• No results found

Biharmonic Maps into S-Space forms

N/A
N/A
Protected

Academic year: 2020

Share "Biharmonic Maps into S-Space forms"

Copied!
10
0
0

Loading.... (view fulltext now)

Full text

(1)

Vol. 11, No. 1, 2018, 150-159

ISSN 1307-5543 – www.ejpam.com Published by New York Business Global

Biharmonic Maps into S-Space Forms

Najma Abdul Rehman1,∗, Mehwish Bari2 1 Department of Mathematics, CIIT, Sahiwal, Pakistan

2 Department of Mathematics, NCBA and E, Bahawalpur, Pakistan

Abstract. We study in this paper the condition on second fundamental form for biharmonicity of submanifolds in S-space forms.

2010 Mathematics Subject Classifications: 53C55, 53C43, 58E20

Key Words and Phrases: S-Manifolds, harmonic map, biharmonic maps

1. Introduction

Harmonic maps between Riemannian manifolds have been discussed for last decades, initiated with the paper of J. Eells and J.H. Sampson [7]. Since harmonic maps have both properties analytic and geometric, they have become an important and interesting re-search field. The study of harmonic maps on Riemannian manifolds with some structures started from the paper of Lichnerowicz [13]. After that, Rawnsley [14] studied structure preserving harmonic maps between f-manifolds. Later on many authors studied harmonic maps (see [6] [10], [11], [12], [15] [16]).

The biharmonic maps theory is an old and attractive subject. They have been studied since 1862 by Maxwell and Airy to describe a mathematical model of elasticity. The Euler-Lagrange equation for bienergy functional was first derived by Jiange in 1986 [8]. After this biharmonic maps were studied by many authors see [2], [3], [5]. In [5], authors have studied the biharmonic submanifolds in complex space form. The objective of this paper is to find condition on second fundamental form for biharmonicity of a map from submanifolds of S-space form to S-space forms. After we recall some well known facts about biharmonic maps and S-manifolds , we prove the main results in third section.

Corresponding author.

Email addresses: najma−ar@hotmail.com(N. A. Rehman),mehwishbari@yahoo.com(M. Bari)

(2)

2. Preliminaries

In this section, we recall some well known facts concerning harmonic maps, biharmonic maps and S-manifolds.

LetF : (M, g)−→(N, h) be a smooth map between two Riemannian manifolds of dimen-sions m and n respectively. The energy density of F is a smooth function e(F) :M −→

[0,∞) given by [7],

e(F)p=

1 2T rg(F

h)(p) = 1 2

m X

i=1

h(F∗pui, F∗pui),

for any p ∈ M and any orthonormal basis {u1, . . . , um} of TpM. If M is a compact

Riemannian manifold, the energy E(F) of F is the integral of its energy density:

E(F) =

Z

M

e(F)υg,

whereυg is the volume measure associated with the metric g on M. A mapF ∈C∞(M, N)

is said to be harmonic if it is a cricital point of the energy functional E on the set of all maps between (M, g) and (N, h). Now, let (M, g) be a compact Riemannian manifold. If we look at the Euler-Lagrange equations for the corresponding variational problem, a map F : M −→ N is harmonic if and only if τ(F) ≡0, where τ(F) is the tension field which is defined by

τ(F) =T rg∇dF,e

where∇e is the connection induced by the Levi-Civita connection on M and the F-pullback

connection of the Levi Civita connection on N.

We take now a smooth variation Fs,t with two parameters s, t ∈ (−, ) such that

F0,0 =F. The corresponding variation vector fields are denoted by V and W.

The second variation formula of E is:

HF(V, W) =

∂2

∂s∂t(E(Fs,t))

(s,t)=(0,0) =

Z

M

h(JF(V), W)υg,

where JF is a second order self-adjoint elliptic operator acting on the space of variation

vector fields along F (which can be identified with Γ(F−1(T N))) and is defined by

JF(V) =− m X

i=1

(∇eui∇eui−∇e∇uiui)V −

m X

i=1

RN(V, dF(ui))dF(ui), (1)

for anyV ∈Γ(F−1(T N)) and any local orthonormal frame{u1, . . . , um} on M. Here RN

is the curvature tensor of (N, h) (see [9] for more details on harmonic maps).

(3)

studied the first and second variation formulas of biharmonic maps. Let us consider the bienergy functional defined by:

E2(F) =

1 2

Z

M

|τ(F)|2 νg, (2)

where|V |2=h(V, V),V Γ(F−1T N).

Then, the first variation formula of the bienergy functional is given by:

d

dt |t=0 E2(Ft) =−

Z

M

h(τ2(F), V)νg, (3)

here

τ2(F) :=J(τ(F)) = ¯4(τ(F))−R(τ(F)), (4)

which is called the bitension field of F and J is given by (1).

A smooth map F of (M, g) into (N, h) is said to be biharmonic if τ2(F) = 0.

As a generalization of both almost complex (in even dimension) and almost contact (in odd dimension) structures, Yano introduced in [17] the notion off-structure on a smooth manifold of dimension 2n+s, i.e. a tensor field of type (1,1) and rank 2n satisfying

f3+f = 0. The existence of such a structure is equivalent to a reduction of the structural group of the tangent bundle to U(n)×O(s). LetN be a (2n+s)-dimensional manifold with anf-structure of rank 2n. If there existsglobal vector fieldsξ1, ξ2, . . . , ξson N such

that:

f ξα= 0, ηα◦f = 0, f2=−I + X

ξα⊗ηα, (5)

whereηαare the dual 1-forms ofξα, we say that thef-structure has complemented frames.

For such a manifold there exists a Riemannian metricg such that

g(X, Y) =g(f X, f Y) +Xηα(X)ηα(Y)

for any vector fieldsX and Y onN. See [1].

An f-structure f is normal, if it has complemented frames and

[f, f] + 2Xξα⊗dηα= 0,

where [f, f] is Nijenhuis torsion off.

Let Ω be the fundamental 2-form defined by Ω(X, Y) =g(X, f Y), X, Y ∈ T(N). A normalf-structure for which the fundamental form Ω is closed,η1∧· · ·∧ηs∧(dηα)n6= 0 for

anyα, anddη1 =· · ·=dηs= Ω is called to be anS-structure. A smooth manifold endowed

(4)

We have to remark that if we take s= 1, S-manifolds are natural generalizations of Sasakian manifolds. In the case s≥2 some interesting examples are given in [1].

If Nis an S-manifold, then the following formulas are true (see [1]):

Xξα =−f X, X ∈T(N), α= 1, . . . , s, (6)

(∇Xf)Y =X{g(f X, f Y)ξα+ηα(Y)f2X}, X, Y ∈T(N), (7)

where∇is the Riemannian connection of g. Let Lbe the distribution determined by the projection tensor −f2 and let K be the complementary distribution which is determined

by f2+I and spanned by ξ1, . . . , ξs. It is clear that ifX ∈L thenηα(X) = 0 for any α,

and if X ∈ K, then f X = 0. A plane section π on N is called an invariant f-section if it is determined by a vector X∈L(x), x∈N, such that{X, f X} is an orthonormal pair spanning the section. The sectional curvature of π is called the f-sectional curvature. If

N is anS-manifold of constant f-sectional curvature k, then its curvature tensor has the form [1]

R(S, T, V, W) =X

α,β

{g(f S, f W)ηα(T)ηβ(V)−g(f S, f V)ηα(T)ηβ(W) +

+g(f T, f V)ηα(S)ηβ(W)−g(f T, f W)ηα(S)ηβ(V)}+

+1

4(k+ 3s){g(f S, f W)g(f T, f V)−g(f S, f V)g(f T, f W)}+

+1

4(k−s){Ω(S, W)Ω(T, V)−Ω(S, V)Ω(T, W)−2Ω(S, T)Ω(V, W)}, (8)

S, T, V, W ∈ T(N). Ω is fundamental 2-form. Such a manifold N(k) will be called an

S-space form. The Euclidean space E2n+s and the hyperbolic space H2n+s are examples of S-space forms.

Let M be an m-dimensional submanifold immersed in N. Then M is an invariant subm-naifold ifξα∈T M for anyαandf V ∈T M for anyV ∈T M. It is said to be anti-invariant

submanifold iff V ∈T M⊥ for anyV ∈T M. For a vector fieldX∈T M⊥, it can be writ-ten asf X =tX+nX, where tX is tangent component of f X,nX is normal component of f X. If ndoes not vanishes, then its an f-structure [4].

Consider the structure vector fieldsξ1, ξ2, . . . , ξsare tangent to M,dim(M)≥s. Then

M is CR-submaifold of N if there are two differentiable distributions D and D⊥ on M,

T M =D+D⊥ such that

• D and D⊥ are mutually orthogonal to each other.

• The distribution D is invariant under f, i.e. f Dp=Dp, for anyp∈M

• The distribution D⊥ is anti invariant under f, i.e. f Dp⊥⊆TpM⊥ for any p∈M.

It can be proved that each hypersurface of N which is tangent to ξ1, ξ2, . . . , ξs, has the

(5)

3. Biharmonic Maps into S-space form

Before the main results recall the following results by Jiang:

Lemma 1. [8] Let f : (Mm, g)→(Nn, h) be an isometric immersion whose mean curva-ture vector fieldH = m1τ(f) is parallel;∇⊥H= 0, whereis the induced connection of

the normal bundle T⊥M by f. Then,

4τ(f) =

m X

i=1

h(4τ(f), df(ei))df(ei)− m X

i,j=1

h(∇eeiτ(f), df(ej))(∇eeidf)(ej),

where {ei} is a locally defined orthonormal frame field of (M, g).

Lemma 2. [8] Let f : (Mm, g)(Nn, h) be an isometric immersion whose mean

curva-ture vector fieldH = m1τ(f) is parallel;∇⊥H= 0, where∇⊥ is the induced connection of the normal bundle T⊥M by f. Then,

4τ(f) =−

m X

j,k=1

h(τ(f), RN(df(ej), df(ek))df(ej)df(ek)−

m X

i,j=1

h(τ(f),(∇eeidf)(ej))(∇eeidf)(ej),

where {ei} is a locally defined orthonormal frame field of (M, g).

Lemma 3. [8] Let f : (Mm, g) → (Nm+1, h) be an isometric immersion which is not harmonic. Then, the condition thatkτ(f)k is constant is equivalent to the one that

Xτ(f)∈Γ(f∗T M), for all X ∈T M,

that is the mean curvature tensor is parallel with respect to 5⊥.

For details and proof of these Lemmas, see [5], [8]. Now the main result of this article;

3.1. Main results

Theorem 1. Let (M, g) be a 2m+s-dimensional submanifold of S-space form N of dimen-sion (2n+s), and Φ : (M, g) →(N, h) be an isometric immersion with non zero constant parallel mean curvature with respect to connection on normal bundle, then necessary and sufficient conditions for Φto be biharmonic is

• kB(Φ)k2= k+3s

4 (2n−1 +s) + 3(k−s)

4 , for M

2m+s to be a hypersurface, 2m=2n-1

• kB(Φ)k2= k+3s

(6)

Proof. Consider an s-manifold with constantf-sectional curvature k. Let{vi}2i=1m+s be

orthonormal basis on M. Then from equation (8) we have

RN(dΦ(vj), dΦ(vk))dΦ(vk) = X

α,β

−f2dΦ(vj)ηα(dΦvk)ηβ(dΦvk)−h(f dΦvj, dΦvk).

.ηα(dΦvk)ξβ +h(f dΦ(vk), f dΦ(vk))ηα(dΦvj)ξβ+

+f2dΦ(vk)ηα(dΦvj)ηβ(dΦvk) +

1

4(k+ 3s)

−f2dΦ(vj).

.h(f dΦvk, f dΦvk) +h(f dΦvj, f dΦvk)f2dΦ(vk)

+1

4(k−s){−f dΦ(vj)h(dΦvk, dΦvk) +f dΦ(vk)

.h(dΦvj, f dΦvk) + 2f dΦ(vk)h(dΦvj, f dΦvk)},

and

RN(dΦ(vj), dΦ(vk))dΦ(vk) =

1

4(k+ 3s){dΦ(vj) +δjk(−dΦvk)}

+ 3

4(k−s)h(dΦvj, f dΦvk)f dΦ(vk).

Then we have

m X

j,k=1

h(τ(Φ), RN(dF(vj), dF(vk))dF(vk))dF(vj)

= 1

4(k+ 3s){h(τ(Φ), dΦ(vj))−δjkh(τ(Φ), dΦvk)}

+ 3

4(k−s){h(dΦvj, f dΦvk)h(τ, f dΦ(ek))}. (9)

Letτ(Φ)∈T M⊥, then

m X

j,k=1

h τ(Φ), RN(dΦ(vj), dΦ(vk)

dΦ(vk))dΦ(vj) = 0.

Let M2m+s be a hyperspace of s-manifold N. Each hypersurface of s-manifolds has the structure of CR-submanifold. For τ(Φ) ∈ T M⊥, we can take f τ(Φ) ∈ ΓT M. Now

dim(M) = 2m+s= 2n−1 +s. For orthonormal basis{dΦ(vk)}mk=1 of dΦ(TxM) at all

points on M, we can write

f τ(Φ) =

2n−1+s X

k=1

h(f τ(Φ), dΦ(vk))dΦ(vk).

By Lemma 2,

∆τ(Φ) =

2n−1+s X

i,j=1

hτ(F),(∇evidΦ)(vj)

(7)

Furthermore, we have

R(τ(Φ)) =

2n−1+s X

k=1

RN(τ(Φ), dΦ(ek))dΦ =

k+ 3s

4 (2n−1 +s)τ(Φ) +

3(k−s)

4 τ(Φ). (10)

Now the necessary and sufficient conditions F to be biharmonic is that

τ2(F) = ¯4τ(F)−R(τ(F)) = 0, (11)

this becomes

2n−1+s X

i,j=1

hτ(F),∇evjdF(vk)

e

vjdF(vk)−

k+ 3s

4 (2n−1 +s)τ(Φ) +

3(k−s) 4 τ(Φ)

= 0(12)

Now let

B(Φ)(ej, ek) = (∇evjdΦ)vk=h(dΦ(vj), dΦ(vk))V =hjkU,

where U is the unit normal vector along F(M). then

τ(F) =

2n−1+s X

r=1

(∇evrdF)(vr) =

2n−1+s X

r=1

hrrU,

where U is the unit normal vector along F(M). Thus, the left hand side of (15) becomes as:

2n−1+s X

j,k,r=1

hrrhjkhjkU −

k+ 3s

4 (2n−1 +s)τ(Φ) +

3(k−s) 4 τ(Φ)

= 0,

2n−1+s X r=1 hrr !   m X j,k=1

hjkhjkU −

k+ 3s

4 (2n−1 +s)U+

3(k−s)

4 U

= 0,

kτ(Φ)k

kB(Φ)k2U

k+ 3s

4 (2n−1 +s)U+

3(k−s)

4 U

= 0.

sinceτ(Φ)6= 0, by assumption, then we have

kB(Φ)k2 −

k+ 3s

4 (2n−1 +s) +

3(k−s) 4

= 0,

and

kB(Φ)k2=

k+ 3s

4 (2n−1 +s) +

3(k−s) 4

(8)

Next letM2m+sbe an invariant submanifold of s-manifold N. Then forτ(Φ)ΓT M,

f τ(Φ)∈ΓT M⊥. For orthonormal basis{dΦ(vk)}mk=1 ofdΦ(TxM) at all pointsx∈M, we

have h(f τ(Φ), dΦ(vk)) = 0, then in this case

R(τ(Φ)) =

2n−1+s X

k=1

RN(τ(Φ), dΦ(ek))dΦ =

k+ 3s

4 (2m+s)τ(Φ). (14)

Then by equation (11),

2m+s X

i,j=1 h

τ(F),∇evjdF(vk)

e

∇vjdF(vk)−

k+ 3s

4 (2m+s)τ(Φ) = 0 (15)

With similar computations as above we have

kτ(Φ)k

kB(Φ)k2 U

k+ 3s

4 (2m+s)U

= 0.

This implies

kB(Φ)k2= k+ 3s

4 (2m+s)

Corollary 1. Let (M, g) be a2n−1-dimensional submanifold of complex space form N of dimension2n, and Φ : (M, g)→(N, h) be an isometric immersion with non zero constant parallel mean curvature with respect to connection on normal bundle, then necessary and sufficient conditions for Φto be biharmonic is

kB(Φ)k2= k

2(n+ 1)

Proof. In equation (13), for s=0 we get result.

Corollary 2. Let (M, g) be a (2n−1) + 1-dimensional submanifold of Sasakian space form N of dimension2n+ 1, and Φ : (M, g)→(N, h)be an isometric immersion with non zero constant parallel mean curvature with respect to connection on normal bundle, then necessary and sufficient conditions for Φ to be biharmonic is

kB(Φ)k2= k

4(2n+ 3) + 3

4(2n−1)

Proof. In equation (13), for s=1 we get result.

Acknowledgements

(9)

References

[1] D.E. Blair,Geometry of manifolds with structural group U(n)×O(s), J. Differential Geom. 4 (1970). 155-167.

[2] A. Balmus, Biharmonic properties and conformal changes., An. Stiint. Univ. Al.I. Cuza Iasi Mat. (N.S.) 50 (2004), 361372.

[3] A. Balmus, C. Oniciuc,Some remarks on the biharmonic submanifolds of S3 and their stability, An. Stiint. Univ. Al.I. Cuza Iasi, Mat. (N.S), 51 (2005), 171190.

[4] Jose L. Cabrerizo, Luis M. Fernandez, Manuel Fernandez (Sevilla), On normal CR-submanifolds of S-manifolds, Colloquim Mathematicum, Vol. LXIV, (1993), FASC. 2.

[5] Toshiyuki Ichiyama, Jun-ichi Inoguchi, Hajim Urakawa Bi-harmonic maps and bi-Yang-Mills fields, Note di Matematica, Note Mat. 1(2008), suppl. n. 1, 233-275.

[6] J. Davidov, A. G. Sergeev,Twistor spaces and harmonic maps, Uspekhi Mat. Nauk, 1993, Volume 48, Issue 3(291), 396.

[7] J. Eells, J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160

[8] G. Y. Jiang2-harmonic maps and their first and second variational formulas,Chinese Ann. Math. Ser. A7(4) (1986), 389-402.

[9] J. Eells and L. Lemaire, Report on harmonic maps, Bull, London Math. Soc. 20 (1988), 385-524.

[10] C. Gherghe, Harmonicity on cosymplectic manifolds, Rocky Mountain J.Math. 40, No.6,(2010), 247-254.

[11] C. Gherghe, S. Ianus, A. M. Pastore,CR-manifolds, harmonic maps and and stability, J.Geom. 71(2001), 42-53.

[12] C. Gherghe, K.Kenmotsu,Energy minimizer maps on C-manifolds, Diff. Geom. Appl. 21(2004), 55-63.

[13] A. Lichnerowicz, Applications harmoniques et varietes khleriennes, Sympos. Math. 3 (1970) 341402.

[14] J. Rawnsley, f -structures, f -twistor spaces and harmonic maps, Lecture Notes in Math., vol. 1164, Springer-Verlag, 1984, pp. 85159.

[15] N.A. RehmanHarmonic Maps on S-Manifolds, An. St. Univ. Ovidius Constanta, Vol 21(3), 2013, 197-208.

(10)

References

Related documents

In the Postsocialist and the Familialistic regimes, the intention probabilities are lower in the crisis period than in 2004, suggesting that young women’s short-term

3 Overexpressed DACT2 induced G2/M cell cycle arrest and promote cell apoptosis in nasopharyngeal carcinoma cells.. a , b The effect of DACT2 on cell cycle in vector-

5 Gambogic acid (GA) sensitized pancreatic cancer cell lines to gemcitabine. In the GA group, PANC-1 and BxPC-3 cells were treated with 1 μ M GA for 12 h, and then placed in

Om bloot naby aan die self-mutileerder te wees, maak ’n groot verskil – hulle voel egter dikwels hierdie tipe betrokkenheid onwaardig en daarom kan die berader geweldig baie vir

In hippocampal CA1 neu- rons, SK channels contribute to the afterhyperpolariza- tion, affecting neuronal excitability, regulating synaptic plasticity and memory [23].. Using

post-coitum (dpc) with maximal expression at 17.5 dpc. Two additional truncated transcripts were also detected by RT-PCR. Immunostaining of fetal testis sections and

The fact that there was no difference in the vitamin A status of infants between groups suggests that in rural communities in Ghana, this method of home fortification did not

The study of the income distribution in 1958, 1962 and 1967 (Statistisk Sentralbyrå, 1972, pages 13-14) included estimates of total gross income, including those not covered by the