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EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1811-1818

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

Curvature Inequalities for Submanifolds of S-space form

Najma Abdul Rehman

Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Pakistan

Abstract. In this paper we establish new results of squared mean curvature and Ricci curvature for the sub manifolds of S-space from that is the generalization of complex and contact structures. Obtained results are discussed for invariant, anti invariant and CR sub manifolds of S-space from.

2010 Mathematics Subject Classifications: 53C40, 53C25

Key Words and Phrases: Curvature, Sub manifolds, S-space form

1. Introduction

One of the main and useful idea in submanifolds conjectures is to derive relationship among squared mean curvature and Ricci curvature of submanifolds, was explained by Chen [6], [7]. After this many authors derived Chen inequalities for complex and contact space forms [1], [11], [12] and on hyper surfaces of a Lorentzian manifold [9].

After the generalization of complex and contact space forms into S-space form [4], it is natural to study the inequalities of Ricci curvature for submanifolds of S-space forms. Geometry of S-space forms were studied by many authors i.e. [10], [13].

In this paper we find relations between squared mean curvature and Ricci curvature for the sub manifolds of S-space form and discuss this relation for invariant, anti invariant and CR sub manifolds of S-space form. After introduction, second section contains basics of S-space forms and submanifolds. Third section contains main results.

2. Preliminaries

This section presents some well known facts related to S-space form and sub manifolds.

Yano[14] presented that almost complex and almost contact structures can be gen-eralized as f - structure on a smooth manifold of dimension 2m+s . The idea for the f-structure is to consider a tensor field with conditionf3+f = 0, of type (1,1) and rank 2m .

DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3435

Email address: najma−ar@hotmail.com (N. A. Rehman)

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Consider manifold M2m+s along an f-structure of rank 2m. We take s structural vector fieldsξ1, ξ2, . . . , ξs on M such as:

f ξα= 0, ηα◦f = 0, f2=−I + X

ξα⊗ηα, (1)

where ηα and ξα are the dual forms to each other, therefore complemented frames exist

on f-structure. For f- manifold we define a Riemannian metricgas g(Y, X) =g(f Y, f Z) +Xηα(Y)ηα(Z)

for vector fieldsY andZ on M [4].

An f-structure f is normal, if there exist complemented frames and [f, f] + 2Xξα⊗dηα= 0,

where [f, f] is Nijenhuis torsion off. Let fundamental 2-form B be defined as B(Y, Z) = g(Y, f Z),Y,Z ∈T(M). f-structure that is normal and dη1 =· · ·=dηs=B is known as

an S-structure. A smooth manifold along with an S-structure known as an S-manifold. Blair described such types of manifolds in [4].

For Sasakian manifolds we take s = 1. For s ≥ 2 we may have some attractive applications discussed in [4].

If Mis anS-manifold, then we consider the formulas [4]:

e

Yξα =−f Y, Y ∈T(M), α= 1, . . . , s, (2)

(∇eYf)Z = X

α

{g(f Y, f Z)ξα+ηα(Z)f2Y}, Y, Z ∈T(M), (3)

where ∇e is the Riemannian connection of g. The projection tensor −f2 determine the

distributionLandf2+I determine the complementary distributionMwhich is determined and spanned by ξ1, . . . , ξs. It can be observe that ifY ∈L thenηα(Y) = 0 for all α, and

for Y ∈ M, we have f Y = 0. A plane section Π on M is said to be f - section if it is established by a vectorY ∈L(p),p∈M, such that{Y, f Y}span the section. We take the Sectional curvature of Π as the f-sectional curvature. If M is an S-manifold of constant f-sectional curvature k, then its curvature tensor is as:

e

R(Y, Z)U = X

α,β n

ηα(Y)ηβ(U)f2Z−ηα(Z)ηβ(U)f2Y −g(f Y, f U)ηα(Z)ξβ

+g(f Z, f U)ηα(Y)ξβ}+

k+ 3s 4

−g(f Z, f U)f2Y +g(f Y, f U)f2Z

+k−s

(3)

Consider immersed submanifold Mm of N2n+s. Then M is an invariant submnaifold if ξα ∈ T M for any α and f Y ∈ T M for any Y ∈ T M. It is said to be anti-invariant

submanifold if f Y ∈ T M⊥ for any Y ∈ T M. For a vector field Y ∈ T M⊥, it can be written as f Y = tY +SY, where tY represents tangent component of f Y, SY shows normal component of fY. IfS does not disappear, then its an f-structure [5].

For a vector field Z ∈ T M, it can be written as f Z = P Z+N Z, where P Z is tangent component off Z,N Z is normal component of fZ.

Letdim(M)≥sand we take the structure vector fieldsξ1, ξ2, . . . , ξsas tangents to M.

M is known as CR-submaifold of N if we have two differentiable distributions Dand D⊥ on M, T M =D+D⊥ satisfying

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

• Iff Dp=Dp, for anyp∈M then the distribution D is known as invariant under f. • Iff D⊥p ⊆TpM⊥, for anyp∈M then the distributionD⊥is known as anti invariant

under f.

The Gauss equation for the submanifold Mm into a (m+q)-dimensional Riemannian manifold Nm+q is

e

R0(Y, Z, U, W) =R0(Y, Z, U, W) +g(h0(Y, U), h0(Z, W))−g(h0(Y, W), h0(Z, U)).

Consider k-plan section of TpM denoted by D and let X be a unit vector in D. We can

choose orthonormal basise1, e2, . . . , ek of D and we considere1 =X, then Ricci curvature RicD of D at X can be define as

RicD(x) =K12+K13+· · ·+K1k,

where Kij represents sectional curvature of 2-plane section spanned by ei,ej. Therefore

the scalar curvature τ of the k-plane section D can be written as

τ(D) = X 1≤i<j≤k

Kij.

.

3. Main Results

Theorem 1. LetMm, be a sub manifold of S-space formN2n+s(K)of constantf-sectional curvature k, with structural vector fields ξ1, ξ2, . . . , ξs tangent to Mm, then

for each unit vector X∈TpM, orthogonal toξ1, ξ2, . . . , ξs

Ric(X)≤ 1

4

m2 kHk2 +(m−1)(k+ 3s) +3

2 kP X k

2(ks)1

2(3s−1)(k+ 3s−4)

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Proof. LetMmbe a sub manifold of S-space formN2n+s(K) with constantf-sectional curvature K. For a point p ∈ M, take a unit vector X ∈TpM. Choose an orthonormal

basis

e1, e2, . . . , em−s, em−s+1, . . . , em−s+s, em−s+s+1, . . . , e2n+s on TpN. Then it is clear that

e1, e2, . . . , em−s, em−s+1, . . . , em−s+s are unit tangent vectors toM at p. We take X=e1. From the curvature tensor of S-space form N2n+s(K), after summation 1i, j m we

have

e

R0(ei, ej, ej, ei) =

1

4(k+ 3s)m(m−1) +

k+ 3s−4

4 (2s−2ms) + 3

4(k−1)kP k 2 .

Using Gauss equation 1

4(k+ 3s)m(m−1) +

k+ 3s−4

4 (2s−2ms) + 3

4(k−1)kP k

2= 2τ m2kHk2+khk2, (6)

where H represents mean curvature, h represents second fundamental form and τ is scalar curvature. Now we take the notation for second fundamental form h as hrij = g(h(ei, ej), er), kh k2=Pmi,j=1g(h(ei, ej), h(ei, ej)) and kP k2=Pmi,j=1g2(P ei, ej). Also

we know that mean curvatureH(p) = m1 Pmi=1h(ei, ei). Then from (6), we have

n2kH k2 = 2τ+ 1 2

2n+s X

r=m+1

(hr11+· · ·+hrmm)2+ (hr11−hr22− · · · −hrmm)2

+ 2 2n+s

X

r=m+1 X

i<j

(hrij)2−2 2n+s

X

r=m+1 X

2≤i<j≤n

hriihrjj

− 1

4(k+ 3s)m(m−1)−

k+ 3s−4

4 (2s−2ms)− 3

4(k−s)kP k 2 (7)

Since sectional curvature for sub manifoldM can be represented as

X

2≤i,j≤n

Kij =

1

4(k+ 3s)(m−2)(m−1)−

k+ 3s−4

4 (2(m−2)s−(s−1))

+ 1

4(k−s) 3kP k

2 3kP e 1 k2

. (8)

By Gauss equation, on S-space form we can write

X

2≤i,j≤n

˜ Kij =

1

4(k+ 3s)(m−2)(m−1)−

k+ 3s−4

4 (2(m−2)s−(s−1))

+ 1

4(k−s) 3kP k

23kP e 1 k2

+ 2m+s

X

r=m+1 X

2≤i<j≤n

(hriihrjj−(hrij)2) (9)

(5)

1 2n

2kH k2 = 2Ric(X) +1 2

2n+s X

r=m+1

(hr11−hr22− · · · −hrmm)2+ 2n+s

X

r=m+1 m X

j=1 (hr1j)2

− 2(m−1)k+ 3s

4 −3kP X k

2 k−s

4 + (3s−1)

k+ 3s−4

4 , (10)

therefore 1 2n

2 kHk22Ric(X)2(m1)k+ 3s

4 −3kP X k

2 k−s

4 + (3s−1)

k+ 3s−4 4 , (11) that gives required result.

Corollary 1. In Theorem 1, equality holds at point p∈M, if (i) H(p) = 0 and X is normal to TpM

(ii) p is totally geodesic point

Proof. (1) WhenH(p) = 0 , we gethr11=hr22=· · ·=hrmm = 0 and when X is normal to TpM, we have hr12 = hr13 = · · · = hr1m = 0, r ∈ {m+ 1, . . . ,2n+s} and from (10)

equality holds.

(2)When p is totally geodesic point,hrij = 0 for all i, j,r∈ {m+ 1, . . . ,2n} and from (10) equality holds.

Corollary 2. Let M be an m-dimensional invariant sub manifold tangent to structure vector fieldsξ1, ξ2, . . . , . . . , ξsin S-space formN2n+s(k), then for each unit vectorX∈TpM

orthogonal to ξ1, ξ2, . . . , . . . , ξs, we have

Ric(X)≤ 1

4

(m−1)(k+ 3s) +3

2(k−s)− 1

2(3s−1)(k+ 3s−4)

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Proof. We know that every invariant sub manifold of S-space form is minimal [14] i.e. mean curvature vanishes and since X in unit tangent vector to M, k f X k=k P X k=k

Xk= 1, therefore from (5), we get result.

Corollary 3. Let M be an m-dimensional anti-invariant sub manifold tangent to structure vector fieldsξ1, ξ2, . . . , . . . , ξsin S-space formN2n+s(k), then for each unit vectorX∈TpM

orthogonal to ξ1, ξ2, . . . , . . . , ξs, we have

Ric(X)≤ 1

4

m2 kHk2 +(m−1)(k+ 3s)−1

2(3s−1)(k+ 3s−4)

. (13)

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Corollary 4. Let M be an m-dimensional CR-submanifold of S-space form N2n+s(K)

then

(i) For each unit vector X ∈Dp,

Ric(X)≤ 1

4

m2kH k2+(m−1)(k+ 3s) +3

2(k−s)− 1

2(3s−1)(k+ 3s−4)

. (14)

(i) For each unit vector X ∈D⊥p,

Ric(X)≤ 1

4

m2 kHk2 +(m−1)(k+ 3s)−1

2(3s−1)(k+ 3s−4)

. (15)

LetMm, be a submanifold ofN2n+s(K) s-space form of constantf-sectional curvature

k, using the results of Theorem 1, we can find a relation between the k-Ricci curvature of Mm and the squared mean curvature k H k2. Before this result first we define shape operator.

Let p ∈ Mm and {e1, e2, . . . , em} an orthonormal basis of TpMm. We choose the ortho

normal basis{e1, e2, . . . , em, em+1, . . . , e2n+s}onTpN2n+s. Then shape operator takes the

form (see [12])

Aem+1 =        

a1 0 . . . 0 0 a2 . . . 0 . . . . . . . . . . . .

0 0 0 am

       

Aer = (h

r

ij),i, j= 1, . . . , m;r =m+ 2, . . . ,2n+s,traceAer = 0.

We can take

m2kHk2=

m X i=1 ai !2 = m X i=1

a2i + 2X

i<j

aiaj ≤m m X

i=1 a2i.

Since we know that 0≤P

i<j(ai−aj)2= (n−1) Pm

i=1a2i −2 P

i<jaiaj, therefore m

X

i=1

a2i ≥mkHk2 . (16)

Proposition 3.1. Let Mm, be a submanifold of s-space form N2n+s(K) of constant f -sectional curvature k, tangent to ξ1, ξ2, . . . , ξs then we have

m2 kH k2 ≥ 2τ

m(m−1)− 1

4(k+ 3s)−

k+ 3s−4 4

(2s−2ms) m(m−1) −

3

4(k−1)

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Proof. We choose orthonormal basise1, e2, . . . , em−s, em−s+1, . . . , em−s+s, em−s+s+1, . . . , e2n+s

on TpN withem+1 parallel to mean curvature vector H(p). From (6)

m2 kHk2 = 2τ+khk2 −1

4(k+ 3s)m(m−1)−

k+ 3s−4

4 (2s−2ms)− 3

4(k−1)kP k 2

= 2τ +

m X

i=1 a2i +

2n X

r=m+2 m X

i,j=1

(hrij)2−1

4(k+ 3s)m(m−1)

− k+ 3s−4

4 (2s−2ms)− 3

4(k−1)kP X k 2

by (16)

m2kHk2 ≥ 2τ+mkH k2 −1

4(k+ 3s)m(m−1)

− k+ 3s−4

4 (2s−2ms)− 3

4(k−1)kP X k 2

which proves the required result.

Remark 1. • Fors= 0, we have the results of Theorem 1, Corollary 3, Corollary 2, Proposition 3.1 for Khaler manifold.

• For s= 1, we have the results of Theorem 1, Corollary 3, Corollary 2, Proposition 3.1 for Sasakian Manifolds.

Acknowledgements

This work is partially supported by Higher Education Commission Pakistan.

References

[1] P. Alegre, A. Carriazo, Y. H. Kim, D. W. Yoon, B.Y. Chens inequality for sub-manifolds of generalized space forms, Indian J. Pure Appl. Math., 38 (2007), no. 3, 185-201.

[2] H. Aytimur and C. Ozgur, Inequalities for submanifolds in statistical manifolds of quasi-constant curvature, Ann. Polon. Math. 121 (2018), no. 3, 197-215.

[3] H. Aytimur and C. Ozgur,Inequalities for submanifolds of Sasaki-like statistical man-ifolds, Turkish J. Math. 42 (2018), no. 6, 3149-3163.

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

[6] B.Y. Chen,Strings of Riemannian invariants, inequalities, ideal immersions and their applications, The Third Pacific Rim Geometry Conference (Seoul, 1996), 760, Monogr. Geom. Topology, 25, Int. Press, Cambridge, MA, 1998.

[7] B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel), 60 (1993), no. 6, 568-578.

[8] A. Friedmann and J. A. Schouten,Uber die Geometrie der halbsymmetrischen Uber-tragungen, Math. Z., 21 (1924), no. 1, 211-223.

[9] M. Glbahar, Erol Kilic and Sadik Kele,Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold, Journal of Inequalities and Applications 2013, 2013:266. [10] S. Ianus, G. E. Vlcu, and R. C. Voicu,Harmonic maps and Riemannian submersions

between manifolds endowed with special structures, Algebra, Geometry and Mathe-matical Physics, vol. 93, pp. 277288, Banach Center Publications, 2011.

[11] T. Imai,Notes on semi-symmetric metric connections, Vol. I. Tensor (N.S.), 24 (1972), 293-296.

[12] A. Mihai and Cihan Ozgur, Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection, Taiwanese J. Math., to appear.

[13] Najma Abdul Rehman, Mehwish Bari, Biharmonic Maps into S-Space forms, Euro-pean Journal of Pure and Applied Mathematics, Vol. 11, No. 1, 2018, 150-159. [14] K. Yano and M. Kon, Structures on manifolds, vol. 3, Series in pure Math., World

Scientific, Singapore, 1984.

References

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