# Chen-Ricci Inequalities with a Quarter Symmetric Connection in Generalized Space Forms

Authors: Ali H. Al-Khaldi; Mohd. Aquib; Mohd Aslam; Meraj Ali Khan
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this article, we obtain improved Chen-Ricci inequalities for submanifolds of generalized space forms with quarter-symmetric metric connection, with the help of which we completely characterized the Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian space form. We also discuss some geometric applications of the obtained results.


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## Body

## 1. Introduction

One of the most basic problems in submanifold theory is to develop a simple relationship between the extrinsic invariants and the intrinsic invariants. The sectional curvature, the scalar curvature, and the Ricci curvature are the main intrinsic invariants while the squared mean curvature is the main extrinsic invariant.Chen obtained the following important bound of the Ricci curvatureRic in terms of the mean curvature H for Lagrangian submanifolds in complex space forms [1]:
(1)Ric≤m−1c+m24H2,where c is the constant holomorphic sectional curvature of the complex space form.Further, he discussed the geometry of a Lagrangian submanifold satisfying the equality case of the inequality under the condition that the dimension of the kernel of the second fundamental form is constant. The inequality (1) is known as the Chen-Ricci inequality. This inequality attracted many researchers due to its geometric importance [2–12].Deng [13] improved the above inequality as
(2)RicU≤m−14c+mH2.In [14], Deng further extended his result for Lagrangian submanifolds in quaternion space forms. In [15], Tripathi improved the inequality in the case of curvature-like tensors. In [6], Mihai and Radulescu obtained the same relation in Sasakian space forms using semisymmetric connection as
(3)RicU+m−2αU,U+trα≤m−14c+3+mH2.As the curvature invariants are of great interest in theoretical physics (see [16]), the above studies motivate us to obtain a complete characterization of Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian space form.


## 2. Preliminaries

LetN be a Riemannian manifold and ∇¯ be a linear connection on N. Then, ∇¯ is said to be a semisymmetric connection if its torsion tensor T satisfies
(4)TU,V=πVU−πUV,for a 1-form π, then the connection ∇¯ is called a semisymmetric connection [17]. Let g be a Riemannian metric on N. If ∇¯g=0, then ∇¯ is called a semisymmetric metric connection on N. The semisymmetric metric connection ∇¯ on N is given by
(5)∇¯UV=∇¯UV+πVU−gU,VΓ,for any U,V on N, where ∇¯ denotes the Levi-Civita connection with respect to Riemannian metric g and Γ is a vector field. Further, ∇¯ is said to be a semisymmetric nonmetric connection if it satisfies
(6)∇¯UV=∇¯UV+πVU.Moreover, the linear connection∇¯ on a Riemannian manifold N with Riemannian metric g is said to be a quarter-symmetric connection if its torsion tensor T is given by
(7)TU,V=∇¯UV−∇¯VU−U,V,which satisfies
(8)TU,V=πVϕU−πUϕV,such that π is a 1-form given by
(9)πU=gU,Γ,where Γ is a vector field and ϕ is a (1,1) tensor field.Then, we can define a special quarter-symmetric connection by(10)∇¯UV=∇¯UV+ψ1πVU−ψ2gU,VΓ,where ψ1 and ψ2 are real constants.Remark 1.
We notice from (5) that [18]
(1)
ifψ1=ψ2=1, then a quarter symmetric connection becomes a semisymmetric metric connection(2)
ifψ1=1 and ψ2=0, then a quarter-symmetric connection becomes a semisymmetric nonmetric connectionRemark 2.
It is also worthy to mention here that the quarter symmetric connections generalized several well-known connections.
The curvature tensorR¯ with respect to ∇¯ is
(11)R¯U,VZ=∇¯U∇¯VZ−∇¯V∇¯UZ−∇¯U,VZ.
In the same way, we can also define the curvature tensorR¯~.Let(12)β1U,V=∇¯UπV−ψ1πUπV+ψ22gU,VπΓ,β2U,V=πΓ2gU,V+πUπV,are 0,2 tensors. Then, the curvature tensor of N is given by [19]
(13)R¯U,V,Z,W=R¯~U,V,Z,W+ψ1β1U,ZgV,W−ψ1β1V,ZgU,W+ψ2β1V,WgU,Z−ψ2β1U,WgV,Z+ψ2ψ1−ψ2gU,Zβ2V,W−ψ2ψ1−ψ2gV,Zβ2U,W.LetM be an m-dimensional submanifold in a Riemannian manifold N. Let ∇ and ∇~ be the induced quarter symmetric-metric connection and Levi-Civita connection, respectively, on M. Then, the Gauss formulas are
(14)∇¯UV=∇UV+ζU,V,U,V∈ΓTM,∇¯UV=∇~UV+ζ~U,V,U,V∈ΓTM,where ζ~ is the second fundamental form that satisfies the relation
(15)ζU,V=ζ~U,V−ψ2gU,VΓ⊥,where Γ⊥ is the normal component of the vector field Γ on M.Moreover, the equation of Gauss is defined by [19]
(16)R¯U,V,Z,W=RU,V,Z,W−gζU,W,ζV,Z+gζV,W,ζU,Z+ψ1−ψ2gζV,Z,Γ⊥gU,W+ψ2−ψ1gζU,Z,Γ⊥gV,W.


## 3. Characterization of Lagrangian Submanifold in Generalized Complex Space Form

A smooth manifoldN endowed with an almost complex structure J and a Riemannian metric g that is compatible with J is called an almost Hermitian manifold. Further, for the Levi-Civita connection ∇ if ∇J=0, then an almost Hermitian manifold is said to be a Kaehler manifold. A Kaehler manifold of constant holomorphic curvature is called a complex space form. The curvature tensor of a complex space form is given by
(17)R¯~U,V,Z,W=c4gV,ZgU,W−gU,ZgV,W+gU,JZgJV,W−gV,JZgJU,W+2gU,JVgJZ,W.However, an almost Hermitian manifoldN is called a generalized complex space form [20–22], denoted by Nf1,f2, if for all vector fields U,V, and Z on N, the Riemannian curvature tensor R¯~ satisfies
(18)R¯~U,V,Z,W=f1gV,ZgU,W−gU,ZgV,W+f2gU,JZgJV,W−gV,JZgJU,W+2gU,JVgJZ,W,where f1 and f2 are smooth functions on N.In fact, we have following fundamental result from Tricerri and Vanhecke [20].Theorem 3 (see [20]).
LetN be a connected almost Hermitian manifold with real dimension 2m>6 and Riemannian curvature R¯~ is of the form (18) such that f2 is not identically zero. Then, N is a complex space form.Remark 4.
From (18), we notice that if f1=f2=c/4, then we recover the complex space form.From (13) and (18), we have
(19)R¯U,V,Z,W=f1gV,ZgU,W−gU,ZgV,W+f2gU,JZgJV,W−gV,JZgJU,W+2gU,JVgJZ,W+ψ1β1U,ZgV,W−ψ1β1V,ZgU,W+ψ2β1V,WgU,Z−ψ2β1U,WgV,Z+ψ2ψ1−ψ2gU,Zβ2V,W−ψ2ψ1−ψ2gV,Zβ2U,W.Lemma 5 (see [13]).
Letf1u1,u2,⋯,um be a function on ℝm defined by
(20)f1u1,u2,⋯,um=u1∑j=2muj−∑j=2muj2.
Ifu1+u2+⋯+um=2ma, then
(21)f1u1,u2,⋯,um≤m−14mu1+u2+⋯+um2,and the equality holds if and only if 1/m+1u1=u2=⋯=um=a, where a is a constant.Lemma 6 (see [13]).
Letf2u1,u2,⋯,um be a function on ℝm defined by
(22)f2u1,u2,⋯,um=u1∑j=2muj−u12.
Ifu1+u2+⋯+um=4a, then
(23)f2u1,u2,⋯,um≤18u1+u2+⋯+um2,
and the equality holds if and only ifu1=a and u2+⋯+um=3a, where a is a constant.LetMm be an m-dimensional submanifold of an almost Hermitian manifold N. Then, Mm is said to be totally real if
(24)JTpMm⊂Tp⊥Mm.Then, we have the following relations [23]:
(25)A~JUV=A~JVU,U,V∈TpM,or equivalently,
(26)ζ~ijk=A~ikj=ζ~jki,∀i,j,k=1,⋯m,where A~k is the shape operator with respect to ∇¯ and
(27)ζ~ijk=gζ~ei,ej,Jek,i,j,k=1,⋯,m.Remark 7.
A totally real submanifold which is of maximal dimension is known as the Lagrangian submanifold [24].Definition 8 (see [25]).
A nontotally geodesic Lagrangian submanifoldMm of a complex space form N2m4c is called H-umbilical if its second fundamental form satisfies
(28)he1,ei=μJen,hei,em=μJei,i=1⋯m−1,hem,em=λJem,hei,ej=0,1≤i≠j≤m−1,for some functions μ and λ with respect to an orthonormal frame e1,⋯,em, where J is the complex structure of N2m4c.Theorem 9.
LetMm be a totally real submanifold of maximal dimension mm≥2 in a connected complex space form Nf1,f2 of dimension 2m with a quarter-symmetric metric connection such that the vector field Γ is tangent to Mm. Then, for any unit tangent vector U to Mm(29)mm−14H2≥RicU−f1m−1−ψ2+ψ11−mβ1U,U+ψ2traceβ1−ψ2ψ1−ψ2β2U,U+traceβ2−m−1ψ1−ψ2πζ,
and the equality holds in (29) identically if and only if either
(1)
Mm is totally geodesic, provided that m>2, or(2)
m=2 and M2 is a H-umbilical Lagrangian surface with λ=3μProof.
AsΓ is tangent to Mm, we have
(30)ζ=ζ~,H=H~.
Let us assume an orthonormal basise1=U,e2,⋯,em⊂TpMm and em+1=Je1,⋯,e2m=Jem⊂Tp⊥Mm at point p∈Mm with unit vector U∈TpMm. Then, by combining (16) and (19) and substituting U=W=ej and V=Z=e1, for j=2,⋯,m, we get
(31)Rej,e1,e1,ej=f1ge1,e1gej,ej−gej,e1ge1,ej+f2gej,Je1gJe1,ej−ge1,Je1gJej,ej+2gej,Je1gJe1,ej+ψ1β1ej,e1ge1,ej−ψ1β1e1,e1gej,ej+ψ2β1e1,ejgej,e1−ψ2β1ej,ejge1,e1+ψ2ψ1−ψ2gej,e1β2e1,ej−ψ2ψ1−ψ2ge1,e1β2ej,ej+gζej,ej,ζe1,e1−gζe1,ej,ζej,e1−ψ1−ψ2gζe1,e1,Γ⊥gej,ej−ψ2−ψ1gζej,e1,Γ⊥ge1,ej.
Taking the summation overj=2,⋯,m, we find
(32)RicU=f1m−1+ψ2+ψ11−mβ1U,U−ψ2traceβ1+ψ2ψ1−ψ2β2U,U−traceβ2−m−1ψ1−ψ2πζ+∑s=1m∑j=2mζ11sζjjs−ζ1js2,which implies
(33)RicU−f1m−1−ψ2+ψ11−mβ1U,U+ψ2traceβ1−ψ2ψ1−ψ2β2U,U−traceβ2+m−1ψ1−ψ2πζ=∑s=1m∑j=2mζ11sζjjs−ζ1js2≤∑s=1m∑j=2mζ11sζjjs−∑j=2mζ1j12−∑j=2mζ1jj2.
From the above equation and (26), it is easy to see that
(34)RicU−f1m−1−ψ2+ψ11−mβ1U,U+ψ2traceβ1−ψ2ψ1−ψ2β2U,U−traceβ2+m−1ψ1−ψ2πζ≤∑s=1m∑j=2mζ11sζjjs−∑j=2mζ11j2−∑j=2mζjj12.
Putting(35)f1ζ111,ζ221,⋯,ζmm1=ζ111∑j=2mζjj1−∑j=2mζjj12,fsζ11s,ζ22s,⋯,ζmms=ζ11s∑j=2mζjjs−ζ11s2,∀s=2,⋯,m,and combining Lemma 5 with the relation mH1=ζ111+ζ221+⋯+ζmm1, we obtain
(36)f1ζ111,ζ221,⋯,ζmm1≤m−14mmH12=mm−14H12.
Then, by Lemma6 for s=2,⋯,m, we get
(37)fsζ11s,ζ22s,⋯,ζmms≤18mHs2=m28Hs2≤mm−14Hs2.
We find from (34), (36), and (37) that
(38)RicU≤f1m−1+ψ2+ψ11−mβ1U,U−ψ2traceβ1+ψ2ψ1−ψ2β2U,U−traceβ2−m−1ψ1−ψ2πζ,+mm−14∥H∥2,which is the desired inequality (29).☐Now, we discuss the equality cases.Case 1.
Form>2, if Je1H, then
(39)Hs=0,for all s>1. Therefore, using Lemma 6, we derive
(40)ζ1j1=ζ11j=mHj4=0,forallj>1,ζjk1=0,forallj,k>1,j≠k.
Further, Lemma5 yields
(41)ζ111=m+1H12,ζjj1=H12,forallj>1.
In (33), we see that RicU=Rice1. In the same way, by deriving Rice2 and making use of the equality, we conclude that
(42)ζ2js=ζjs2=0,foralls≠2,j≠2,s≠j.
In consequence, we find(43)ζ112n+1=ζ222=⋯=ζmm2=H22=0.
We see that the equality holds for every unit tangent vectors. The above conclusion is also valid forζjks. Thus,
(44)ζ2j2=ζ22j=Hj2=0,∀j≥3.
Then, the only possible nonzero entries forζjk2 (resp., for ζjks) are
(45)ζ122=ζ212=ζ221=H12respectivelyζ1ss=ζs1s=ζss1=H12,∀s≥3.
SubstitutingU=Z=e2 and V=W=ej,j=2,⋯,m in (16), we derive
(46)R~e2,ej,e2,ej=Re2,ej,e2,ej−H122,∀j≥3.
On the other hand, if we substituteU=Z=e2 and V=W=e1 in (16), we get
(47)R~e2,e1,e2,e1=Re2,e1,e2,e1−m+1H122+H122.
Using (46) and (47), we find
(48)Rice2−f1n−1−ψ1+ψ2−nψ1β1e2,e2+ψ1traceβ1−ψ2ψ1−ψ2mβ2e2,e2−traceβ2+m−1ψ1−ψ2πζ=2m−1H122.
Moreover, the equality case of (29) implies that
(49)Rice2−f1m−1−ψ1+ψ2−mψ1β1U,U+ψ1traceβ1−ψ2ψ1−ψ2mβ2U,U−traceβ2+m−1ψ1−ψ2πζ=mm−1H122.
Using the factm≠1,2, by (48) and (49), it is easy to see that H1=0. This implies that Mm is a totally geodesic in N2m4c.Case 2.
In casem=2, M2 is nontotally geodesic, then ζe1,e1=λe3,ζe2,e2=μe3,ζe1,e2=μe4, together with λ=3μ. This proves that M2 is H-umbilical surface.The above theorem gives the following results.Corollary 10.
LetMm be a totally real submanifold of maximal dimension mm≥2 in a connected complex space form Nf1,f2 of dimension 2m with a semisymmetric metric connection such that the vector field Γ is tangent to Mm. Then, for any unit tangent vector U to Mm(50)RicU≤mm−14H2+f1m−1+2−mβ1U,U−traceβ1,and the equality holds in (50) identically if and only if either
(1)
Mm is totally geodesic, provided m>2, or(2)
m=2 and M2 is a H-umbilical Lagrangian surface with λ=3μProof.
Using the factψ1=ψ2=1 together with Theorem 9, the result directly follows.☐Remark 11.
It is worthy to mention here that Corollary10 together with Remark 4 is the main result for complex case of the paper [26].Corollary 12.
LetMm be a totally real submanifold of maximal dimension mm≥2 in a connected complex space form Nf1,f2 of dimension 2m with a semisymmetric nonmetric connection such that the vector field Γ is tangent to Mm. Then, for any unit tangent vector U to Mm(51)RicU≤mm−14∥H∥2+f1m−1+1−mβ1U,U+m−1πζ,and the equality holds in (51) identically if and only if either
(1)
Mm is totally geodesic, provided n>2, or(2)
M2 is a H-umbilical Lagrangian surface with λ=3μProof.
Using the factψ1=1 and ψ2=0 together with Theorem 9, the result directly follows.☐


## 4. Characterization of Legendrian Submanifold in Generalized Sasakian Space Form

Let a2m+1-dimensional almost contact metric manifold N2m+1 furnished with the almost complex structure φ,ξ,η,g, where φ is a (1,1) tensor field, ξ is the structure vector field, η, the 1-form, and g is the Riemannian metric on N2m+1. Then, following relations hold good:
(52)φ2=−I+η⊗ξ,ηξ=1,gφU,φV=gU,V−ηUηV.It also follows from the above relations that(53)φξ=0,ηφU=0,ηU=gU,ξ,gφU,V+gU,φV=0,for all vector fields U,V on N.LetN,φ,ξ,η,g be an almost contact metric manifold whose curvature tensor satisfies [27]
(54)R¯~U,VZ=f1gV,ZU−gU,ZV+f2gU,φZφV−gV,φZφU+2gU,φVφZ+f3ηUηZV−ηVηZU+gU,ZηVξ−gV,ZηUξ,for all vector fieldsU,V,Z on N, where f1,f2,f3 are differentiable functions on N. Then, Nf1,f2,f3 is said to be a generalized Sasakian space form.Remark 13.
The generalized Sasakian space forms are [27]
(1)
Sasakian space forms iff1=c+3/4,f2=f3=c−1/4(2)
Kenmotsu space forms iff1=c−3/4 and f2=f3=c+1/4(3)
cosymplectic space forms iff1=f2=f3=c/4From (13) and (54), we have
(55)R¯U,VZ=f1gV,ZU−gU,ZV+f2gU,φZφV−gV,φZφU+2gU,φVφZ+f3ηUηZV−ηVηZU+gU,ZηVξ−gV,ZηUξ+ψ1β1U,ZgV,W−ψ1β1V,ZgU,W+ψ2β1V,WgU,Z−ψ2β1U,WgV,Z+ψ2ψ1−ψ2gU,Zβ2V,W−ψ2ψ1−ψ2gV,Zβ2U,W.A submanifoldMm of an almost contact manifold N2n+1 normal to ξ is called a C-totally real submanifold. On such a submanifold, φ maps any tangent vector to Mm at p∈Mm into the normal space Tp⊥Mm . In particular, if n=m, i.e., Mm has maximum dimension, then it is a Legendrian submanifold. For a Legendrian submanifold Mm, if e1,⋯,em and em+1=φe1,⋯,e2m=φem,e2m+1=ξ be tangent orthonormal frame and normal orthonormal frame, respectively, on Mm. One has
(56)A~JUV=A~JVU,U,V∈TpM,or equivalently,
(57)ζ~ijk=A~ikj=ζ~jki,∀i,j,k=1,⋯,m,where A~k is the shape operator with respect to ∇¯ and
(58)ζ~ijk=gζ~ei,ej,Jek,i,j,k=1,⋯,m.Definition 14 (see [28]).
A nontotally geodesic Legendrian submanifoldMm of a Sasakian space form N2m+14c is called H-umbilical if its second fundamental form satisfies
(59)he1,e1=λϕe1,he2,e2=⋯=hem,em=μϕe1,he1,ej=μϕej,hej,ek=0,j≠k,j,k=2⋯,m,for some functions μ and λ with respect to an orthonormal frame e1,⋯,em, where ϕ is the contact structure of N2m+14c.Theorem 15.
LetMm be a totally real submanifold of maximal dimension mm≥2 in a generalized Sasakian space form Nf1,f2,f3 of dimension 2m+1 with a quarter-symmetric metric connection such that the vector field Γ is tangent to Mm. Then, for any unit tangent vector U to Mm(60)RicU≤mm−14H2+f1m−1+ψ2+ψ11−mβ1U,U−ψ2traceβ1+ψ2ψ1−ψ2β2U,U+traceβ2+m−1ψ1−ψ2πζand the equality holds in (60) identically if and only if either
(1)
Mm is totally geodesic, provided m>2, or(2)
m=2 and M2 is a H-umbilical Legendrian surface with λ=3μProof.
AsΓ is tangent to Mm, we have
(61)ζ=ζ~,H=H~.
Let us assume an orthonormal basise1=U,e2,⋯,em⊂TpMm and em+1=φe1,⋯,e2m=φem,e2m+1=ξ⊂Tp⊥Mm at point p∈Mm with unit vector U∈TpMm. Then, by combining (16) and (55) and substituting U=W=ej and V=Z=e1 and summing over j=2,⋯,m, we compute
(62)RicU=f1m−1+ψ2+ψ11−mβ1U,U−ψ2traceβ1+ψ2ψ1−ψ2β2U,U−traceβ2−m−1ψ1−ψ2πζ+∑s=1m∑j=2mζ11sζjjs−ζijs2.
From (62) and (57), we deduce
(63)RicU−f1m−1−ψ2+ψ11−mβ1U,U+ψ2traceβ1−ψ2ψ1−ψ2β2U,U−traceβ2+m−1ψ1−ψ2πζ≤∑s=1m∑j=2mζ11sζjjs−∑j=2mζ11j2−∑j=2mζjj12.
Putting(64)f1ζ111,ζ221,⋯,ζmm1=ζ111∑j=2mζjj1−∑j=2mζjj12,fsζ11s,ζ22s,⋯,ζmms=ζ11s∑j=2mζjjs−ζ11s2,∀s=2,⋯,m,and by using the fact mH1=ζ111+ζ221+⋯+ζmm1 together with the Lemma 5, we see that
(65)f1ζ111,ζ221,⋯,ζmm1≤m−14mmH12=mm−14H12.
Application of Lemma6 for s=2,⋯,m, gives
(66)fsζ11s,ζ22s,⋯,ζmms≤18mHs2=m28Hs2≤mm−14Hs2.
Equations (65), (67), and (68) yield the following relation
(67)RicU−f1m−1−ψ2+ψ11−mβ1U,U+ψ2traceβ1−ψ2ψ1−ψ2β2U,U−traceβ2+m−1ψ1−ψ2πζ≤mm−14∑s=1mHs2=mm−14∥H∥2.
Thus, we derive(68)RicU≤f1m−1+ψ2+ψ11−mβ1U,U−ψ2traceβ1+ψ2ψ1−ψ2β2U,U−traceβ2+m−1ψ1−ψ2πζ,+mm−14∥H∥2,which is the desired inequality (60).☐Now, we discuss the equality cases.Case 1.
Form>2, if Je1H. Then,
(69)Hs=0,for all s>1. Therefore, using Lemma 6, we derive
(70)ζ1j1=ζ11j=mHj4=0,forallj>1,(71)ζjk1=0,forallj,k>1,j≠k.
Further, Lemma5 yields
(72)ζ111=m+1H12,ζjj1=H12,forallj>1.
In (63), we see that RicU=Rice1. In the same way, by deriving Rice2 and making use of the equality, we conclude that
(73)ζ2js=ζjs2=0,foralls≠2,j≠2,s≠j.
In consequence, we find(74)ζ112m+1=ζ222=⋯=ζmm2=H22=0.
We see that the equality holds for every unit tangent vectors. The above conclusion is also valid forζjks. Thus,
(75)ζ2j2=ζ22j=Hj2=0,∀j≥3.
Then, the only possible nonzero entries forζjk2 (resp., for ζjks) are
(76)ζ122=ζ212=ζ221=H12resp.ζ1ss=ζs1s=ζss1=H12,∀s≥3.
SubstitutingU=Z=e2 and V=W=ej,j=2,⋯,m in (16), we obtain
(77)R~e2,ej,e2,ej=Re2,ej,e2,ej−H122,∀j≥3.
On the other hand, if we putU=Z=e2 and V=W=e1 in (16), we get
(78)R~e2,e1,e2,e1=Re2,e1,e2,e1−m+1H122+H122.
From (77) and (78), it follows that
(79)Rice2−f1m−1−ψ1+ψ2−mψ1β1e2,e2+ψ1traceβ1−ψ2ψ1−ψ2mβ2e2,e2−traceβ2+m−1ψ1−ψ2πζ=2m−1H122.
Moreover, using the equality case of (29), we see that
(80)Rice2−f1m−1−ψ1+ψ2−mψ1β1U,U+ψ1traceβ1−ψ2ψ1−ψ2mβ2U,U−traceβ2+m−1ψ1−ψ2πζ=mm−1H122.
Indeedm≠1,2, with (81) and (84), we find H1=0. This implies that Mm is a totally geodesic in N2m+1c.Case 2.
In the casem=2, M2 is nontotally geodesic, then ζe1,e1=λe3,ζe2,e2=μe3,ζe1,e2=μe4 together with λ=3μ. This proves that M2 is H-umbilical surface.Remark 16.
If we consider the structure vector fieldξ tangential to the submanifold M, then we have the following result.Theorem 17.
LetMm+1 be a totally real submanifold of maximal dimension m+1m≥2 in a generalized Sasakian space form Nf1,f2,f3 of dimension 2m+1 with a quarter-symmetric metric connection such that the vector field Γ is tangent to Mm+1. Then, for any unit tangent vector U to Mm+1,
(81)RicU≤mm+14H2+f1m−m−1η2e1+1f3+ψ2+ψ1mβ1U,U−ψ2traceβ1+ψ2ψ1−ψ2β2U,U+traceβ2+mψ1−ψ2πζ,
and the equality holds in (81) identically if and only if either
(1)
Mm+1 is totally geodesic, provided m>2, or(2)
m=2 and M2 is a H-umbilical Legendrian surface with λ=3μProof.
We obtain the proof on the same lines of the proof for Theorem15 additionally assuming an orthonormal basis
(82)e1=U,e2,⋯,em,em+1,em+2,⋯,e2m+1,such that e1,e2,⋯,em,em+1∈TpM.
As a consequence of Theorem15, we obtain the following results.☐Corollary 18.
LetMm be a totally real submanifold of maximal dimension mm≥2 in a Sasakian space form Nc of dimension 2m+1 with a quarter-symmetric metric connection such that the vector field Γ is tangent to Mm. Then, for any unit tangent vector U to Mm(83)RicU≤mm−14H2+c+34m−1+ψ2+ψ11−mβ1U,U−ψ2traceβ1+ψ2ψ1−ψ2β2U,U+traceβ2+m−1ψ1−ψ2πζ,and the equality holds in (83) identically if and only if either
(1)
Mm is totally geodesic, provided m>2, or(2)
m=2 and M2 is a H-umbilical Legendrian surface with λ=3μProof.
The proof follows immediately from Theorem15 by putting f1=c+3/4,f2=f3=c−1/4.☐Corollary 19.
LetMm be a totally real submanifold of maximal dimension mm≥2 in a Kenmotsu space form Nc of dimension 2m+1 with a quarter-symmetric metric connection such that the vector field Γ is tangent to Mm. Then, for any unit tangent vector U to Mm(84)RicU≤mm−14H2+c−34m−1+ψ2+ψ11−mβ1U,U−ψ2traceβ1+ψ2ψ1−ψ2β2U,U+traceβ2+m−1ψ1−ψ2πζ,
and the equality holds in (84) identically if and only if either
(1)
Mm is totally geodesic, provided m>2, or(2)
m=2 and M2 is a H-umbilical Legendrian surface with λ=3μProof.
The proof follows immediately from Theorem15 by replacing f1=c−3/4 and f2=f3=c+1/4.☐Corollary 20.
LetMm be a totally real submanifold of maximal dimension mm≥2 in a cosymplectic space form Nc of dimension 2m+1 with a quarter-symmetric metric connection such that the vector field Γ is tangent to Mm. Then, for any unit tangent vector U to Mm(85)RicU≤mm−14H2+c−34m−1+ψ2+ψ11−mβ1U,U−ψ2traceβ1+ψ2ψ1−ψ2β2U,U+traceβ2+m−1ψ1−ψ2πζ,
and the equality holds in (85) identically if and only if either
(1)
Mm is totally geodesic, provided m>2, or(2)
m=2 and M2 is a H-umbilical Legendrian surface with λ=3μProof.
The proof follows immediately from Theorem15 by substituting f1=f2=f3=c/4.☐Corollary 21.
LetMm be a totally real submanifold of maximal dimension mm≥2 in a generalized Sasakian space form Nf1,f2,f3 of dimension 2m+1 with a semisymmetric metric connection such that the vector field Γ is tangent to Mm. Then, for any unit tangent vector U to Mm(86)RicU≤mm−14∥H∥2+f1m−1+2−mβ1U,U−traceβ1,and the equality in (60) holds identically if and only if either
(1)
Mm is totally geodesic, provided m>2, or(2)
m=2 and M2 is a H-umbilical Legendrian surface with λ=3μProof.
Using the fact thatψ1=ψ2=1 together with Theorem 15, the result directly follows.☐Remark 22.
It is worthy to mention here that Corollary21 together with Remark 13 (1) is the main result of the paper [6].Corollary 23.
LetMm be a totally real submanifold of maximal dimension mm≥2 in a generalized Sasakian space form Nf1,f2,f3 of dimension 2m+1 with a semisymmetric nonmetric connection such that the vector field Γ is tangent to Mm. Then, for any unit tangent vector U to Mm(87)RicU≤mm−14∥H∥2+f1m−1+1−mβ1U,U+m−1πζ,and the equality in (87) holds identically if and only if either
(1)
Mm is totally geodesic, provided m>2, or(2)
m=2 and M2 is a H-umbilical Legendrian surface with λ=3μProof.
Using the fact thatψ1=1 and ψ2=0 together with Theorem 15, the result directly follows.☐Remark 24.
All the above cases for Theorem15 can be seen in the case of Theorem 17 as well.Remark 25.
Examples of totally geodesic submanifolds,H-umbilical Lagrangian submanifolds, and H-umbilical Legendrian submanifolds, i.e., examples of submanifolds attaining the equality case of the inequalities stated in this article, can be found in [25, 29, 30].



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*Source: 3221643-2021-10-11.xml*