The pinching problem for totally real minimal submanifolds in CPn(c) has been studied by many mathematicians

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Tomus 41 (2005), 107 – 116

ON TOTALLY REAL MINIMAL SUBMANIFOLDS IN COMPLEX PROJECTIVE SPACE

XIAOLI CHAO AND YAOWEN LI

In this paper, we obtain some pinching theorems for totally real minimal submanifolds in complex projective space.

§1. Introduction

Let CPⁿ(c) be an n-dimensional complex projective space with the Fubini- Study metric of constant holomorphic sectional curvaturec(c >0). The pinching problem for totally real minimal submanifolds in CPⁿ(c) has been studied by many mathematicians. Montiel, Ros and Urbano [MRU] proved a pinching result about Ricci curvature condition. Recently, Matsuyama [M1,2] has discussed the scalar curvature case which give a positive answer for Ogiue’s conjecture [O]. Now, in this paper, we give a pinching condition for the norm of the second fundamental form under which the submanifolds is totally geodesic.

Throughout this paper, we use the similar notations and formulas as those used in [MRU]. LetM be an n-dimensional compact Riemannian manifold. We denote byU M the unit tangent bundle overM and byU Mpits fibre at p∈M. For any continuous functionf:U M →R, we have

Z

U M

f dv= Z

M

Z

U Mp

f dv_pdp

where dp, dvp and dv stand for the canonical measures on M, U Mp and U M respectively.

IfT is a k-covariant tensor onM and∇T is covariant derivative, then we have ([R1])

(1.1)

Z

U M

nXⁿ

i=1

(∇T)(ei, e_i, v,· · ·.v)o dv= 0

2000Mathematics Subject Classification: 53C40.

Key words and phrases: totally real submanifolds, pinching, totally geodesic.

This work is supported by National Natural Science Foundation of China (No. 10226001, 10301008) and Support program for outstanding young teachers of Southeast University.

Received May 7, 2003.

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wheree1, . . . , e_n is an orthonormal basis ofT_pM,p∈M.

Suppose now thatM is isometrically immersed in an (n+p)-dimensional Rie- mannian manifoldM^n+p. We denote byh,ithe metric ofMas well as that induced onM. Letσ be the second fundamental form of the isometrically immersion and A_ξ the Weingarten endomorphism for a normal vectorξ. IfT_pMandT_p^⊥Mdenote the tangent and normal spaces toM atp, one can define

L:T_pM→T_pM and T:T_p^⊥M×T_p^⊥M→R by the expressions

Lv=

n

X

i=1

A_σ(v,e_i₎e_i and T(ξ, η) = traceA_ξA_η

where e1, . . . , e_n is an orthonormal basis of T_pM. Then L is a self-adjoint linear map andT a symmetric bilinear map.

There are many submanifolds satisfying T = kh,i. Obviously, hypersurfaces represent a trivial case. InCP^n+p(c), a Kaehler submanifold of order{k1, k2}for some natural numbers k1 and k2 is one submanifold of this type ([R3]). In this paper, we have a pinching theorem for this kind of submanifolds as following:

Theorem 3.1. LetMⁿ be a totally real minimal submanifold withT =kh,iin CP^n+p(c). If

σ

2< nc(n+ 2p)(n+ 4)

4(n+ 2)(n+ 4) +n(n+ 4)²+ 4n, thenM must be totally geodesic.

§2. Some Lemmas

In this section, we will prove some lemmas which will be used later. First, we give the following modified version of Simons’ formula which generalizes a result from [MRU]. Now we suppose thatM is a curvature-invariant submanifold ofM, i.e.,R(X, Y)Z∈TpM for allX,Y, Z∈TpM, beingR the curvature operator of M.

Lemma 2.1 [LC]. LetM be an n-dimensional compact curvature-invariant submanifold with parallel mean curvature vector isometrically immersed in an(n+p)- dimensional Riemannian manifoldM^n+p. Then we have

0 = Z

U M

nXⁿ

i=1

(∇σ)(ei, v, v)

2+

n

X

i=1

hσ(ei, ei), Aσ(v,v)vi + (n+ 4)

Aσ(v,v)v

2−4hLv, Aσ(v,v)vi −2T(σ(v, v), σ(v, v)) +hXⁿ

i=1

R(ei, v, σ(v, ei), σ(v, v)) + 2

n

X

i=1

R(ei, v, v, A_σ(v,e_i₎v)io dv

Remark. When the immersion is minimal, Lemma 2.1 is due to [MRU].

Remark. It’s clear that submanifolds in real space forms, Kahler, and totally real submanifolds in complex space forms are curvature-invariant.

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Lemma 2.2. LetM be ann-dimensional compact submanifold isometrically immersed in a Riemannian manifold M^n+p. Then, for ∀p∈M, we have:

i)

Z

U Mp

hLv, Aσ(v,v)vidvp

= 2

n+ 2 Z

U Mp

Lv

2dvp+ 1 n+ 2

Z

U Mp

hσ(v, v), ξidvp

ii) Z

U Mp

σ(v, v)

2dv_p

= 2

n+ 2 Z

U Mp

hLv, vidvp+ 1 n+ 2

Z

U Mp

n

X

i=1

hσ(v, v), σ(ei, ei)idvp

iii)

Z

U Mp

hLv, vidv_p= 1 n

Z

U Mp

σ

2dv_p

iv)

Z

U Mp

hσ(v, v), ηidvp= 1 n

Z

U Mp

n

X

i=1

hσ(ei, e_i), ηidv_p Whereξ=Pn

i=1σ(ei, Le_i)andη is a fixed vector in normal bundle.

Proof. Letα¹ be the 1-form onU Mp defined by

α¹(e) =hLv, Aσ(v,v)ei, v∈U M_p, e∈T_vU M_p

For anyv ∈U M_p, lete1, . . . , e_n−1, e_n=v be an orthonormal basis ofT_pM. Then (δα¹)(v) =−(n+ 2)hLv, Aσ(v,v)vi+ 2

Lv

2+hσ(v, v), ξi. Integrating it overU M_p, we obtain i).

ii), iii) and iv) are obtained by using the same technique for the 1-formsα²,α³ andα⁴onU M_pdefined by

α²_v(e) =hσ(v, v), σ(v, e)i α³_v(e) =hLv, ei

α⁴_v(e) =hσ(v, e), ηi

Lemma 2.3. LetM be ann-dimensional compact submanifold isometrically immersed in a Riemannian manifold M^n+p. Then we have

Z

U Mp

Aσ(v,v)v

2dvp≥ 2 n+ 2

Z

U Mp

hLv, Aσ(v,v)vidvp

+ 1

n+ 2 Z

U Mp

hAσ(ei,ei)v, Aσ(v,v)vidvp

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Proof. Let 4 denote the Laplace operator on Sⁿ⁻¹. Then, for the function f :U M_p→T_pM defined byf(v) =A_σ(v,v)v, we have

(4f)(v) =−3(n+ 1)Aσ(v,v)v+ 4Lv+ 2Aσ(ei,ei)v .

SinceU Mpis a (n−1)-dimensional sphere, the first eigenvalue of−4=∇∇_eiei−

∇ei∇ei isn−1. Then

− Z

U Mp

h4f, fidv_p≥(n−1) Z

U Mp

f

2dv_p

and the lemma follows.

Letαbe a 1-form onU M_p defined by

α_v(e) =hAσ(v,v)e, A_σ(v,v)vi

wherev∈U M_p, ande∈T_vU M_p. Ife1, . . . , e_n−1is an orthnormal basis ofT_vU M_p, then the codifferential ofαis

(δα) =

n

X

i=1

ei·αv(ei)

=−(n+ 4)

Aσ(v,v)v

2+ 2hLv, Aσ(v,v)vi +T(σ(v, v), σ(v, v)) + 2

n

X

i=1

hAσ(v,v)e_i, A_σ(v,e_i₎vi,

where e1, . . . , e_n−1, e_n =v is an orthonormal basis ofT_pM. Now integrating the above equality overU M_pand using divergence theorem, we have

2 Z

U Mp

nXⁿ

i=1

hA_σ(v,v)e_i, A_σ(v,e_i₎vio dv_p

= (n+ 4) Z

U Mp

A_σ(v,v)v

2dv_p−2 Z

U Mp

hLv, Aσ(v,v)vidv_p

− Z

U Mp

T(σ(v, v), σ(v, v))dv_p (2.1)

In a similar way, for the 1-formαdefined by

α_v(e) =hAσ(v,e)v, A_σ(v,v)vi, we have

(δα)(v) =

n

X

i=1

{2

A_σ(v,e_i₎v

2+hAσ(v,ei)v, A_σ(v,v)e_ii +hAσ(ei,ei)v, A_σ(v,v)vi} −(n+ 4)

f(v)

2+hLv, f(v)i.

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Integrating this and using (2.1), we get 2

Z

U Mp

n

X

i=1

Aσ(v,ei)v

2dvp= Z

U Mp

nn+ 4 2

f(v)

2− hAnHv, f(v)i

+1

2T(σ(v, v), σ(v, v))o dvp

(2.2)

By (2.1),(2.2) and 2

n

X

i=1

hAσ(v,ei)v, Aσ(v,v)eii ≤a

n

X

i=1

Aσ(v,ei)v

2+1 a

n

X

i=1

Aσ(v,v)ei

2

=a

n

X

i=1

A_σ(v,e_i₎v

21

aT(σ(v, v), σ(v, v)), (2.3)

By (2.1),(2.2) and (2.3), we have, for∀b >0, Z

U Mp

n

n+ 4−b(n+ 4) 4

f(v)

2−2hLv, f(v)i

− 1 + b

4+1 b

T(σ(v, v), σ(v, v))o

dv≤0. (2.4)

Now, we can prove the following lemma:

Lemma 2.4. LetMⁿ →M^n+p be a compact Riemannian immersion. Then we have

Z

U Mp

(n+ 2)hAHv, f(v)idv_p (1)

= Z

U Mp

n2

n

X

i=1

hAHe_i, A_σ(v,e_i₎vi+T(H, σ(v, v))o dv_p

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Z

U Mp

hAHv, Lvidv_p= Z

U Mp

n

X

i=1

hAHe_i, A_σ(v,e_i₎vidv_p

Z

U Mp

hAHv, Lvidv_p= 1 n Z

U Mp

n

X

i=1

hAHe_i, Le_iidv_p (3)

= 1 n Z

U Mp

hH·ξidvp

(4)

Z

U Mp

T(H < σ(v, v))dvp= Z

U Mp

T(H, H)dvp

(6)

Z

U Mp

(n+ 2)T(σ(v, v), σ(v, v))dv_p (5)

= Z

U Mp

{nT(H, σ(v, v)) + 2

n

X

i=1

T(σ(v, ei), σ(v, ei))}dvp

(6) Z

U Mp

n

X

i=1

T(σ(v, ei), σ(v, ei))dvp= 1 n

Z

U Mp

n

X

i,j=1

T(σ(ei, ej), σ(ei, ej))dvp

(7) Z

U Mp

hAHv, f(v)idv_p = Z

U Mp

n 1

n+ 2T(H, H) + 2

n(n+ 2)hH, ξio dv_p

Z

U Mp

T(σ(v, v), σ(v, v))dv_p = Z

U Mp

n n

n+ 2T(H, H) (8)

+ 2

n(n+ 2)

n

X

i,j=1

T(σ(ei, e_j), σ(ei, e_j))o dv_p

Z

U Mp

(2−b(n+ 4) 4 )

f(v)

2dvp

(9)

≤ Z

U Mp

n1 + b 4+1

b

T(σ(v, v), σ(v, v))− 1 + b

2

nhAHv, f(v)io dvp,

for each b.

Proof. By taking some proper 1-form on U Mp respectively as above, we can obtain (1)∼(6) and then (7) and (8) as their corollaries. Using Lemma 2.3, (2.4)

implies (9).

Remark. When b(> 0) is small, (9) gives a estimation of the upper bound of f(v)

2.

§3. Totally real submanifolds with T = kh,iin complex projective spaces

There are many submanifolds satisfying T = kh,i. Obviously, hypersurfaces represent a trivial case. InCP^n+p(c), a Kaehler submanifold of order{k1, k2}for

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some natural numbersk1andk2is one submanifold of this type ([R3]). LetMⁿbe a totally real minimal submanifold withT =kh,iimmersed inCP^n+p(c). Then

P(R) =

n

X

i=1

R(ei, v, σ(v, ei), σ(v, v)) + 2

n

X

i=1

R(ei, v, v, Aσ(v,ei)v)

=c

2hLv, vi −c 2

σ(v, v)

2+c 4

n

X

i=1

hσ(v, v), J eii²

−c 4

n

X

i=1

hJ v, σ(ei, e_i)ihJ v, σ(v, v)i. (3.1)

Now, we define a mapg¹:U M_p→T_pM by

g¹(v) =Aσ(v,v)v−Lv . By a direct computation, we have

(−4g¹)(v) = 3(n+ 1)f(v)−(n+ 3)Lv−2nAHv . Here4 is the Laplacian ofU Mp. SinceR

U Mpg¹(v)dvp = 0, we get Z

U Mp

h(−4g¹)(v), g¹(v)i ≥(n−1) Z

U Mp

g¹(v)

2.

Then, the above relation gives Z

U Mp

{(2n+ 4) f(v)

2−(2n+ 8)hLv, f(v)i

−2nhf(v), AHvi+ 4 Lv

2+ 2nhLv, AHvi}dvp≥0. (3.2)

In a similar way, for the 1-formg²(v) =f(v) +Lv, we have Z

U Mp

{(2n+ 4) f(v)

2−2nhLv, f(v)i

−2nhf(v), A_Hvi −4 Lv

2−2nhLv, A_Hvi}dv_p≥0. (3.3)

By (3.2) and (3.3), we get Z

U Mp

{(2n+ 4) f(v)

2−(2kn+ 4k+ 4)hLv, f(v)i

−2nhf(v), AHvi+ 4k Lv

2−2nkhLv, AHvi}dv_p≥0. (3.4)

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SinceM is minimal, by (3.4) withk=−_n+2² , we have Z

U Mp

f(v)

2dvp≥ 4 (n+ 2)²

Z

U Mp

Lv

2dvp.

From this and Lemma 2.2 i) we get (3.5)

Z

U Mp

f(v)

2dv_p≥ 2 n+ 2

Z

U Mp

hLv, f(v)idv_p. From (3.1),(3.5) and Lemma 2.1 we have

0 = Z

U M

nXⁿ

i=1

(∇σ)(ei, v, v)

2+ (n+ 4) f(v)

2

−4hLv, f(v)i −2T(σ(v, v), σ(v, v)) +hc

2hLv, vi −c 2

σ(v, v)

2+c 4

n

X

i=1

hσ(v, v), J eii²io dv

≥ Z

U M

nXⁿ

i=1

(∇σ)(ei, v, v)

2+nc 4

σ(v, v)

2

−n f(v)

2−2T(σ(v, v), σ(v, v))o (3.6) dv .

Assuming now thatM is minimal, and puttingb= _n+4⁴ in formula (9) of Lemma 2.4,we obtain

(3.7) Z

U Mp

f(v)

2dv_p≤ 1 + 1

n+ 4+n+ 4 4

Z

U Mp

T(σ(v, v), σ(v, v))dv_p.

By (3.6), (3.7) and the fact thatT = σ

2

2p+ngwe get 0≥

Z

U M

nXⁿ

i=1

(∇σ)(ei, v, v)

2

+hnc

4 −n(1 +_n+4¹ +ⁿ⁺⁴₄ ) + 2 2p+n

σ

2i

· σ(v, v)

2o dv .

From this we immediately have

Theorem 3.1. Let Mⁿ be a totally real minimal submanifold with T = kh,i in CP^n+p(c). If

(3.8)

σ

2< nc(n+ 2p)(n+ 4)

4(n+ 2)(n+ 4) +n(n+ 4)²+ 4n,

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thenM must be totally geodesic.

Remark. Xia [X] gave a pinching constant ^nc₆ without the assumption: T =kh,i.

When p > ⁿ⁽ⁿ⁺⁴⁾₁₂ +²₃+_3(n+4)ⁿ −ⁿ₆, our pinching constant is larger than Xia’s.

Remark. When the target manifold is the quaternionic space form QP^n+p(c), we have also a corresponding result, i.e., changing the factor n+ 2pin (3.8) to 3n+ 4p. So our result is better than that of [Sh1] in case when pis large enough.

Remark. B. Y. Chen and K. Ogiue ([CO]) had proved that, for a submanifold M of nonflat complex space form, M is curvature-invariant if and only if M is holomorphic or totally real submanifold. So we can use Lemma 2.1 in the proof of Theorem 3.1.

Acknowledgment. The authors would like to thank the referee for careful read- ing of the manuscript and very helpful suggestions.

References

[CO] Chen, B. Y. and Ogiue, K.,On totally real submanifolds, Trans. Amer. Math. Soc.193 (1994), 257–266.

[LC] Li, Y. W and Chao, X. L.,A modified version of Simons formula, preprint (2001).

[M1] Matsuyama, Y.,Curvature pinching for totally real submanifolds of complex projective space, J. Math. Soc. Japan52(2000), 51–64.

[M2] Matsuyama, Y., On totally real submanifolds of a complex projective space, Nihonkai Math. J.13(2002), 153–157.

[MRU] Montiel, S., Ros, A. and Urbano, F., Curvature pinching and eigenvalue rigidity for minimal submanifolds, Math. Z.191(1986), 537-548.

[O] Ogiue, K., Recent topics of differential geometry, Mathematics: the publication of the Mathematical Society of Japan39(1987), 51–64.

[R1] Ros, A.,A characterization of seven compact Kahler submanifolds by holomorphic pinching, Ann. Math.121(1985), 377–382.

[R2] Ros A.,Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z.187 (1984), 393–404.

[R3] Ros, A.,On spectral geometry of Kaehlar submanifolds, J. Math. Soc. Japan36(1984), 433–448.

[Si] Simons, J.,Minimal verieties in Riemannian manifolds, Ann. Math.88(1968), 62–65.

[Sh1] Shen, Y. B., Totally real minimal submanifolds in quaternionic projective space, Chin.

Ann. Math.14B(1993), 297–306.

[Sh2] Shen, Y. B.,On scalar curvature of totally real minimal submanifolds, Chin. Ann. Math.

12A(1991), 573–577.

[Sh3] Shen, Y. B.,On curvature pinching for minimal and Kahler submanifolds with isotropic second fundamental form, Chin. Ann. Math.12B(1991), 454–463.

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[Ub] Urbano F.,Totally real submanifolds, Geometry and Topology of Submanifolds, Proceed- ings (1987), 198–208.

[X] Xia, C. Y., On the minimal submanifolds inC P^m(c) and Sⁿ(1), Kodai Math. J. 15 (1992), 141–153.

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