# CR-Submanifolds of Generalizedf.p.k.-Space Forms

Authors: Mahmood Jaafari Matehkolaee
Journal: Geometry
 (2013)

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## Abstract

We study sectional curvature, Ricci tensor, and scalar curvature of submanifolds of generalizedf.p.k.-space forms. Then we give an upper bound for foliate ξα-horizontal (and vertical) CR-submanifold of a generalized f.p.k.-space form and an upper bound for minimal ξα-horizontal (and vertical) CR-submanifold of a generalized f.p.k.-space form. Finally, we give the same results for special cases of generalized f.p.k.-space forms such as S-space forms, generalized Sasakian space forms, Sasakian space forms, Kenmotsu space forms, cosymplectic space forms, and almost C(α)-manifolds.


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## Body

## 1. Introduction

In 1978, Bejancu introduced and studied CR-submanifolds of a Kähler manifold [1, 2]. Since then, many papers appeared on this topic with ambient manifold such as Sasakian space form [3], cosymplectic space form [4], and Kenmotsu space form [5, 6]. Recently Falcitelli and Pastore [7] introduced generalized globally framed f-space forms. Globally framed f-manifolds are studied from the point of view of the curvature and are introduced and the interrelation with generalized Sasakian and generalized complex space forms is pointed out. In this paper, we study CR-submanifolds of generalized f-space forms.The theory of a submanifold of a Sasaki manifold was investigated from two different points of view: one is the case where submanifolds are tangent to the structure vector and the other is the case where those are normal to the structure vector [8].In the class off-structures introduced in 1963 by Yano [9], the so-called f-structures with complemented frames, also called globally framed f-structures or f-structures with parallelizable kernel (briefly f.p.k.-structures) [10–13] are particulary interesting. An f.p.k.-manifold is a manifold M2n+s on which an f-structure is defined, that is a (1,1)-tensor field φ satisfying φ3+φ=0, of rank 2n, such that the subbundle kerφ is parallelizable. Then, there exists a global frame {ξi}, i∈{1,…,s}, for the subbundle kerφ, with dual 1-form ηi, satisfying φ2=-I+ηi⊗ξi,ηi(ξj)=δji, from which φξi=0, ηi∘φ=0 follow. An f.p.k.-structure on a manifold M2n+s is said to be normal if the tensor field N=[φ,φ]+2dηi⊗ξi vanishes, [φ,φ] denoting the Nijenhuis torsion of φ. It is known that one can consider a Riemannian metric g on M2n+s associated with an f.p.k.-structure (φ,ξi,ηi), such that g(φX,φY)=g(X,Y)-∑i=1s‍ηi(X)ηi(Y), for any X,Y∈Γ(T2n+sM), and the structure (φ,ξi,ηi,g) is then called a metric f.p.k.-structure. Therefore, T2n+sM splits as complementary orthogonal sum of its subbundles Imφ and kerφ. We denote their respective differentiable distributions by D and D⊥.LetΩ denote the 2-form on M2n+s defined by Ω(X,Y)=g(X,φY), for any X,Y∈Γ(T2n+sM).Several subclasses have been studied from different points of view [10, 11, 14–16], also dropping the normality condition, and, in this case, the term almost precedes the name of the considered structures or manifolds. As in [10], a metric f.p.k.-structure is said a 𝒦-structure if it is normal and the fundamental 2-form Ω is closed; a manifold with a 𝒦-structure is called a 𝒦-manifold. In particular, if dηi=Ω, for all i∈{1,…,s}, the 𝒦-structure is said to be an 𝒮-structure and M2n+s an 𝒮-manifold. Finally, if dηi=0 for all i∈{1,…,s}, then the 𝒦-structure is called a 𝒞-structure and M2n+s is said to be a 𝒞-manifold. Obviously, if s=1, a 𝒦-manifold M2n+1 is a quasi Sasakian manifold, a 𝒞-manifold is a cosymplectic manifold, and an 𝒮-manifold is a Sasakian manifold.The purpose of the present paper is to study Ricci tensor, sectional curvature, and scalar curvature of submanifolds of a generalizedf.p.k.-space form. In Section 2, we state definitions of f.p.k.-space form, its curvature tensor, ξα-horizontal CR-submanifold, and ξα-vertical CR-submanifold. Section 3 is devoted to the study sectional curvature of submanifold of an f.p.k.-space form. Finally, in Section 4, we investigate Ricci tensor and scalar curvature of submanifold of an f.p.k.-space form and obtain upper bound for scalar curvature.


## 2. Preliminaries

We recall that the Levi-Civita connection∇ of a metric f.p.k.-manifold satisfies the following formula [10, 11]:
(1)2g((∇Xφ)Y,Z)=3dΩ(X,φY,φZ)-3dΩ(X,Y,Z)+g(N(Y,Z),φX)+Nj(2)(Y,Z)ηjX+dηj(φY,X)ηj(Z)-2dηj(φZ,X)ηj(Y),
where Nj(2) is given by Nj(2)(X,Y)=2dηj(φX,Y)-2dηj(φY,X).Furthermore, for𝒮-manifolds we have ∇Xξj=-φX, j=1,…,s, [10]. Putting ξ-=∑j=1s‍ξj, η-=∑j=1s‍ηj is its dual form with respect to g and
(2)(∇Xφ)Y=g(φX,φY)ξ-+η-(Y)φ2X.
We remark that (2) together with £ξig=0 and £ξiηj=0, i,j∈{1,…,s}, characterizes the 𝒮-manifolds among the metric f.p.k.-manifolds.A metricf.p.k.-manifold (M2n+s,φ,ξi,ηi,g) has pointwise constant (p.c.) φ-sectional curvature if at any p∈M2n+s, c(p)=Rp(X,φX,X,φX) does not depend on the φ-section spanned by {X,φX}, for any unit X∈Dp. Several results involving the pointwise constancy of the φ-sectional curvatures of an almost contact metric manifold (i.e., for s=1) are recently obtained in [17–19]. We refer to [20] for a systematic exposition of the classical curvature results on contact metric manifolds.We recall some known results.Proposition 1 (see [6]).
A Sasaki manifold(M2n+s,φ,ξ,η,g) has p.c. φ-sectional curvature c if and only if its curvature tensor field verifies
(3)R(X,Y,Z)=14(c+3){g(Y,Z)X-g(X,Z)Y}+14(c-1){g(X,φZ)φY-g(Y,φZ)φX+2g(X,φY)φZ+η(X)η(Z)Y-η(Y)η(Z)X+g(X,Z)η(Y)ξ-g(Y,Z)η(X)ξ},
for any X,Y,Z tangent to M2n+1.A Sasaki manifoldM2n+1 with constant φ-sectional curvature c∈ℝ is called a Sasakian space form and denoted by M2n+1(c). It is well known that, if n≥2, a Sasaki manifold M2n+1 with p.c. φ-sectional curvature c is a Sasakian space form. As examples of Sasakian space forms, we mention ℝ2n+1 and S2n+1, with standard Sasakian structures [14].Definition 2 (see [10]).
An almost contact metric manifold(M2n+s,φ,ξ,η,g) is a generalized Sasakian space form, denoted by (M2n+s,f1,f2,f3), if it admits three smooth functions f1, f2, f3 such that its curvature tensor field verifies that, for any X,Y,Z∈TM(4)R(X,Y,Z)=f1{g(Y,Z)X-g(X,Z)Y}+f2{g(X,φZ)φY-g(Y,φZ)φX+2g(X,φY)φZ}+f3{η(X)η(Z)Y-η(Y)η(Z)X+g(X,Z)η(Y)ξ-g(Y,Z)η(X)ξ},Remark 3.
Any generalized Sasakian space form has p.c.φ-sectional curvature c=f1+3f2. Obviously, a Sasaki manifold of p.c. φ-sectional curvature c satisfies (4) with f1=(1/4)(c+3) and f2=f3=(1/4)(c-1). A cosymplectic manifold with p.c. φ-sectional curvature c satisfies (4) with f1=f2=f3=(1/4)c.Proposition 4 (see [10, 21]).
AnS-manifold M2n+s has p.c. φ-sectional curvature c if and only if its curvature tensor field verifies
(5)R(X,Y,Z)=14(c+3s){g(φX,φZ)φ2Y-g(φY,φZ)φ2X}+14(c-s){g(Z,φY)φX-g(Z,φX)φY+2g(X,φY)φZ}+{η-(X)η-(Z)φ2Y-η-(Y)η-(Z)φ2X+g(φY,φZ)η-(Y)ξ--g(φX,φZ)η-(X)ξ-}.
for any X,Y,Z tangent to M2n+1.AnS-manifold M2n+s with constant φ-sectional curvature c∈ℝ is called an S-space-form and denoted by M2n+s(c). Moreover, it is also well known that if n≥2, then an S-manifold with p.c. φ-sectional curvature c is an S-space form. We remark that for s=1 (5) reduces to (3).Definition 5.
In [22], Oubiña introduced the notion of a trans-Sasakian manifold. An almost contact metric manifold M is called trans-Sasakian manifold if there exist two functions α and β on M such that [22–24]
(6)(∇Xφ)(Y)=α{g(X,Y)ξ-η(Y)X}+β{g(φX,Y)ξ-η(Y)φX},
for vector fields X,Y on M. From (6) it is easy to see that
(7)∇Xξ=-αφX+β(X-η(X))ξ.
In particular, if β=0, then M is said to be an α-Sasakian manifold. Sasakian manifolds appear as examples of α-Sasakian manifolds with α=1.
On the other hand, ifα=0, then M is said to be a β-Kenmotsu manifold. Kenmotsu manifolds, defined in [25], are particular examples with β=1.
Another important kind of trans-Sasakian manifolds is that of cosymplectic manifolds obtained forα=β=0.Proposition 6 (see [25]).
An almost contact metric manifold is said to be an almostC(α)-manifold if its Riemannian curvature tensor verifies
(8)R(X,Y,Z,W)=R(X,Y,φZ,φW)+α{g(X,W)g(Y,Z)-g(X,Z)g(Y,W)+g(X,φZ)g(Y,φW)-g(X,φW)g(Y,φZ)},
for vector fields X, Y, Z, and  W on M, where α is a real number. Moreover, if such a manifold has constant φ-sectional curvature equal to c, then its curvature tensor is given by
(9)R(X,Y)Z=14(c+3α2){g(Y,Z)X-g(X,Z)Y}+14(c-α2){g(X,φZ)φY-g(Y,φZ)φX+2g(X,φY)φZ+η(X)η(Z)Y-η(Y)η(Z)X+g(X,Z)η(Y)ξ-g(Y,Z)η(X)ξ},
and so, it is a generalized Sasakian space form with f1=(1/4)(c+3α2) and f2=f3=(1/4)(c-α2).Letℱ denote any set of smooth function Fij on M2n+s such that Fij=Fji for any i,j∈{1,2,…,s}.Definition 7 (see [7]).
A generalizedf.p.k.-space form, denoted by M2n+s(F1,F2,ℱ), is a metric f.p.k.-manifold (M2n+s,φ,ξi,ηi,g) which admits smooth functions F1, F2, and ℱ such that its curvature tensor field verifies
(10)R(X,Y,Z)=F1{g(φX,φZ)φ2Y-g(φY,φZ)φ2X}+F2{g(Z,φY)φX-g(Z,φX)φY+2g(X,φY)φZ}+∑i,j=1s‍Fij{ηi(X)ηj(Z)φ2Y-ηi(Y)ηj(Z)φ2X+ηi(X)ξjg(φY,φZ)-ηi(Y)ξjg(φX,φZ)}.
Fors=1, we obtain a generalized Sasakian space form M2n+1(f1,f2,f3) with f1=F1, f2=F2 and f3=F1-F11. In particular, if the given structure is either Sasakian, Kenmotsu, or possibly cosymplectic, then (10) holds with F11=1,F1=(1/4)(c+3),F2=(1/4)(c-1), and f3=F1-F11=(1/4)(c-1)=f2 in the first case, F11=-1, F1=(1/4)(c-3), F2=(1/4)(c+1) and f3=F1-F11=(1/4)(c+1)=f2 in the second case, and F11=0, F1=(1/4)c, F2=(1/4)c, and f3=(1/4)c in the last case.Definition 8.
LetM be an m-dimensional submanifold immersed in M-. M is said to be an invariant submanifold if ξα∈TM for any 1≤α≤s and φX∈TM for any X∈TM. On the other hand, it is said to be an anti-invariant submanifold if φX∈T⊥M for any X∈TM.
Anm-dimensional Riemannian submanifold M of a generalized f.p.k.-space form M-2n+s is called a CR-submanifold if ξα's are tangent to M (so, dimM≥s) and there exist two differentiable distributions D and D⊥ on M satisfying(i)
TM=D⊕D⊥ (direct sum),(ii)
the distributionD is invariant under φ, that is, φDx=Dx for any x∈M,(iii)
the distributionD⊥ is anti-invariant under φ, that is, φDx⊥⊆Tx⊥M for any x∈M.
We denote by2p and q the real dimensions of Dx and Dx⊥, respectively, for any x∈M. Then, if p=0, we have an anti-invariant submanifold tangent to ξ1,…,ξs, and if q=0, we have an invariant submanifold.
As an example, it is easy to prove that each hypersurface ofM- which is tangent to ξ1,…,ξs inherits the structure of CR-submanifold of M-. Also, pseudoumbilical, totally contact umbilical, totally contact geodesic, totally umbilical, and totally geodesic hypersurfaces of a generalized S-space form are also generalized S-space forms, and, moreover, the bundle space of a principal toroidal bundle over a Kählerian manifold and the warped product of R times a generalized S-space form are generalized S-space forms, too [26].Definition 9.
Theφ-sectional curvature H of M determined by a unit vector X∈D orthogonal to ξα's is the sectional curvature of the plane section spanned by X and φX. Also, we denote by Ric(X,Y) (and K(X,Y)) Ricci tensor (and sectional curvature) determined by (orthonormal) vector fields {X,Y}, respectively.Definition 10.
A CR-submanifoldM of a generalized f.p.k.-space form M-2n+s is said to be D-totally geodesic (resp., D⊥- totally geodesic) if h(X,Y)=0 for any X,Y∈D (resp., X,Y∈D⊥), and it is said to be (D,D⊥)-mixed totally geodesic if h(X,Y)=0 for any X∈D, Y∈D⊥.
Also, CR-submanifoldM is said to be minimal if μ=0, where μ is the mean curvature vector, defined by μ=(1/(2n+s))trace(h).Definition 11.
LetM be a CR-submanifold with horizontal distribution D and vertical distribution D⊥. The pair (D,D⊥) is called ξα-horizontal if ξαx∈Dx for any x∈M, and in a similar way the pair (D,D⊥) is called ξα-vertical if ξαx∈Dx⊥ for any x∈M.Definition 12.
LetdimM=n=2p+s+q and {e1,e2,…,e2p,ξ1,ξ2,…,ξs,e1′,e2′,…,eq′} be a local field of orthonormal frames on TM such that in case when M is ξα-horizontal, {e1,e2,…,e2p,ξ1,ξ2,…,ξs} is a local frame field on D and {e1′,e2′,…,eq′} is a local frame field on D⊥.
LetM be an ξα-horizontal CR-submanifold of M-. The mean curvature vector field μ of M in M- is defined by
(11)μ=12p+s+q{∑i=12p+s‍h(ei,ei)+∑i=1q‍h(ei′,ei′)}.
If μ=0, then M is said to be minimal. Now, we will define
(12)μD=12p+s∑i=12p+s‍h(ei,ei),μD⊥=1q∑i=1q‍h(ei′,ei′).
If μD=0, then the CR-submanifold M is said to be D-minimal, and if μD⊥=0, then it is said to be D⊥-minimal. Similar definitions can be given for ξα-vertical CR-submanifolds.We denote byP and Q the projection morphisms of TM on D and D⊥, respectively. We call D (resp., D⊥) the horizontal (resp., vertical) distribution. Then for any vector field X tangent to M, we have:
(13)X=PX+QX,
where PX and QX belong to the distribution D (horizontal part) and D⊥ (vertical part), respectively. Also, for a vector field N normal to M, we put:
(14)φN=tN+fN,
where tN and fN denote the horizontal and normal component of φN, respectively.Definition 13.
LetM be a CR-submanifold of an ambient manifold M-, with horizontal distribution D. Then D is called involutive (or integrable) if [X,Y]∈D for any X,Y∈D where [X,Y] is Lie bracket of X,Y. Also, M is a foliate if D is involutive (or integrable).LetM be an m-dimensional submanifold immersed in a generalized f.p.k.-space form M-2n+s. The Gauss-Weingarten formulas are given by
(15)∇-XY=∇XY+h(X,Y);X,Y∈TM,∇-XN=-ANX+∇X⊥N;X∈TM,N∈T⊥M,
where ∇⊥ is the connection in the normal bundle, h is the second fundamental form of M and AN the Weingarten endomorphism associated with N. Then AN and h are related by:
(16)g(ANX,Y)=g(h(X,Y),N).We denote byR- and R the curvature tensor fields associated with ∇- and ∇, respectively. The Gauss equation is given by
(17)R-(X,Y,Z,W)=R(X,Y,Z,W)+g(h(X,Z),h(Y,W))-g(h(X,W),h(Y,Z)),
where X,Y,Z,and W belong to TM.


## 3. Sectional Curvature of Submanifolds

LetM be a submanifold of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ). Then from the equation of Gauss, we have
(18)R(X,Y,Z,W)=R-(X,Y,Z,W)-g(h(X,Z),h(Y,W))+g(h(X,W),h(Y,Z))=F1{g(φX,φZ)g(φ2Y,W)-g(φY,φZ)g(φ2X,W)}+F2{g(Z,φY)g(φX,W)-g(Z,φX)g(φY,W)+2g(X,φY)g(φZ,W)}+∑i,j=1s‍Fij{ηi(X)ηj(Z)g(φ2Y,W)-ηi(Y)ηj(Z)g(φ2X,W)+ηi(X)ηj(W)g(φY,φZ)-ηi(Y)ηj(W)g(φX,φZ)}-g(h(X,Z),h(Y,W))+g(h(X,W),h(Y,Z)),
for any X,Y,Z,and W tangent to M.LetKM(X,Y) be the sectional curvature determined by orthonormal vectors X and Y. Then from (18), we have
(19)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+3F2g2(X,φY)+F1{(∑k=1s‍ηk(X)ηk(Y))2(1-∑k=1s‍ηk(X)2)(1-∑k=1s‍ηk(Y)2)-(∑k=1s‍ηk(X)ηk(Y))2}+∑i,j=1sFij{ηi(X)ηj(X)(1-∑k=1s‍ηk(Y)2)+ηi(Y)ηj(Y)(1-∑k=1s‍ηk(X)2)+2ηi(X)ηj(Y)∑k=1s‍ηk(X)ηk(Y)}.
Thus we have the following theorem.Theorem 14.
LetM be a submanifold of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ). Then the sectional curvature of M determined by orthonormal tangent vectors {X,Y} is given by
(20)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+3F2g2(X,φY)+F1{-(∑k=1s‍ηk(X)ηk(Y))2(1-∑k=1s‍ηk(X)2)(1-∑k=1s‍ηk(Y)2)-(∑k=1s‍ηk(X)ηk(Y))2}+∑i,j=1s‍Fij{ηi(X)ηj(X)(1-∑k=1s‍ηk(Y)2)+ηi(Y)ηj(Y)(1-∑k=1s‍ηk(X)2)+2ηi(X)ηj(Y)∑k=1s‍ηk(X)ηk(Y)}.From this we have the following corollaries for the sectional curvature of submanifold determined by orthonormal tangent vectors{X,Y}.Corollary 15.
The sectional curvature of a submanifold of anS-space form M-2n+s(c) is given by
(21)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+34(c-s)g2(X,φY)+14(c+3s)×{(∑k=1s‍ηk(X)ηk(Y))2(1-∑k=1s‍ηk(X)2)(1-∑k=1s‍ηk(Y)2)-(∑k=1s‍ηk(X)ηk(Y))2}+∑i,j=1s‍{ηi(X)ηj(X)(1-∑k=1s‍ηk(Y)2)+ηi(Y)ηj(Y)(1-∑k=1s‍ηk(X)2)+2ηi(X)ηj(Y)∑k=1s‍ηk(X)ηk(Y)}.Proof.
We will get the result by usingFij=1, for all 1≤i,j≤s; F1=(1/4)(c+3s); F2=(1/4)(c-s) in (20).Corollary 16.
The sectional curvature of a submanifold of a generalized Sasakian space formM-(c) is given by
(22)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+f1+3f2g2(X,φY)-f3(η2(X)+η2(Y)).Proof.
We will get the result by usings=1, F1=f1, F2=f2, and F11=f1-f3 in (20).Corollary 17.
The sectional curvature of a submanifold of a Sasakian space formM-(c) is given by
(23)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c+3)+14(c-1)×(3g2(X,φY)-η2(X)-η2(Y)).Proof.
We get the result by usingf1=(1/4)(c+3), f2=f3=(1/4)(c-1) in (22).Corollary 18.
The sectional curvature of a submanifold of a Kenmotsu space formM-(c) is given by
(24)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c-3)+14(c+1)×(3g2(X,φY)-η2(X)-η2(Y)).Proof.
We get the result by usingf1=(1/4)(c-3) and f2=f3=(1/4)(c+1) in (22).Corollary 19.
The sectional curvature of a submanifold of a cosymplectic space formM-(c) is given by
(25)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14c(1+3g2(X,φY)-η2(X)-η2(Y)).Proof.
By takingf1=f2=f3=(1/4)c in (22), we obtain the above.Corollary 20.
The sectional curvature of a submanifold of an almostC(α)-manifold M-(c) is given by
(26)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c+3α2)+14(c-α2)(3g2(X,φY)-η2(X)-η2(Y)).Proof.
By  gettingf1=(1/4)(c+3α2), f2=f3=(1/4)(c-α2) in (22), we obtain (26).Proposition 21.
IfM is a ξα-horizontal CR-submanifold of a generalized f.p.k.-space form M-2n+s, then the sectional curvature of M determined by X,Y∈D⊥ is given by
(27)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+F1.Proof.
From (20) and by replacing ηk(X)=0=ηk(Y); 1≤k≤s and g(X,φY)=0 we get the result immediately.Corollary 22.
IfM is a ξα-horizontal CR-submanifold of an S-space form M-2n+s(c), then the sectional curvature of M determined by X,Y∈D⊥ is given by
(28)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c+3s).Corollary 23.
IfM is a ξ-horizontal CR-submanifold of a generalized Sasakian space form M-(c), then the sectional curvature of M determined by X,Y∈D⊥ is given by
(29)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+f1.Corollary 24.
IfM is a ξ-horizontal CR-submanifold of a Sasakian space form M-(c), then the sectional curvature of M determined by X,Y∈D⊥ is given by
(30)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c+3).Corollary 25.
IfM is a ξ-horizontal CR-submanifold of a Kenmotsu space form M-(c), then the sectional curvature of M determined by X,Y∈D⊥ is given by
(31)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c-3).Corollary 26.
IfM is a ξ-horizontal CR-submanifold of a cosymplectic space form M-(c), then the sectional curvature of M determined by X,Y∈D⊥ is given by
(32)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14c.Corollary 27.
IfM is a ξ-horizontal CR-submanifold of a C(α)-manifold M-(c), then the sectional curvature of M determined by X,Y∈D⊥ is given by
(33)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c+3α2).Proposition 28.
IfM is a ξα-vertical CR-submanifold of a generalized f.p.k.-space form M-2n+s, then the sectional curvature of M determined by X,Y∈D is given by
(34)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+F1+3F2g2(X,φY).Corollary 29.
IfM is a ξα-vertical CR-submanifold of an S-space form M-2n+s(c), then the sectional curvature of M determined by X,Y∈D is given by
(35)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c+3s)+34(c-s)g2(X,φY).Corollary 30.
IfM is a ξ-vertical CR-submanifold of space form M-(c), then the sectional curvature of M determined by X,Y∈D,(i)
whereM-(c) is a generalized Sasakian space form, is given by:
(36)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+f1+3f2g2(X,φY),(ii)
whereM-(c) is a Sasakian space form, is given by
(37)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c+3)+34(c-1)g2(X,φY),(iii)
whereM-(c) is a Kenmotsu space form, is given by
(38)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c-3)+34(c+1)g2(X,φY),(iv)
whereM-(c) is a cosymplectic space form, is given by
(39)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14c(1+3g2(X,φY)),(v)
whereM-(c) is a C(α)-manifold, is given by
(40)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+14(c+3α2)+34(c-α2)g2(X,φY).Proposition 31.
Theφ-sectional curvature of a CR-submanifold of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ), determined by X∈TM, is given by
(41)H(X)=g(h(X,X),h(φX,φX))-∥h(X,φX)∥2+F1+3F2.Proof.
By usingηk(X)=0, for all 1≤k≤s in (20), we will get the result.Proposition 32.
Theφ-sectional curvature of a CR-submanifold of a generalized Sasakian space form M-(c), determined by X∈TM is given by
(42)H(X)=g(h(X,X),h(φX,φX))-∥h(X,φX)∥2+f1+3f2.Corollary 33.
Theφ-sectional curvature of a CR-submanifold of either an S-space form, a Sasakian space form, a Kenmotsu space form, a cosymplectic space form, or an almost C(α)-manifold M-2n+s, determined by X∈TM, is given by:
(43)H(X)=g(h(X,X),h(φX,φX))-∥h(X,φX)∥2+c.We recall the following Lemma [27].Lemma 34.
LetM be a foliate ξα-horizontal CR-submanifold of a S-space form M-2n+s(c); then
(44)h(φX,φY)=-h(X,Y);X,Y∈D.Proposition 35.
IfM is a foliate ξα-horizontal CR-submanifold of a S-space form M-2n+s(c); then
(45)H(X)≤c;X∈D,
and the equality holds if and only if M is D-totally geodesic.Corollary 36.
IfM is a foliate ξ-horizontal CR-submanifold of a generalized Sasakian space form M-(c), then
(46)H(X)≤f1+3f2;X∈D,
and the equality holds if and only if M is D-totally geodesic.Corollary 37.
If M is a foliateξ-horizontal CR-submanifold of either a Sasakian space form, a Kenmotsu space form, a cosymplectic space form, or an almost C(α)-manifold form M-(c), then
(47)H(X)≤c;X∈D,
and the equality holds if and only if M is D-totally geodesic.Proposition 38.
IfM be a D⊥-minimal ξα-horizontal CR-submanifold of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ); then M is D⊥-totally geodesic iff
(48)KM(X,Y)=F1,
for any X,Y∈D⊥.Proof.
LetM is D⊥-minimal ξα-horizontal CR-submanifold of generalized f.p.k.-space form M-2n+s, then by definition of D⊥-minimal, we have:
(49)∑i=1q‍h(ei′,ei′)=0,
where {e1′,e2′,…,eq′} is a local frame field on D⊥. Therefore, for any X∈D⊥(50)h(X,X)=0.
On the other hand, from (20), we have for X,Y∈D⊥(51)KM(X,Y)=g(h(X,X),h(Y,Y))-∥h(X,Y)∥2+F1.
Hence, M is D⊥-totally geodesic if and only if for any X,Y∈D⊥(52)KM(X,Y)=F1.Proposition 39.
IfM is a D-minimal ξα-vertical CR-submanifold of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ), then M is D-totally geodesic if and only if
(53)KM(X,Y)=F1,
for any X,Y∈D with g(X,φY)=0.Proposition 40.
IfM is a ξα-horizontal CR-submanifold and (D,D⊥)-mixed totally geodesic of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ), then
(54)KM(X,Y)=g(h(X,X),h(Y,Y))+F1(1-∑k=1s‍ηk(X)2)+∑i,j=1s‍Fijηi(X)ηj(X),
for any X∈D and Y∈D⊥.Proof.
By usingηk(Y)=0, for all 1≤k≤s and g(X,φX)=0 and h(X,Y)=0, we arrive at the aforementioned equation, easily.Proposition 41.
IfM is a ξα-vertical CR-submanifold and (D,D⊥)-mixed totally geodesic of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ), then
(55)KM(X,Y)=g(h(X,X),h(Y,Y))+F1(1-∑k=1s‍ηk(Y)2)+∑i,j=1s‍Fijηi(Y)ηj(Y),
for any X∈D and Y∈D⊥.Proof.
By usingηk(X)=0, for all 1≤k≤s,   g(X,φX)=0,h(X,Y)=0 and (20), we arrive at the abovementioned equation, easily.


## 4. The Ricci Tensor and Scalar Curvature of a Submanifold

LetM be a submanifold of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ). Then it is straightforward to calculate the Ricci tensor of M as follows
(56)Ric(X,Y)=∑k=1n‍g(R(ek,X)Y,ek)=∑k=1n‍F1{g(φek,φY)g(φ2X,ek)-g(φX,φY)g(φ2ek,ek)}+∑k=1n‍F2{g(Y,φX)g(φek,ek)-g(Y,φek)g(φX,ek)+2g(ek,φX)g(φY,ek)}+∑k=1n‍∑i,j=1s‍Fij{ηi(ek)ηj(Y)g(φ2X,ek)-ηi(X)ηj(Y)g(φ2X,ek)+ηi(ek)ηj(ek)g(φX,φY)-ηi(X)ηj(ek)g(φek,φY)}-g(h(ek,Y),h(X,ek))+g(h(ek,ek),h(X,Y))=F1{∑k=1n‍(-1+∑α=1s‍(ηα(ek))2)-g(φX,φY)-g(φX,φY)×∑k=1n‍(-1+∑α=1s‍(ηα(ek))2)}+3F2{g(X,Y)-∑α=1s‍ηα(X)ηα(Y)}+∑i,j=1s‍Fij{-∑k=1n‍ηi(ek)ηj(Y)g(φX,φek)+ηi(X)ηj(Y)∑k=1n‍g(φek,φek)+g(φX,φY)∑k=1n‍ηi(ek)ηj(ek)-ηi(X)∑k=1n‍ηj(ek)g(φY,φek)}+∑k=1n‍[g(h(ek,ek),h(X,X))-g(h(X,ek),h(Y,ek))]=F1(n-s-1)×(g(X,Y)-∑α=1s‍ηα(X)ηα(Y))+3F2(g(X,Y)-∑α=1s‍ηα(X)ηα(Y))+∑i,j=1s‍Fij(n-s)ηi(X)ηj(Y)+(∑i=1s‍Fii)(g(X,Y)-∑α=1s‍ηα(X)ηα(Y))+∑k=1n‍[g(h(X,X),h(ek,ek))-g(h(X,ek),h(Y,ek))].
Also, the scalar curvature ρ of a submanifold M of M-2n+s(F1,F2,ℱ) is then given by
(57)ρ=∑t=1n‍Ric(et,et)=(F1(n-s-1)+3F2)×(∑t=1n‍g(et,et)-∑t=1n‍∑α=1s‍(ηα(et))2)+∑i,j=1s‍Fij(n-s)∑t=1n‍ηi(et)ηj(et)+(∑i=1s‍Fii)×(∑t=1n‍g(et,et)-∑t=1n‍∑α=1s‍(ηα(et))2)+∑t=1n‍∑k=1n‍[g(h(et,et),h(ek,ek))-g(h(et,ek),h(et,ek))]=(n-s)((n-s-1)F1+3F2)+(n-s)∑i=1s‍Fii+(n-s)∑i=1s‍Fii+∑i,j=1n‍[g(h(ei,ei),h(ej,ej))-g(h(ei,ej),h(ei,ej))].
Thus, we obtain the following.Theorem 42.
LetM be a submanifold of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ). Then the Ricci tensor and scalar curvature of M (resp.) are given by
(58)Ric(X,Y)=((n-s-1)F1+3F2+∑i=1s‍Fii)g(φX,φY)+(n-s)∑i,j=1s‍Fijηi(X)ηj(Y)+∑k=1n‍[g(h(X,X),h(ek,ek))-g(h(X,ek),h(Y,ek))],ρ=(n-s)((n-s-1)F1+3F2+2∑i=1s‍Fii)+∑i,j=1n‍[g(h(ei,ei),h(ej,ej))-g(h(ei,ej),h(ei,ej))].Theorem 43.
LetM be a minimal CR-submanifold of a generalized f.p.k.-space form M-2n+s. Then
(59)Ric(X,Y)-((n-s-1)F1+3F2+∑i=1s‍Fii)g(φX,φY)-(n-s)∑i,j=1s‍Fijηi(X)ηj(Y)
is negative semidefinite and
(60)ρ≤(n-s)×((n-s-1)F1+3F2+2∑i=1s‍Fii).Theorem 44.
LetM be a ξα-horizontal (resp., ξα-vertical) CR-submanifold of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ). Then the Ricci tensor of M for any X,Y∈D⊥ (resp., X,Y∈D) is given by
(61)Ric(X,Y)=((n-s-1)F1+3F2+∑i=1s‍Fii)g(X,Y)+∑i=1n‍[g(h(X,Y),h(ei,ei))-g(h(X,ei),h(Y,ei))].Theorem 45.
LetM be minimal ξα-horizontal (resp., ξα-vertical) CR-submanifold of a generalized f.p.k.-space form M-2n+s(F1,F2,ℱ). Then for any X,Y∈D⊥ (resp., X,Y∈D)
(62)Ric(X,Y)-((n-s-1)F1+3F2+∑i=1s‍Fii)g(X,Y)
is negative semidefinite.Corollary 46.
One has for Ricci tensor of minimalξα-horizontal CR-submanifold Table 1.Table 1
Manifold
Ricci tensor of Minimalξα-horizontal CR-submanifold
M
-
2
n
+
s
(
F
1
,
F
2
,
ℱ
)
(
(
n
-
s
-
1
)
F
1
+
3
F
2
+
∑
i
=
1
s
F
i
i
)
g
(
φ
X
,
φ
Y
)
+
(
n
-
s
)
∑
i
,
j
=
1
s
F
i
j
η
i
(
X
)
η
j
(
Y
)
M
-
2
n
+
s
1
4
(
3
c
+
s
+
(
n
-
s
-
1
)
(
c
+
3
s
)
)
g
(
φ
X
,
φ
Y
)
+
(
n
-
s
)
∑
i
,
j
=
1
s
η
i
(
X
)
η
j
(
Y
)
M
-
(
f
1
,
f
2
,
f
3
)
(
(
n
-
1
)
f
1
+
3
f
2
-
f
3
)
g
(
X
,
Y
)
-
η
(
X
)
η
(
Y
)
(
3
f
2
+
(
n
-
2
)
f
3
)
M
-
Sas
(
c
)
1
4
g
(
X
,
Y
)
(
n
(
c
+
3
)
+
c
-
5
)
-
1
4
(
c
-
1
)
(
n
+
1
)
η
(
X
)
η
(
Y
)
M
-
Ken
(
c
)
1
4
g
(
X
,
Y
)
(
n
(
c
-
3
)
+
c
+
5
)
-
1
4
(
c
+
1
)
(
n
+
1
)
η
(
X
)
η
(
Y
)
M
-
cosym
(
c
)
1
4
c
(
n
+
1
)
(
g
(
X
,
Y
)
-
η
(
X
)
η
(
Y
)
)
M
-
C
(
α
)
(
c
)
1
4
g
(
X
,
Y
)
(
n
(
c
+
3
α
2
)
+
c
-
5
α
2
)
-
1
4
(
n
+
1
)
(
c
-
α
2
)
η
(
X
)
η
(
Y
)
WhereM-2n+s(F1,F2,ℱ), M-2n+s, M-(f1,f2,f3), M-Sas(c), M-Ken(c), M-cosym(c), and M-C(α)(c) denote generalized f.p.k.-space form, S-space form, generalized Sasakian space form, Sasakian space form, Kenmotsu space form, cosymplectic space form, and C(α)-space form, respectively.Corollary 47.
One has for scalar curvature of minimalξα-horizontal CR-submanifold Table 2.Table 2
Manifold
Scalar curvature of Minimalξα-horizontal CR-submanifold
M
-
2
n
+
s
(
F
1
,
F
2
,
ℱ
)
(
n
-
s
)
(
(
n
-
s
-
1
)
F
1
+
3
F
2
+
2
∑
i
=
1
s
F
i
i
)
M
-
2
n
+
s
1
4
(
n
-
s
)
(
3
c
+
5
s
+
(
n
-
c
-
1
)
(
c
+
3
s
)
)
M
-
(
f
1
,
f
2
,
f
3
)
(
n
-
1
)
(
n
f
1
+
3
f
2
-
2
f
3
)
M
-
Sas
(
c
)
1
4
(
n
-
1
)
(
c
-
1
+
n
(
c
+
3
)
)
M
-
Ken
(
c
)
1
4
(
n
-
1
)
(
c
+
1
+
n
(
c
-
3
)
)
M
-
cosym
(
c
)
1
4
c
(
n
2
-
1
)
M
-
C
(
α
)
(
c
)
1
4
(
n
-
1
)
(
c
-
α
2
+
n
(
c
+
3
α
2
)
)
WhereM-2n+s(F1,F2,ℱ), M-2n+s, M-(f1,f2,f3), M-Sas(c), M-Ken(c), M-cosym(c), and M-C(α)(c) denote generalized f.p.k.-space form, S-space form, generalized Sasakian space form, Sasakian space form, Kenmotsu space form, cosymplectic space form, and C(α)-space form, respectively.Remark 48.
Similar results can be written for minimalξα-vertical CR-submanifolds, easily.



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*Source: 654780-2013-04-30.xml*