω_kj+h^k_lj^∗ω_k^∗_i^∗)−h^s_lj^∗J_s^∗_i,kω_k+hⁱ_sj^∗J_s^∗_l,kω_k

= ∇hⁱ_lj^∗+hⁱ_sj^∗Js^∗l,kωk−h^s_lj^∗Js^∗i,kωk . After sorting the above equality, we get

(1.23) h^l_ijk^∗ −hⁱ_ljk^∗ = h^s_ij^∗Js^∗l,k−h^s_lj^∗Js^∗i,k . It follows from (1.14), (1.22) and (1.23) that

(1.24) g^³∇σ(e_k, ej, ei), e_l^∗´ = g^³∇σ(e_k, ej, e_l), ei^∗

´+ g^³σ(ej, ei), G(e_l, e_k)^´

−g^³σ(ej, el), G(ei, ek)^´.

From (1.24) we have, for anyX=X_ke_k, Y =Yjej, Z =Ziei and W =W_le_l, g^³∇σ(X, Y, Z), JW^´ = g^³∇σ(X, Y, W), JZ^´+ g^³σ(Y, Z), G(W, X)^´

−g^³σ(Y, W), G(Z, X)^´ . Therefore we get the following

Proposition 1.4. LetM^m be a Lagrangian submanifold of a nearly K¨ahler manifold ( ¯M^2m, g, J). Let σ be the second fundamental form of M^m. Define a skew-symmetric tensor field G of type (1,2) by G(X, Y) = ( ¯∇XJ)Y for any vector fields X and Y tangent to M^m, where ∇¯ is the Levi–Civita connection with respect tog. Then

(1.25) g^³∇σ(X, Y, Z), JW^´ = g^³∇σ(X, Y, W), JZ^´+ g^³σ(Y, Z), G(W, X)^´

−g^³σ(Y, W), G(Z, X)^´, for any vector fieldsX,Y,Z and W tangent to M^m.

In the next section, we shall give an application to Proposition 1.4.

2 – A pinching problem on totally real submanifolds in S⁶

In this section, we proceed to show an application of Propositions 1.1 and 1.4 to the theory of submanifold. Suppose that ¯M^2m is a nearly K¨ahler manifold of constant curvature 1. Then the curvature tensor of ¯M^2m can be represented by (2.1) R¯abcd = δadδbc−δacδbd .

In this case, we have from (1.12) that

(2.2) h^l_ijk^∗ −h^l_ikj^∗ = ¯Rl^∗ijk = 0 . It is known that the Laplacian ofh^α_ij is

(2.3)

∆h^l_ij^∗ = h^l_ijkk^∗ = (TrAl^∗),ij+ (Ar^∗Al^∗Ar^∗)ij−(TrAl^∗Ar^∗)h^r_ij^∗

+ (TrAr^∗) (Al^∗Ar^∗)ij −(Al^∗Ar^∗Ar^∗)ij+ (Ar^∗Al^∗Ar^∗)ij

−(Ar^∗Ar^∗Al^∗)ij + n h^l_ij^∗−(TrAl^∗)δij .

Multiplyingh^l_ij^∗ on both-sides of (2.3) and taking sum withi,j andl, we derive

(2.4) X

i,j,l

h^l_ij^∗∆h^l_ij^∗ = h^l_ij^∗(n H_l^∗)_,ij + (TrA_i^∗) Tr(A_j^∗A_j^∗A_i^∗) + n S−(TrA_i^∗)²

−^X

i,j

nN(Ai^∗Aj^∗−Aj^∗Ai^∗) + (Si^∗j^∗)^2o,

where we denote S =^P_i,j,k(h^k_ij^∗)² the square of the length of the second funda- mental form ofMⁿ, Si^∗j^∗= Tr(Ai^∗Aj^∗) and N(Ai^∗Aj^∗−Aj^∗Ai^∗) =−Tr(Ai^∗Aj^∗− Aj^∗Ai^∗)². Assume that M^m is minimal in ¯M^2m. Then TrAi^∗ = 0 for alli. And (2.4) becomes

(2.5) ^X

i,j,k

h^k_ij^∗∆h^k_ij^∗ = n S−^X

i,j

nN(Ai^∗Aj^∗−Aj^∗Ai^∗) + (Si^∗j^∗)^2o.

The Laplacian of S satisfies

(2.6) 1

2∆S = ^X

i,j,k,l

(h^l_ijk^∗ )² +^X

i,j,k

h^k_ij^∗∆h^k_ij^∗ .

From now on, we assume ¯M to be the 6-dimensional unit sphere with the standard induced metric from 7-dimensional Euclidean space R⁷. It is known that we can identify R⁷ with the set of all purely imaginary Cayley numbers.

ThenS⁶can be equipped with an almost complex structure by the cross product of Cayley numbers. Moreover S⁶ is nearly K¨ahlerian but non-K¨ahlerian. This enables us to study the pinching problem on the square of the length of the second fundamental form of 3-dimensional totally real submanifolds inS⁶ in new viewpoint.

The main technique is to give a lower estimate to the right-hand side of (2.6).

Up to now, the best estimation on the second part of the right-hand side of (2.5) was given by A.M. Li and J.M. Li [7].

Lemma 2.1(Li’s [7]). LetA1, A2, ..., Apbe symmetric n×n-degree matrices, wherep≥2. Denote Sαβ = Tr(A^T_αAβ)and S =S11+S22+· · ·+Spp. Then

(2.7) ^X

α,β

nN(AαAβ−AβAα) + (Sαβ)^2o ≤ 3 2S² ,

where the equality holds if and only if one of the following conditions holds:

(1) A1=A2 =· · ·=Ap = 0;

(2) Only two of Aα’s are different from zero. If we assume A1 6= 0, A2 6= 0 and A₃ = · · · = A_p = 0, then S₁₁= S₂₂. Furthermore there exists an orthogonal n×n-degree matrix U such that

U A₁U^T = sS

4







1 0 0

0 −1 ... 0 · · · 0





, U A₂U^T = sS

4







0 1 0

1 0 ... 0 · · · 0





 .

In the rest of this section, we shall give a lower estimate to|∇σ|²=^P_i,j,k,l(h^l_ijk^∗ )². It is easy to see that JG(ei, ej) lies in TxM³ and is perpendicular to ei and e_j for any i6=j. So we can choose e₁,e₂ ande₃ such that (cf. Ejiri [3])

(2.8) JG(e₁, e₂) =e₃, JG(e₂, e₃) =e₁, JG(e₃, e₁) =e₂ . It follows from (2.8) that

(2.9) J12,3^∗ =J23,1^∗=J31,2^∗ =−1; and J_ij,k^∗= 0, otherwise . Fixinge1 and choosinge2 and e3 suitably, we can assumeh¹₂₃^∗ = 0. Denote (2.10) h¹₁₂^∗=x1, h¹₁₃^∗ =x2, h¹₂₂^∗ =x3, h¹₃₃^∗ =x4, h²₂₃^∗ =x5, h²₃₃^∗ =x6 .

Then it follows from (1.18) thatAi^∗’s can be expressed as A1^∗ =





−x3−x4 x1 x2

x1 x3 0 x2 0 x4



,

A2^∗ =





x1 x3 0

x3 −x1−x6 x5

0 x5 x6



,

A3^∗ =





x2 0 x4

0 x5 x6

x4 x6 −x2−x5



. And the square of the length ofσ, say S, is expressed as

(2.12) S = 4 (x²₁+x²₂+x²₃+x²₄+x²₅+x²₆) + 2 (x₃x₄+x₁x₆+x₂x₅) . Let us consider the representations of{h^l_ijk^∗ }. Put

(2.13) h¹₁₂₃^∗ =y1, h¹₁₂₂^∗ =y2, h¹₁₃₃^∗ =y3, h¹₁₁₂^∗ =y4 , h¹₂₃₃^∗ =y5, h¹₁₁₃^∗ =y6, h¹₂₂₃^∗ =y7 .

SinceM³ is minimal implies h^l_11k^∗ +h^l_22k^∗ +h^l_33k^∗ = 0 for anyk, we have (2.14) h¹₁₁₁^∗ =−y2−y3, h¹₂₂₂^∗ =−y4−y5, h¹₃₃₃^∗ =−y6−y7 . So we have

(2.15) ^X

i,j,k

(h¹_ijk^∗)² = ^X

i

(h¹_iii^∗)² + 3^X

i6=j

(h¹_iij^∗)² + 6 (h¹₁₂₃^∗ )² =

= 6y²₁ + 4 (y₂²+y²₃+y²₄+y₅²+y₆²+y²₇) + 2 (y2y3+y4y5+y6y7) . On the other hand, it follows from (1.23) that

(2.16) h²₁₁₁^∗ = y4+x2, h²₁₁₂^∗ = y2, h²₁₁₃^∗ = y1+x4 ,

h²₁₂₂^∗ =−y4−y5+x5, h²₁₂₃^∗ = y7+x6, h²₁₃₃^∗ = y5−x2−x5 . Putting

(2.17) h²₂₃₃^∗ =y8, h²₂₂₃^∗ =y9 , then we have

(2.18) h²₂₂₂^∗ =−y2−y8, h²₃₃₃^∗ =−x4−y1−y9 .

Therefore we infer from (2.16), (2.17) and (2.18),

(2.19) ^X

i,j,k

(h²_ijk^∗)² = ^X

i

(h²_iii^∗)² + 3^X

i6=j

(h²_iij^∗)² + 6 (h²₁₂₃^∗ )² =

= 4y₁²+ 4y²₂+ 4y₄²+ 6y²₅+ 6y₇²+ 4y₈²+ 4y²₉+ 2y1y9+ 2y2y8+ 6y4y5

+ 8x4y1+ (2x2−6x5)y4−(6x2+ 12x5)y5+ 12x6y7+ 2x4y9

+ 4x²₂+ 4x²₄+ 6x²₅+ 6x²₆+ 6x2x5 . By the same procedure, we obtain

(2.20)

h³₁₁₁^∗ = y6−x1, h³₁₁₂^∗ = y1−x3, h³₁₁₃^∗ =y3 ,

h³₁₂₂^∗ = y7+x1+x6, h³₁₂₃^∗ = y5−x5, h³₁₃₃^∗ =−x6−y6−y7 , h³₂₂₂^∗ = y₉+x₃, h³₂₂₃^∗ =y₈, h³₂₃₃^∗ =−y₁−y₉, h³₃₃₃^∗ =−y₃−y₈ . Therefore we infer from (2.20) that

(2.21) ^X

i,j,k

(h³_ijk^∗)² = ^X

i

(h³_iii^∗)² + 3^X

i6=j

(h³_iij^∗)² + 6 (h³₁₂₃^∗ )² =

= 6y₁²+ 4y²₃+ 6y₅²+ 4y²₆+ 6y₇²+ 4y₈²+ 4y²₉+ 2y3y8+ 6y1y9+ 6y6y7

−2x1y6+ 2x3y9+ 6x1y7+ 6x6y6−6x3y1+ 12x6y7−12x5y5

+ 4x²₁+ 4x²₃+ 6x²₅+ 6x²₆+ 6x₁x₆ . DenoteF =|∇σ|². Then

F = ^X

i,j,k,l

(h^l_ijk^∗ )² = ^X

i,j,k

(h¹_ijk^∗)²+^X

i,j,k

(h²_ijk^∗ )²+^X

i,j,k

(h³_ijk^∗)² .

It follows from (2.15), (2.19) and (2.21) that

(2.22)

F = 16y₁²+ 8y₂²+ 8y²₃+ 8y₄²+ 16y₅²+ 8y²₆+ 16y²₇+ 8y₈²+ 8y²₉ + 2y2y3+ 2y2y8+ 2y3y8+ 8y1y9+ 8y4y5+ 8y6y7

+ (8x4−6x3)y1+ (2x2−6x5)y4−(6x2+ 24x5)y5

+ (6x6−2x1)y6+ (6x1+ 24x6)y7+ (2x4+ 2x3)y9

+ 4x²₁+ 4x²₂+ 4x²₃+ 4x²₄+ 12x²₅+ 12x²₆+ 6x1x6+ 6x2x5 .

It is not difficult to check that the only critical point of F with respect to (y1, y2, ..., y9) is P0 = (y₁⁰, y₂⁰, ..., y₉⁰), where

(2.23)

y⁰₁ = 1

4(x₃−x₄), y₂⁰= 0, y⁰₃ = 0 , y⁰₄ =−1

4x2, y⁰₅ = 1

4(x2+ 3x5), y₆⁰= 1 4x1 , y⁰₇ =−1

4(x1+ 3x6), y⁰₈ = 0, y₉⁰=−1 4x3 .

Furthermore we can see thatP0 is the minimum point ofF. Substituting (2.23) into (2.22) and recalling (2.12), we obtain the minimum value ofF:

(2.24)

Fmin = 3x²₁+ 3x²₂+ 3x²₃+ 3x²₄+ 3x²₅+ 3x²₆ +3

2x5x2+ 3

2x6x1+3 2x3x4

= 3 4S .

Therefore we get the following lower estimate to|∇σ|²:

(2.25) |∇σ|² = ^X

i,j,k,l

(h^l_ijk^∗ )² ≥ 3 4S . It follows from (2.5), (2.6), (2.7) and (2.25) that

(3.26) ∆S ≥ 3S

µ5 2 −S

¶ . Inequality (3.26) implies the following

Proposition 2.1. LetM³be a closed3-dimensional totally real submanifold of S⁶. Denote by S the square of the length of the second fundamental form of M³. IfS <5/2, thenM³ is totally geodesic.

ACKNOWLEDGEMENTS – The author is very grateful to Professor A. Machado for his kind help and good advice. He would like to express his heartfelt gratitude to faculty and staffs of CMAF/UL for their hospitality during his stay and study in Portugal.

And he wish to thank referee for his (her) earnest recommendation.

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Zhong Hua Hou,

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, Liaoning – PEOPLE’S REPUBLIC OF CHINA

E-mail: [email protected] and

CMAF, Universidade de Lisboa,

Av. Prof. Gama Pinto, 2, 1649-003 Lisboa – PORTUGAL E-mail: [email protected]