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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection

Asif Ahmad / He Guoqing / Tang Wanxiao / Zhao Peibiao
Published Online: 2018-04-18 | DOI: https://doi.org/10.1515/math-2018-0034

Abstract

In this paper, we obtain Chen’s inequalities for submanifolds in (κ, μ)-contact space form endowed with a semi-symmetric metric connection.

Keywords: (κ, μ)-contact space form; Semi-symmetric metric connection; Chen’s inequalities

MSC 2010: 53C40; 53B05

1 Introduction

Since Chen [1, 2] proposed the so-called Chen invariants, and chen’s inequalities, the study of Chen invariants and inequalities has been an active field over the past two decades. There are many works studying Chen’s inequalities for different submanifolds in various ambient spaces (one can see [3, 4, 5, 6, 7, 8, 9, 10, 11, 12] for details).

In 1924, Friedmann and Schoutenn [13] first introduced the notion of a semi-symmetric linear connection on a differentiable manifold. Later many scholars studied the geometric and analytic problems on a manifold associated with a semi-symmetric metric connection. For instance, H. A. Hayden [14] studied the geometry of subspaces with a semi-symmetric metric connection. K. Yano [15] studied some geometric properties of a Riemannian manifold endowed with a semi-symmetric metric connection. In [16], Z. Nakao studied submanifolds of a Riemannian manifold with a semi-symmetric metric connection.

Recently, T. Koufogiorgos [17] introduced the notation of (κ, μ)-contact space form, in particular, when κ = 1, (κ, μ)-contact space form is the well known class of Sasakian space form, and A. Mihai and C. Özgür [18, 19] arrived at Chen’s inequalities for submanifolds of Sasakian space forms, real(complex) space forms with a semi-symmetric metric connection. On the other hand, P. Zhang et al [20] also established Chen’s inequalities for submanifolds of a Riemannian manifold of quasi-constant curvature with a semi-symmetric metric connection.

Motivated by the celebrated works above, a natural question is: can we establish the Chen’s inequalities for submanifolds in a (κ, μ)-contact space form endowed with a semi-symmetric metric connection?

In this paper, we will consider and derive Chen’s first inequality and Chen-Ricci inequalities for submanifolds of a (κ, μ)-contact space form with a semi-symmetric metric connection.

2 Preliminaries

Let Nn+p be an (n+p)-dimensional Riemannian manifold with a Riemannian metric g and ▽ be a linear connection on Nn+p. If the torsion tensor T of ▽ satisfies T¯(X¯,Y¯)=ϕ(Y¯)X¯ϕ(X¯)Y¯

for a 1-form ϕ, then the connection ▽ is called a semi-symmetric connection. Furthermore, if ▽ satisfies ▽g = 0, then ▽ is called a semi-symmetric metric connection.

Let ▽ denote the Levi-Civita connection associated with the Riemannian metric g. In [15] K.Yano obtained a relation between the semi-symmetric metric connection ▽ and the Levi-Civita connection ▽ which is given by ¯X¯Y¯=¯X¯Y¯+ϕ(Y¯)X¯g(X¯,Y¯)U,X¯,Y¯X(Nn+p)

where U is a vector field given by g(U, X) = ϕ(X).

Let Mn be an n-dimensional submanifold of Nn+p with the semi-symmetric metric connection ▽ and the Levi-Civita connection ▽. On Mn we consider the induced semi-symmetric metric connection denoted by ▽ and the induced Levi-Civita connection denoted by ▽. The Gauss formula with respect to ▽ and ▽ can be written as ¯XY=XY+σ(X,Y),¯XY=XY+σ(X,Y),X,YX(Mn),

where σ is the second fundamental form of Mn and σ is a (0, 2)-tensor on Mn. According to [16] σ is also symmetric.

Let R and R denote the curvature tensors with respect to ▽ and ▽ respectively. We also denote the curvature tensor associated with ▽ and ▽ by R and R, repectively. From [16], we know R¯(X,Y,Z,W)=R¯(X,Y,Z,W)α(Y,Z)g(X,W)+α(X,Z)g(Y,W)α(X,W)g(Y,Z)+α(Y,W)g(X,Z),(1)

for all X, Y, Z, W ∈ 𝓧(Mn), where α is a (0, 2)-tensor field defined by α(X,Y)=(¯Xϕ)Yϕ(X)ϕ(Y)+12ϕ(U)g(X,Y).

Denote by λ the trace of α.

In [16] the Gauss equation for Mn with respect to the semi-symmetric metric connection is given by R¯(X,Y,Z,W)=R(X,Y,Z,W)+g(σ(X,Z),σ(Y,W))g(σ(X,W),σ(Y,Z))(2)

for all X, Y, Z, W ∈ 𝓧(Mn).

In Nn+p we can choose a local orthonormal frame {e1, e2,…, en, en+1, …, en+p} such that {e1, e2, …, en} are tangent to Mn. Setting σijr = g(σ(ei, ej), er), then the squared length of σ is given by σ2=i,j=1ng(σ(ei,ej),σ(ei,ej))=r=nn+pi,j=1n(σijr)2.

The mean curvature vector of Mn associated to ▽ is H=1ni=1nσ(ei,ei). The mean curvature vector of Mn associated to ▽ is defined by H=1ni=1nσ(ei,ei).

According to the formula (7) in [16], we have the following result.

Lemma 2.1

([16]). If U is a tangent vector field on Mn, then H = H and σ = σ.

Let πTpMn be a 2-plane section for any pMn and K(π) be the sectional curvature of Mn associated to the semi-symmetric metric connection ▽. The scalar curvature τ associated to the semi-symmetric metric connection ▽ at p is defined by τ(p)=1i<jnK(eiej).(3)

Let Lk be a k-plane section of TpMn and {e1, e2,…, ek} be any orthonormal basis of Lk. The scalar curvature τ(Lk) of Lk associated to the semi-symmetric metric connection ▽ is given by τ(Lk)=1i<jkK(eiej).(4)

We denote by (inf K)(p) = inf{K(π)∣πTpMn, dimπ = 2}. In [1] B.-Y. Chen introduced the first Chen invariant δM(p) = τ(p) − (inf K)(p), which is certainly an intrinsic character of Mn.

Suppose L is a k-plane section of TpM and X is a unit vector in L, we choose an orthonormal basis {e1, e2,…, ek} of L, such that e1 = X. The Ricci curvature Ricp of L at X is given by RicL(X)=K12+K13+...+K1k(5)

where Kij = K(eiej). We call RicL(X) a k-Ricci curvature.

For each integer k, 2 ⩽ kn, the Riemannian invariant Θk [2] on Mn is defined by Θk(p)=1k1infL,X{RicL(X)},pMn(6)

where L is a k-plane section in TpMn and X is a unit vector in L.

Recently, T. Koufogiorgos [17] introduced the notion of (κ, μ)-contact space form, which contains the well known class of Sasakian space forms for κ = 1. Thus it is worthwhile to study relationships between intrinsic and extrinsic invariants of submanifolds in a (κ, μ) -contact space form with a semi-symmetric metric connection ▽.

A (2m+1)-dimensional differentiable manifold is called an almost contact metric manifold if there is an almost contact metric structure (φ, ξ, η, g) consisting of a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a compatible Riemannian metric g satisfying φ2=I+ηξ,η(ξ)=1,φξ=0,ηφ=0, g(X,φY)=g(φX,Y),g(X,ξ)=η(X),X,YX(M^).(7)

An almost contact metric structure becomes a contact metric structure if = Φ, where Φ(X, Y) = g(X, φY) is the fundamental 2-form of .

In a contact metric manifold , the (1, 1)-tensor field h defined by 2h = 𝓛ξφ is symmetric and satisfies hξ=0,hφ+φh=0,¯ξ=φφh,trace(h)=trace(φh)=0

where ▽ is a Levi-civita connection associated with the Riemannian metric g.

The (κ, μ)-nullity distribution of a contact metric manifold is a distribution N(κ,μ):pNp(κ,μ)={ZTpM^R^(X,Y)Z=κ[g(Y,Z)Xg(X,Z)Y]+μ[g(Y,Z)hXg(X,Z)hY]}

where κ and μ are constants. If ξN(κ, μ), is called a (κ, μ)-contact metric manifold. Since in a (κ, μ)-contact metric manifold one has h2 = (κ − 1)φ2, therefore κ ⩽ 1 and if κ = 1 then the structure is Sasakian.

The sectional curvature (X, φX) of a plane section spanned by a unit vector orthogonal to ξ is called a φ-sectional curvature. If the (κ, μ)-contact metric manifold has constant φ-sectional curvature C, then it is called a (κ, μ)-contact space form and it is denoted by (C). The curvature tensor of (C) is given by [17]. R¯(X,Y)Z=c+34{g(Y,Z)Xg(X,Z)Y}+c+34κ4{η(X)η(Z)Yη(Y)η(Z)X+g(X,Z)η(Y)ξg(Y,Z)η(X)ξ)}+c14{2g(X,φY)φZ+g(X,φZ)φYg(Y,φZ)φX}+12{g(hY,Z)hXg(hX,Z)hY+g(φhX,Z)φhYg(φhY,Z)φhX}g(X,Z)hY+g(Y,Z)hX+η(X)η(Z)hYη(Y)η(Z)hXg(hX,Z)Y+g(hY,Z)Xg(hY,Z)η(X)ξ+g(hX,Z)η(Y)ξ+μ{η(Y)η(Z)hXη(X)η(Z)hY+g(hY,Z)η(X)ξg(hX,Z)η(Y)ξ}(8)

for all X, Y, Z ∈ 𝓧(), where c+2κ = −1 = κμ if κ < 1.

For a vector field X on a submanifold M of a (κ, μ)-contact form (c), let PX be the tangential part of φX. Thus, P is an endomorphism of the tangent bundle of M and satisfies g(X, PY) = −g(PX, Y) for all X, Y ∈ 𝓧(M). (φh)TX and hTX are the tangential parts of φhX and hX respectively. Let {e1, e2, …, en} be an orthonormal basis of TpM. We set ϑ2=i,j=1ng(ei,ϑej)2,ϑ{P,(φh)T,hT}.

Let πTpM be a 2-plane section spanned by an orthonormal basis {e1, e2}. Then, β(π) is given by β(π)=e1,Pe22

is a real number in [0, 1], which is independent of the choice of orthonormal basis {e1, e2}. Put γ(π)=(η(e1))2+(η(e2))2,θ(π)=η(e1)2g(hTe2,e2)+η(e2)2g(hTe1,e1)2η(e1)η(e2)g(hTe1,e2).

Then γ(π) and θ(π) are also real numbers and do not depend on the choice of orthonormal basis {e1, e2}, of course, γ(π) ∈ [0, 1]

3 Chen first inequality

For submanifolds of a (κ, μ) -contact space form endowed with a semi-symmetric metric connection ▽, we establish the following optimal inequality relating the scalar curvature and the squared mean curvature associated with the semi-symmetric metric connection, which is called Chen first inequality [1]. At first we recall the following lemma.

Lemma 3.1

([20]). Let f(x1, x2, …, xn) (n ≥ 3) be a function in Rn defined by f(x1,x2,...,xn)=(x1+x2)i=3nxi+3i<jnxixj.

If x1+x2+⋯+xn = (n − 1)ε, then we have f(x1,x2,...,xn)(n1)(n2)2ε2

with the equality holding if and only if x1+x2 = x3 = ⋯ = xn = ε.

Theorem 3.2

Let M ba an n-dimensional (n ≥ 3) submanifold of a (2m+1)-dimensional (κ, μ)-contact form M̂(c) endowed with a semi-symmetric metric connectionsuch that ξTM. For each 2-plane section πTpM, we have τ(p)K(π)n2(n2)2(n1)H2+18n(n3)(c+3)+(n1)κ+3(c1)8{P22β(π)}+14(c+34κ)γ(π)(μ1)θ(π)12{2trace(h|π)+det(h|π)det((φh)|π)}+(μ+n2)trace(hT)+14{(φh)T2hT2(trace(φh)T)2+(trace(hT))2}(n1)λ+trace(α|π).(9)

If U is a tangent vector field to M, then the equality in (9) holds at pM if and only if there exists an orthonormal basis {e1, e2, …, en} of TpM and an orthonormal basis {en+1, …, e2m+1} of TpM such that π=Span{e1,e2}

and the forms of shape operators ArAer (r = n+1,…, 2m+1) become An+1=σ11n+1000σ22n+1000(σ11n+1+σ22n+1)In2,Ar=σ11rσ12r0σ12rσ11r0000n2,r=n+2,,2m+1.

Proof

Let πTpM be a 2-plane section. We choose an orthonormal basis {e1, e2,…, en} for TpM and {en+1,…, e2m+1} for TpM such that π = Span{e1, e2}.

Setting X = W = ei, Y = Z = ej, ij, i, j = 1,…, n, and using (1), (2), (8), we have Rijji=c+34+c+34κ4{η(ei)2η(ej)2}+c14{3g(ei,φej)2}+12{g(ei,φhej)2g(ei,hej)2+g(ei,hei)g(ej,hej)g(ei,φhei)g(ej,φhej)}+g(ei,hei)+2η(ei)η(ej)g(ei,hej)g(hei,ei)η(ej)2g(hej,ej)η(ei)2+g(hej,ej)+μ{g(hei,ei)η(ej)2+g(hej,ej)η(ei)22η(ei)η(ej)g(ei,hej)}+g(σ(ei,ei),σ(ej,ej))g(σ(ei,ej),σ(ei,ej))α(ei,ei)α(ej,ej).(10)

From (10) we get τ=18{n(n1)(c+3)+3(c1)P22n(n1)(c+34κ)}+14{(φh)T2hT2(trace(φhT))2+(trace(hT))2}+(μ+n2)trace(hT)(n1)λ+r=12m+11i<jn[σiirσjjr(σijr)2].(11)

On the other hand, using (10) we have R1221=14{c+3+3(c1)β(π)(c+34κ)γ(π)+4(μ1)θ(π)}+12{det(h|π)det((φh)|π)+2trace(h|π)}trace(α|π)+r=n+12m+1[σ11rσ22r(σ12r)2].(12)

From (11) and (12) it follows that τK(π)=18n(n3)(c+3)+(n1)κ+3(c1)8[P22β(π)]+14(c+34κ)γ(π)(μ1)θ(π)12{2trace(h|π)+det(h|π)det(φh|π)}+(μ+n2)trace(hT)+14{(φh)T2hT2(trace(φh)T)2+(trace(hT))2}(n1)λ+trace(α|π)+r=n+12m+1[(σ11r+σ22r)3inσiir+3i<jnσiirσjjr3jn(σ1jr)22i<jn(σijr)2)]18n(n3)(c+3)+(n1)κ+3(c1)8[p22β(π)]+14(c+34κ)γ(π)(μ1)θ(π)12{2trace(h|π)+det(h|π)det(φh|π)}+(μ+n2)trace(hT)+14{(φh)2hT2(trace(φh)T)2+(trace(hT))2}(n1)λ+trace(α|π)+r=n+12m+1[(σ11r+σ22r)3inσiir+3i<jnσiirσjjr].(13)

Let us consider the following problem: max{fr(σ11r,...,σnnr)=(σ11r+σ22r)3inσiir+3i<jnσiirσjjr|σ11r+...+σnnr=kr}

where kr is a real constant. From Lemma 3.1, we know frn22(n1)(kr)2,(14)

with the equality holding if and only if σ11r+σ22r=σiir=krn1,i=3,...,n.(15)

From (13) and (14), we have τK(π)n2(n2)2(n1)H2+18n(n3)(c+3)+(n1)κ+3(c1)8[P22β(π)]+14(c+34κ)γ(π)(μ1)θ(π)12[2trace(h|π)+det(h|π)det((φh)|π)]+(μ+n2)trace(hT)+14[(φh)T2hT2(trace(φh)T)2+(trace(hT))2](n1)λ+trace(α|π).

If the equality in (9) holds, then the inequalities given by (13) and (14) become equalities. In this case we have 3in(σ1ir)2=0,2i<jn(σijr)2=0,r.σ11r+σ22r=σiir,3in,r.

Since U is a tangent vector field to M, we know σ = σ. By choosing a suitable orthonormal basis, the shape operators take the desired forms.

The converse is easy to show. □

For a Sasakian space form (c), we have κ = 1 and h = 0. So using Theorem 3.2, we have the following corollary.

Corollary 3.3

Let M be an n-dimensional (n ≥ 3) submanifold in a Sasakian space form M̂(c) endowed with a semi-symmetric metric connection such that ξTM. Then, for each point pM and each plane section πTpM, we have τK(π)n2(n2)2(n1)H2+18n(n3)(c+3)+(n1)+3(c1)8[P22β(π)]+c14γ(π)(n1)λ+trace(α|π).(16)

If U is a tangent vector field to M, then the equality in (16) holds at pM if and only if there exists an orthonormal basis {e1, e2, …, en} of TpM and orthonormal basis {en+1, …, e2m+1} of TpM such that π=Span{e1,e2}

and the forms of shape operators ArAer (r = n+1,…, 2m+1) become An+1=σ11n+1000σ22n+1000(σ11r+σ22r)In2,Ar=σ11rσ12r0σ12rσ11r0000n2,r=n+2,,2m+1.

Since in case of non-Sasakian (κ, μ)-contact space form, we have κ < 1, thus c = −2κ − 1 and μ = κ+1. Putting these values in (9), we can have a direct corollary to Theorem 3.2.

Corollary 3.4

Let Let M be an n-dimensional (n ≥ 3) submanifold in a non-Sasakian (κ, μ)-contact space form M̂(c) with a semi-symmetric metric connection such that ξTM. Then, for each point pM and each plane section πTpM, we have τK(π)n2(n2)2(n1)H214n(n3)(κ1)+(n1)κ34(κ+1)P2+12[3(κ+1)β(π)(3κ1)γ(π)2κθ(π)]12[2trace(h|π)+det(h|π)det((φh)|π)]+(κ+n1)trace(hT)+14[(φh)T2hT2(trace(φh)T)2+(trace(hT))2](n1)λ+trace(α|π).(17)

If U is a tangent vector field to M, then the equality in (17) holds at pM if and only if there exists an orthonormal basis {e1, e2, …, en} of TpM and orthonormal basis {en+1,…, e2m+1} of TpM such that π=Span{e1,e2}

and the forms of shape operators ArAer(r = n+1,…, 2m+1) become An+1=σ11n+1000σ22n+1000(σ11r+σ22r)In2,Ar=σ11rσ12r0σ12rσ11r0000n2,r=n+2,,2m+1.

4 Ricci and k-Ricci curvature

In this section, we establish inequalities between Ricci curvature and the squared mean curvature for submanifolds in a (κ, μ)-contact space form with a semi-symmetric metric connection. These inequalities are called Chen-Ricci inequalities [2].

First we give a lemma as follows.

Lemma 4.1

([20]). Let f(x1, x2, …, xn) be a function in Rn defined by f(x1,x2,...,xn)=x1i=2nxi.

If x1+x2+ …+xn = 2ε, then we have f(x1,x2,...,xn)ε2,

with the equality holding if and only if x1 = x2+ … +xn = ε.

Theorem 4.2

Let M be an n-dimensional (n ⩾ 2) submanifold of a (2m+1)-dimensional (κ, μ)-contact space form M̂(c) endowed with a semi-symmetric metric connection such that ξTM. Then for each point pM,

  1. For each unit vector X in TpM, we have Ric(X)n24H2+(n1)(c+3)4+3(c1)4PX2c+34κ4[1+(n2)η(X)2]+12[(φhX)T2(hX)T2g(φhX,X)trace((φh)T)+g(hX,X)trace(hT)]+(μ+n3)g(hX,X)+[1+(μ1)η(X)2]trace(hT)(n2)α(X,X)λ.(18)

  2. If H(p) = 0, a unit tangent vector XTpM satisfies the equality case of (18) if and only if XN(p) = {X ∈ {TpM|σ(X, Y) = 0, ∀YTpM}.

  3. The equality of (18) holds identically for all unit tangent vectors if and only if either

    1. n ≠ 2, σijr = 0, i, j = 1, 2, …, n; r = n+1, …, 2m+1;

      or

    2. n = 2, σ11r=σ22r,σ12r=0,r=3,...,2m+1..

Proof

  1. Let XTpM be an unit vector. We choose an orthonormal basis e1, …, en, en+1 …, e2m+1 such that e1, …, en are tangential to M at p with e1 = X.

    Using (10), we have Ric(X)=(n1)(c+3)4c+34κ4[1+(n2)η(X)2]+3(c1)4PX2+12[(φhX)T2(hX)T2+g(hX,X)trace(hT)g(φhX,X)trace((φh)T)]+(μ+n3)g(X,hX)+(1η(X)2+μη(X)2)trace(hT)(n2)α(X,X)λ+r=n+12m+1i=2n[σ11rσiir(σ1ir)2](n1)(c+3)4+3(c1)4PX2c+34κ4[1+(n2)η(X)2]+12[(φhX)T2(hX)T2g(φhX,X)trace(φh)T+g(hX,X)trace(hT)]+(μ+n3)g(hX,X)+[1η(X)2+μη(X)2]trace(hT)(n2)α(X,X)λ+r=n+12m+1i=2nσ11rσiir.(19)

    Let us consider the function fr: RnR, defined by fr(σ11r,σ22r,...,σnnr)=i=2nσ11rσiir.

    We consider the problem max{fr|σ11r+...+σnnr=kr},

    where kr is a real constant. From Lemma 4.1, we have fr(kr)24,(20)

    with equality holding if and only if σ11r=i=2nσiir=kr2.(21)

    From (19) and (20) we get Ric(X)n24H2+(n1)(c+3)4+3(c1)4PX2c+34κ4[1+(n2)η(X)2]+12[(φhX)T2(hX)T2g(φhX,X)trace(φh)T+g(hX,X)trace(hT)]+(μ+n3)g(hX,X)+[1+(μ1)η(X)2]trace(hT)(n2)α(X,X)λ.

  2. For a unit vector XTpM, if the equality case of (18) holds, from (19), (20) and (21) we have σ1ir=0,i1,r.σ11r+σ22r+...+σnnr=2σ11r,r.

    Since H(p) = 0, we know σ11r=0,r.

    So we get σ1jr=0,r.

    i.e XN(p)

    The converse is trivial.

  3. For all unit vectors XTpM, the equality case of (18) holds if and only if 2σiir=σ11r+...+σnnr,i=1,...,n;r=n+1,...,2m+1.σijr=0,ij,r=n+1,...,2m+1.

    Thus we have two cases, namely either n ≠ 2 or n = 2.

    In the first case we have σijr=0,i.j=1,...,n;r=n+1,...,2m+1.

    In the second case we have σ11r=σ22r,σ12r=0,r=3,...,2m+1.

    The converse part is straightforward. □

Corollary 4.3

Let M be an n-dimensional (n ⩾ 2) submanifold in a Sasakian space form M̂(c) endowed with a semi-symmetric metric connection such that ξTM. Then for each point pM and for each unit vector XTpM, we have Ric(X)n24H2+(n1)(c+3)4+3(c1)4PX2c14[1+(n2)η(X)2](n2)α(X,X)λ.

Corollary 4.4

Let M be an n-dimensional (n ⩾2) submanifold in a non-Sasakian space form M̂(c) endowed with a semi-symmetric metric connection such that ξTM. Then for each point pM and for each unit vector XTpM, we have Ric(X)n24H2+(n1)(κ+1)23(κ+1)2PX213κ2[1+(n2)η(X)2]+12[(φhX)T2(hX)T2g(φhX,X)trace(φh)T+g(hX,X)trace(hT)]+(κ+n2)g(hX,X)+[1+κη(X)2]trace(hT)(n2)α(X,X)λ.

Theorem 4.5

Let M be an n-dimensional (n ⩾ 3) submanifold in a (2m+1)-dimensional (κ, μ)-contact space form M̂(c) endowed with a semi-symmetric connection such that ξTM. Then we have n(n1)H2n(n1)Θk(p)14{n(n1)(c+3)+3(c1)P22(n1)(c+34κ)}12{(φh)T2hT2(trace(φ(h)T))2+(trace(hT))2}2[μ+(n2)]trace(hT)+2(n1)λ.

Proof

Let {e1, …, en} be an orthonormal basis of TpM. We denote by Li1, …, ik the k-plane section spanned by ei1, …, eik. From (4) and (5), it follows that τ(Li1,...,eik)=12i{i1,...,ik}RicLi1,...,ik(ei)(22)

and τ(p)=1Cn2k21i1<...<iknτ(Li1,...,ik).(23)

Combining (6), (22) and (23), we obtain τ(p)n(n1)2Θk(p).(24)

We choose an orthonormal basis {e1, …, en} of TpM such that en+1 is in the direction of the mean curvature vector H(p) and {e1, …, en} diagonalizes the shape operator An+1. Then the shape operators take the following forms: An+1=a10...00a2...0.....0.....0.....000...an,(25) traceAr=0,r=n+2,...,2m+1.(26)

From (10), we have 2τ=14{n(n1)(c+3)+3(c1)P22(n1)(c+34κ)}+12{(φh)T2hT2(trace(φh)T)2+(trace(hT))2}+2[μ+(n2)]trace(hT)2(n1)λ+n2H2σ2.(27)

Using (25) and (27), we obtain n2H2=2τ+i=1nai2+r=n+22m+1i,j=1n(σijr)214{n(n1)(c+3)+3(c1)P22(n1)(c+34κ)}12{(φh)T2hT2(trace(φh)T)2+(trace(hT))2}2[μ+(n2)]trace(hT)+2(n1)λ.(28)

On the other hand from (25) and (26), we have (nH)2=(ai)2ni=1nai2.(29)

From (28) and (29), it follows that n(n1)H22τ14{n(n1)(c+3)+3(c1)P22(n1)(c+34κ)}12{(φh)T2hT2(trace(φh)T)2+(trace(hT))2}2[μ+(n2)]trace(hT)+2(n1)λ+r=n+22m+1i,j=1n(σijr)22τ14{n(n1)(c+3)+3(c1)P22(n1)(c+34κ)}12{(φh)T2hT2(trace(φh)T)2+(trace(hT))2}2[μ+(n2)]trace(hT)+2(n1)λ.

Using (24), we obtain n(n1)H2n(n1)Θk(p)14{n(n1)(c+3)+3(c1)P22(n1)(c+34κ)}12{(φh)T2hT2(trace(φh)T)2+(trace(hT))2}2[μ+(n2)]trace(hT)+2(n1)λ.

Corollary 4.6

Let M be an n-dimensional (n ⩾ 3) submanifold in a Sasakian space form M̂(c) endowed with a semi-symmetric connection such that ξTM. Then for each point pM and for each unit vector XTpM, we have n(n1)H2n(n1)Θk(p)14{n(n1)(c+3)+3(c1)P22(n1)(c1)}+2(n1)λ.

Corollary 4.7

Let M be an n-dimensional (n ⩾ 3) submanifold in a non-Sasakian space form M̂(c) endowed with a semi-symmetric connection such that ξTM. Then for each point pM and for each unit vector XTpM, we have n(n1)H2n(n1)Θk(p)14{2n(n1)(κ+1)6(κ+1)P2+4(n1)(3κ1)}12{(φh)T2hT2(trace(φh)T)2+(trace(hT))2}2(κ+n1)trace(hT)+2(n1)λ.

Acknowledgement

Supported by NNSF of CHINA(11371194).

References

  • [1]

    Chen B.-Y., Some pinching and classfication theorems for minimal submanifolds, Arch. Math., 1993, 60, 568-578. CrossrefGoogle Scholar

  • [2]

    Chen B.-Y., Relations between Ricci curvature and sharp operator for submanifolds with arbitrary codimensions, Glasg. Math. J., 1999, 41, 33-41. CrossrefGoogle Scholar

  • [3]

    Oiagǎ A., Mihai I., Chen B.-Y., Inequalities for slant submanifolds in complex space forms, Demonstr. Math., 1999, 32, 835-846. Google Scholar

  • [4]

    Arslan K., Ezentas R., Mihai I., Özgür C., Certain inequalities for submanifolds in (κ, μcontact space forms, Bulletin of the Australian Mathematical Society, 2001, 64(2), 201-212. CrossrefGoogle Scholar

  • [5]

    Tripathi M.M., Certain basic inequalities for submanifolds in (κ, μcontact space forms, In: Recent advances in Riemannian and Lorentzian geomtries (Baltimore, MD, 2003), Contemp. Math., vol. 337, 187-202, Amer. Math. Soc., Province, RI(2003). Google Scholar

  • [6]

    Mihai A., Chen B.-Y., Inequalities for slant submanifolds in generalized complex space forms, Radovi Mathematicki, 2004, 12, 215-231. Google Scholar

  • [7]

    Li G., Wu C., Slant immersions of complex space forms and Chen’s inequality, Acta Mathematica Scientia, 2005, 25B(2), 223-232. Google Scholar

  • [8]

    Chen B.-Y., Dillen F., δ-invariants for Lagrangian submanifolds of complex space forms, in Riemannian Geometry and Applications proceedings RIGA 2011, Ed. Univ. Bucuresti, Bucharest, 2011, 75-94. Google Scholar

  • [9]

    Chen B.-Y., Dillen F., Optimal general inequalities for Lagrangian submanifolds in complex space forms, J. Math. Anal. Appl., 2011, 379, 229-239. CrossrefWeb of ScienceGoogle Scholar

  • [10]

    Özgür C., Chen B.Y., inequalities for submanifolds a Riemannian manifold of a quasiconstant curvature, Turk. J. Math., 2011, 35, 501-509. Google Scholar

  • [11]

    Chen B.-Y., An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Int. J. Math., 2012, 23, no. 3, 1250045. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    Chen B.-Y., Dillen F., Van der Veken J., Vrancken L., Curvature inequalities for Lagrangian submanifolds: The final solution, Differential Geom. Appl., 2013, 31, 808-819. CrossrefWeb of ScienceGoogle Scholar

  • [13]

    Friedmann A., Schouten J.A., Über die Geometrie der halbsymmetrischen Übertragungen, Math. Z., 1924, 21, no. 1, 211-223. CrossrefGoogle Scholar

  • [14]

    Hayden H.A., Subspaces of a space with torsion, Proc. London Math. Soc., 1932, 34, 27-50. Google Scholar

  • [15]

    Yano K., On semi-symmetric metric connection, Rev. Roumaine de Math. Pures et Appl., 1970, 15, 1579-1586. Google Scholar

  • [16]

    Nakao Z., Submanifolds of a Riemannian manifold with semi-symmetric metric connection, Proc. AMS., 1976, 54, 261-266. CrossrefGoogle Scholar

  • [17]

    Koufogiorgos T., Contact Riemannian manifolds with constant φ-sectional curvature, Tokyo J. Math., 1997, 20, no.1, 13-22. CrossrefGoogle Scholar

  • [18]

    Mihai A., Özgür C., Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection, Taiwan. J. Math., 2010, 4(14), 1465-1477. Web of ScienceGoogle Scholar

  • [19]

    Mihai A., Özgür C., Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi-symmetric metric connections[J], Rocky Mountain J. Math., 2011, 41, no. 5, 1653-1673. Web of ScienceCrossrefGoogle Scholar

  • [20]

    Zhang P., Zhang L., Song W., Chen’s inequalities for submanifolds of a Riemannian manifold of Quasi-constant curvature with a semi-symmetric metric connection, Taiwan. J. Math., 2014, 18, no. 6, 1841-1862. Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2017-09-24

Accepted: 2018-02-27

Published Online: 2018-04-18


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 380–391, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0034.

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© 2018 Ahmad et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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