Mehmet Atc.eken
Certain results on invariant submanifolds of an almost Kenmotsu (κ, μ, ν) -space
Received: 12 May 2021 / Accepted: 3 August 2021 / Published online: 28 August 2021
© The Author(s) 2021
Abstract In the present paper, we study invariant submanifolds of almost Kenmotsu structures whose Rie- mannian curvature tensor has(κ, μ, ν)-nullity distribution. Since the geometry of an invariant submanifold inherits almost all properties of the ambient manifold, we research how the functions κ, μand ν behave on the submanifold. In this connection, necessary and sufficient conditions are investigated for an invariant submanifold of an almost Kenmotsu(κ, μ, ν)-space to be totally geodesic under some conditions.
Mathematics Subject Classification 53B20·53C17·53C21·53C42
1 Introduction
It is well known that a (2n+1)-dimensional contact metric manifoldM admits an almost contact metric structure(φ, ξ, η,g), i.e., it admits a global vector fieldξ, called the characteristic vector field or the Reeb vector field, its dual isη, a tensorφof type(1,1)and the Riemannian metric tensorgsuch that
φ2X = −X+η(X)ξ, η(ξ)=1, η◦φ=0, (1)
and
g(φX, φY)=g(X,Y)−η(X)η(Y), (2) for allX,Y ∈(TM), where(TM)denotes the set of differentiable vector fields onM[2].
The manifoldMtogether with the structure tensor(φ, ξ, η,g)is called a contact metric manifold and we will denote it byM(2n+1)(φ, ξ, η,g)in the rest of this paper.
By ∇, we denote the Levi-Civita connection of g, then the Riemannian curvature tensor of R of M2n+1(φ, ξ, η,g)is given by
R(X,Y)=∇X∇Y −∇Y∇X−∇[X,Y], for allX,Y ∈(TM).
On the other hand, we define the tensor field (1,1)-type byh 2h X =(ξφ)X,
for allX ∈(TM), whereξis the Lie-derivative in the direction ofξ. Then the tensor fieldhis self-adjoint and satisfies
φh+hφ=0, tr h =trφh=0, hξ =0. (3)
M. Atc.eken (
B
)Aksaray University: Aksaray Universitesi, Aksaray, Turkey E-mail: mehmet.atceken382@gmail.com
We have also these formulas for a contact metric manifold
∇Xξ =φX−φh X, ∇ξφ=0. (4)
A contact metric manifold for whichξ is Killing vector field is called a K-contact manifold. It is well known that a contact metric manifold is K-contact iffh=0.
The(κ, μ)-nullity distribution of a contact metric manifoldM2n+1(φ, ξ, η,g)for the pair(κ, μ)∈R2is distribution
M(κ, μ): p−→ Mp(κ, μ)= {Zp∈TM(p):
R(X,Y)Z =κ{g(Y,Z)X −g(X,Z)Y} +μ{g(Y,Z)h X−g(X,Z)hY}},
for allX,Y ∈(TM). So if the characteristic vector fieldξ belongs to the(κ, μ)-nullity distribution, then R(X,Y)ξ =κ{η(Y)X−η(X)Y} +μ{η(Y)h X−η(X)hY},
and the manifold M2n+1(φ, ξ, η,g)is called (κ, μ)-contact metric manifold. If κ and μ are non-constant smooth functions on M, then the manifold M2n+1(φ, ξ, η,g) is called generalized (κ, μ)-contact metric manifold [3].
Going beyond generalized(κ, μ)-space, T. Koufogiorgos, M. Markellos and V. J. Papantoniou introduced in [2] the notion of(κ, μ, ν)-contact metric manifold, its Riemannian curvature tensorRis given by
R(X,Y)ξ =κ{η(Y)X −η(X)Y} +μ{η(Y)h X−η(X)hY}
+ν{η(Y)φh X−η(X)φhY}, (5) for allX,Y ∈(TM), whereκ, μ, νare smooth functions onM2n+1.
It is well known that an almost contact metric manifold is an almost Kenmotsu ifdη = 0 andd = 2η , where (X,Y)= g(X, φY)is the fundamental 2-form ofM2n+1. If an almost Kenmotsu manifold M2n+1(φ, η, ξ,g)has a(κ, μ, ν)-nullity distribution, it is called an almost Kenmotsu(κ, μ, ν)-space [5].
Proposition 1.1 GivenM2n+1(φ, η, ξ,g)an almost Kenmotsu(κ, μ, ν)-space, then
h2=(κ+1)φ2, κ≤ −1, (6)
ξ(κ)=2(κ+1)(ν−2) (7)
(∇Xφ)Y =g(φX+h X,Y)ξ−η(Y)(φX +h X) (8)
∇Xξ = −φ2X −φh X (9)
S(X, ξ)=2nκη(X) (10)
R(ξ,X)Y =κ{g(X,Y)ξ −η(Y)X} +μ{g(h X,Y)ξ −η(Y)h X}
+ν{g(φh X,Y)ξ−η(Y)φh X}. (11) They proved that this type of manifold is intrinsically related to the harmonicity of the Reeb vector on contact metric 3-manifolds. Some authors have studied manifolds satisfying condition (5) but a non-contact metric structure. In this connection, P. Dacko and Z. Olszak defined an almost cosymplectic(κ, μ, ν)-space as an almost cosymplectic manifold that satisfies (5), but withκ, μandνfunctions varying exclusively in the direction ofξ in [6]. Later examples have been given for this type of manifold [7].
In modern analysis, the geometry of submanifolds has become a subject of growing interest for its significant applications in applied mathematics and theoretical physics. For instance, the notion of invariant submanifold is used to discuss properties of non-linear autonomous systems. Also, the notion of geodesic plays an important role in the theory of relativity. For totally geodesic submanifolds, the geodesics of the ambient manifolds remain geodesics in the submanifolds. Hence, totally geodesic submanifolds have also importance in mathematics as well as in physical sciences. There have been several papers on contact metric manifolds which admit covector fieldξ tangent to the submanifold. In this connection, we refer to [11,12,14,16].
Now, in this part of the study, we are especially interested in an invariant submanifold of an almost Kenmotsu (κ, μ, ν)-space to be pseudoparallel, 2-pseudoparallel, Ricci-generalized pseudoparallel and 2-generalized pseudoparallel.
Pseudoparallel submanifolds have been studied in different structures and working on [8–10]. On the other hand, the study of the geometry of invariant submanifolds was introduced by Bejancu and Papaghuic [10]. In general, the geometry of an invariant submanifold inherits almost all properties of the ambient manifold.
In the present paper, we generalize the ambient space and investigate the conditions under which invariant pseudoparallel submanifolds of an almost Kenmotsu(κ, μ, ν)-space are totally geodesic.
Now, letMbe an immersed submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1. By(T M)and (T⊥M), we denote the tangent and normal subspaces ofMinM. Then the Gauss and Weingarten formulae are, respectively, given by
∇XY = ∇XY +σ (X,Y), (12)
and
∇XV = −AVX + ∇⊥XV, (13) for allX,Y ∈(T M)andV ∈(T⊥M), where∇and∇⊥are the induced connections onMand(T⊥M) andσ and Aare called the second fundamental form and shape operator of M, respectively,(T M)denotes the set differentiable vector fields onM. They are related by
g(AVX,Y)=g(σ (X,Y),V). (14) The first covariant derivative of the second fundamental formσ is defined by
(∇Xσ )(Y,Z)= ∇⊥Xσ (Y,Z)−σ (∇XY,Z)−σ (Y,∇XZ), (15) for all X,Y,Z ∈ (T M). If∇σ = 0, then the submanifold is said to be its second fundamental form is parallel.
By R, we denote the Riemannian curvature tensor of the submanifoldM, we have the following Gauss equation
R(X,Y)Z =R(X,Y)Z+Aσ (X,Z)Y −Aσ (Y,Z)X+(∇Xσ )(Y,Z)
−(∇Yσ )(X,Z), (16)
for allX,Y,Z ∈(T M). R·σ is given by
(R(X,Y)·σ )(U,V)= R⊥(X,Y)σ (U,V)−σ (R(X,Y)U,V)
−σ (U,R(X,Y)V), (17) where the Riemannian curvature tensor of normal bundle(T⊥M)is given
R⊥(X,Y)= [∇⊥X,∇Y⊥] − ∇[⊥X,Y]
On the other hand, the concircular curvature tensor for Riemannian manifold(M2n+1,g)is given by C(X,Y)Z = R(X,Y)Z
− τ
2n(2n+1){g(Y,Z)X −g(X,Z)Y}, (18) whereτ denotes the scalar curvature ofM.
Similarly, the tensorC·σ is defined by
(C(X,Y)·σ )(U,V)= R⊥(X,Y)σ (U,V)−σ (C(X,Y)U,V)
−σ (U,C(X,Y)V), (19) for allX,Y,U,V ∈(T M).
For a(0,k)-type tensor fieldT,k≥1 and a(0,2)-type tensor field Aon a Riemannian manifold(M,g), Q(A,T)-tensor field is defined by
Q(A,T)(X1,X2, . . . ,Xk;X,Y)= −T((X ∧AY)X1,X2, . . . ,Xk)
. . .−T(X1,X2, . . . ,Xk−1, (X∧AY)Xk), , (20) for allX1,X2, . . . ,Xk,X,Y ∈(T M)[8], where
(X∧AY)Z = A(Y,Z)X−A(X,Z)Y. (21)
Definition 1.2 LetMbe a submanifold of a Riemannian manifold(M,g). If there exist functionsL1,L2,L3 andL4onMsuch that
R·σ =L1Q(g, σ ), (22)
R·∇σ =L2Q(g,∇σ ), (23)
R·σ =L3Q(S, σ ) (24)
R·∇σ =L4Q(S,∇σ ), (25) then M is, respectively, pseudoparallel, 2-pseudoparallel, Ricci-generalized pseudoparallel and 2-Ricci- generalized pseudoparallel submanifold. In particular, if L1 = 0(resp., L2 = 0), then M is said to be semiparallel(resp. 2-semiparallel) [9].
Kowalczyk studied the semi-Riemannian manifolds satisfying Q(S,R) = 0 and Q(S,g)=0 [17]. Also, De and Majhi investigated the invariant submanifolds of Kenmotsu manifolds and showed that geometric conditions of invariant submanifolds of Kenmotsu manifolds are totally geodesic [13]. Recently, Hu and Wang obtained the geometric conditions of invariant submanifolds of a trans-Sasakian manifold to be totally geodesic [15]. Furthermore, the geometry of invariant submanifolds of different manifolds was studied by many geometers [8,9,11–16].
Motivated by the above studies, we make an attempt to study the invariant submanifolds of an almost Kenmotsu (κ, μ, ν)-space satisfying some geometric conditions such as R ·σ = L1Q(g, σ ), R·∇σ = L2Q(g,∇σ ),R·σ =L3Q(S, σ )andR·∇σ =L4Q(S,∇σ ).
Finally, we show that the submanifold is either totally geodesic or the functionsLi, κ, μandνfunctions are restricted.
2 Invariant Submanifolds of an almost Kenmotsu(κ, μ, ν)-Space
Now, let M2n+1(φ, ξ, η,g)be an almost Kenmotsu(κ, μ, ν)-space andM be an immersed submanifold of M2n+1. Ifφ(TxM) ⊆ TxM, for each point at x ∈ M, then M is said to be an invariant submanifold of M2n+1(φ, ξ, η,g)with respect toφ. From (3), one can easily see that an invariant submanifold with respect toφis also invariant with respect toh.
Proposition 2.1 Let M be an invariant submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1(φ, ξ, η,g) such thatξ is tangent to M. Then the following equalities hold on M;
R(X,Y)ξ =κ[η(Y)X−η(X)Y] +μ[η(Y)h X−η(X)hY]
+ν[η(Y)φh X−η(X)φhY] (26) (∇Xφ)Y =g(φX+h X,Y)ξ −η(Y)(φX+h X) (27)
∇Xξ = −φ2X−φh X (28)
φσ (X,Y)=σ (φX,Y)=σ (X, φY), σ (X, ξ)=0, (29) where∇,σ and R denote the induced Levi-Civita connection on M, the shape operator and Riemannian curvature tensor of M, respectively.
Proof We omit the proof as it is a result of direct calculations.
In the rest of this paper, we will assume that M is an invariant submanifold of an almost Kenmotsu (κ, μ, ν)-spaceM2n+1(ϕ, ξ, η,g). In this case, from (5), we have
ϕh X= −hϕX, (30)
for allX ∈(T M), that is,Mis also invariant with respect to the tensor fieldh.
We need the following lemma to guarantee that the second fundamental formσ is not always identically zero.
Lemma 2.2 Let M be an invariant submanifold of an almost Kenmotsu(κ, μ, ν)-space M2n+1(φ, ξ, η,g). Then the second fundamental formσ of M is parallel if and only if M is totally geodesic providedκ =0.
Proof Let us suppose thatσ is parallel. From (16), we have
(∇Xσ )(Y,Z)= ∇⊥Xσ (Y,Z)−σ (∇XY,Z)−σ (Y,∇XZ)=0, (31) for all vector fields X,Y andZ onM2n+1. Setting Z =ξ in (31) and taking into account (28) and (29), we have
σ (∇Xξ,Y)= −σ (φ2X+φh X,Y)=0, that is,
σ (X,Y)−φσ (h X,Y)=0. (32)
Writingh XofXin (30) and using (7) and (27), we obtain
σ (h X,Y)−φσ (h2X,Y)=0,
σ (h X,Y)+(1+κ)φσ (X,Y)=0. (33) From (32) and (33), we conclude thatκσ (X,Y)=0, which proves our assertion. The converse is obvious.
Theorem 2.3 Let M be an invariant submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1(φ, η, ξ,g). If M is a pseudoparallel submanifold of an almost Kenmotsu(κ, μ, ν)-space M2n+1(φ, η, ξ,g), then M is either totally geodesic or the function L1satisfies
L1=κ∓
(κ+1)(ν2−μ2), μ.ν(κ+1)=0. (34) Proof LetMbe an invariant pseudoparallel submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1(φ, η, ξ,g). That means
L1Q(g, σ )(U,V;X,Y)=(R(X,Y)·σ )(U,V), for allX,Y,U,V ∈(T M), which implies that
−L1{σ ((X∧gY)U,V)+σ (U, (X∧gY)V)} =R⊥(X,Y)σ (U,V)
−σ (R(X,Y)U,V)−σ (U,R(X,Y)V).
SubstitutingX =U =ξ in the last equality, and taking into account Proposition2.1, we have L1σ (Y,V)= −σ (R(ξ,Y)ξ,V)=κσ (Y,V)+μσ (hY,V)+νσ (φhY,V), that is,
(L1−κ)σ (V,Y)=μσ (hY,V)+νφσ (hY,V). (35) SubstitutinghY forY in (35), by view of (6) and (29), we have
(L1−κ)σ (hY,V)=μσ (h2Y,V)+νφσ (h2Y,V)
= −(κ+1)[μσ (Y,V)+νφσ (Y,V)]. (36) From (35) and (36), one can easily see that
[(κ+1)(μ2−ν2)+(L1−κ)2]σ (Y,V)+2(κ+1)μνφσ (Y,V)=0. This tell us that Mis either totally geodesic submanifold or
(κ+1)(μ2−ν2)+(L1−κ)2=μν(κ+1)=0, (37)
which is equivalent to (34).
From Theorem2.3, we have the following corollary.
Corollary 2.4 Let M be an invariant submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1(φ, η, ξ,g). M is a semiparallel submanifold if and only if M is totally geodesic provided
(κ+1)(μ2−ν2)+κ2=0 or μν(κ+1)=0.
Theorem 2.5 Let M be an invariant submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1(φ, η, ξ,g). If M is a Ricci-generalized pseudoparallel submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1(φ, η, ξ,g), then M is either totally geodesic or the function L3satisfies
L3= 1 2n
1∓1
κ
(κ+1)(ν2−μ2)
, μ·ν=0. (38)
Proof SinceMis an invariant Ricci-generalized pseudoparallel, there exists a functionL3onMsuch that (R(X,Y)·σ )(U,V)=L3Q(S, σ )(U,V;X,Y),
for allX,Y,U,V ∈(T M). This yields to
−L3{σ ((X∧SY)U,V)+σ (U, (X∧SY)V)} =R⊥(X,Y)σ (U,V)
−σ (R(X,Y)U,V)−σ (U,R(X,Y)V). (39) Expanding by (39) and insertingX =V =ξ, making use of (29) and Proposition2.1, we obtain
2nκL3σ (U,Y)= −σ (R(ξ,Y)ξ,U)=κσ (Y,U)+μσ (hY,U) +νφσ (hY,U),
that is,
κ(2n L3−1)σ (U,Y)=μσ (hY,U)+νφσ (hY,U). (40) IfhY is taken instead ofY in (40) and by virtue of (6) and Proposition2.1, we reach at
κ(2n L3−1)σ (U,hY)= −(κ+1)[μσ (hY,U)+νφσ (hY,U)]. (41) From (40) and (41), we conclude that
[κ2(2n L3−1)2+(κ+1)(μ2−ν2)]σ (U,V)+2μνφσ (U,V)=0.
This completes the proof.
Theorem 2.6 Let M be an invariant submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1(φ, η, ξ,g). If M is a 2-pseudoparallel submanifold ofM2n+1(φ, η, ξ,g), then M is either totally geodesic or the function L2satisfies the condition
L2=κ∓
(κ+1)(ν2−μ2), μν(κ+1)=0. (42)
Proof If Mis an invariant 2-pseudoparallel of an almost Kenmotsu(κ, μ, ν)-space M2n+1(φ, η, ξ,g), then there exits a functionL2such that
L2Q(g,∇σ )(U,V,Z;X,Y)=(R(X,Y)·∇σ )(U,V,Z), for allX,Y,U,V,Z ∈(T M), which implies that
−L2{(∇(X∧gY)Uσ )(V,Z)+(∇Uσ )((X ∧gY)V,Z) +(∇Uσ )(V, (X∧gY)Z} =R⊥(X,Y)(∇Uσ )(V,Z)
−(∇R(X,Y)Uσ )(V,Z)−(∇Uσ )(R(X,Y)V,Z)
−(∇Uσ )(V,R(X,Y)Z). (43)
Here, substitutingξ forX =V in (43), we have
−L2{(∇(ξ∧gY)Uσ )(ξ,Z)+(∇Uσ )((ξ∧gY)ξ,Z)+(∇Uσ )(ξ, (ξ∧gY)Z}
= R⊥(ξ,Y)(∇Uσ )(ξ,Z)−(∇R(ξ,Y)Uσ )(ξ,Z)
−(∇Uσ )(R(ξ,Y)ξ,Z)−(∇Uσ )(ξ,R(ξ,Y)Z). (44) Now we will calculate them separately. In view of (15), (21), (28) and (29), we can derive
(∇(ξ∧gY)Uσ )(ξ,Z)= −σ (∇(ξ∧gY)Uξ,Z)
=σ (φ2(ξ ∧gY)U+φh(ξ ∧gY)U,Z)
= −σ (g(Y,U)ξ −η(U)Y,Z)−η(U)σ (φhY,Z)
=η(U){σ (Y,Z)−σ (φhY,Z)}. (45)
In the same way,
(∇Uσ )((ξ∧gY)ξ,Z)=(∇Uσ )(η(Y)ξ−Y,Z)
= −σ (∇Uη(Y)ξ,Z)−(∇Uσ )(Y,Z)
= −σ (U[η(Y)]ξ +η(Y)∇Uξ,Z)−(∇Uσ )(Y,Z)
=η(Y)σ (φ2U+φhU,Z)−(∇Uσ )(Y,Z)
=η(Y){σ (φhU,Z)−σ (U,Z)} −(∇Uσ )(Y,Z), (46) (∇Uσ )(ξ, (ξ∧gY)Z)= −σ (∇Uξ, (ξ∧gY)Z)
=σ (φ2U+φhU,g(Y,Z)ξ−η(Z)Y)
=η(Z){σ (U,Y)−σ (φhU,Y)}. (47)
For the right hand side of (44), by view of Proposition2.1, (15) and (17), we obtain
R⊥(ξ,Y)(∇Uσ )(ξ,Z)=R⊥(ξ,Y){∇U⊥σ (ξ,Z)−σ (∇Uξ,Z)−σ (ξ,∇UZ)}
= −R⊥(ξ,Y)σ (∇Uξ,Z)=R⊥(ξ,Y)σ (φ2U+φhU,Z)
=R⊥(ξ,Y){σ (φhU,Z)−σ (U,Z)}. (48) Also, making use of (3) and (26), we derive
(∇R(ξ,Y)Uσ )(ξ,Z)= −σ (∇R(ξ,Y)Uξ,Z)=σ (φ2R(ξ,Y)U+φh R(ξ,Y)U,Z)
= −σ (−κη(U)Y −μη(U)hY−νη(U)φhY,Z)
+σ (−κη(U)φhY −μη(U)φh2Y −νη(U)φhφhY,Z)
=η(U){κσ (Y,Z)+μσ (hY,Z)+νφσ (hY,Z)
−κφσ (hY,Z)+(κ+1)μφσ (Y,Z)+ν(κ+1)σ (Y,Z)}, (49) (∇Uσ )(R(ξ,Y)ξ,Z)
=(∇Uσ )(κ[η(Y)ξ−Y] −μhY −νφhY,Z). (50) Finally,
(∇Uσ )(ξ,R(ξ,Y)Z)= −σ (∇Uξ,R(ξ,Y)Z)=σ (φ2U+φhU,R(ξ,Y)Z)
= −σ (−κη(Z)Y −μη(Z)hY −νη(Z)φhY,U) +σ (−κη(Z)Y −μη(Z)hY −νη(Z)φhY, φhU)
=η(Z){κσ (Y,U)+μσ (hY,U)+νφσ (hY,U)
−κσ (Y, φhU)+μ(κ+1)φσ (U,Y)
−ν(κ+1)σ (U,Y)}. (51)
Consequently, the values of (45–51) are put in (44), we have
L2{η(U)σ (Y,Z)−η(U)φσ (hY,Z)−η(Y)σ (U,Z) +η(Y)φσ (hU,Z)+η(Z)σ (U,Y)−η(Z)φσ (hU,Y)
−(∇Uσ )(Y,Z)}
= R⊥(ξ,Y)σ (U,Z)−R⊥(ξ,Y)φσ (hU,Z) +η(U){κσ (Y,Z)+μσ (hY,Z)+νφσ (hY,Z)
−κφσ (hY,Z)+μ(κ+1)φσ (Y,Z)
+ν(κ+1)σ (Y,Z)} +η(Z){κσ (Y,U)+μσ (hY,U) +νφσ (hY,U)−κφσ (Y,hU)+μ(κ+1)φσ (U,Y)
−ν(κ+1)σ (U,Y)}
+(∇Uσ )(κ[η(Y)ξ−Y] −μhY −νφhY,Z). (52) TakingZ =ξ in (52) and taking into account Proposition2.1, (52) reduce to
L2{σ (U,Y)−φσ (hU,Y)−(∇Uσ )(Y, ξ)} =κσ (Y,U)+μσ (hY,U) +νφσ (hY,U)−κφσ (Y,hU)+μ(κ+1)φσ (U,Y)
−ν(κ+1)σ (U,Y)
+(∇Uσ )(κ[η(Y)ξ−Y] −μhY−νφhY, ξ), (53)
where, by direct calculations, one can easily see that
(∇Uσ )(κ[η(Y)ξ −Y] −μhY−νφhY, ξ)
= −σ (∇Uξ, κ[η(Y)ξ −Y] −μhY−νφhY)
=σ (φ2U +φhU, κ[η(Y)ξ−Y] −μhY −νφhY)
= −σ (U, κ[η(Y)ξ −Y] −μhY−νφhY) +σ (φhU, κ[η(Y)ξ−Y] −μhY −νφhY)
=κσ (Y,U)+μσ (hY,U)+νφσ (U,hY)−κφσ (Y,hU)
+μ(κ+1)φσ (U,Y)−ν(κ+1)σ (U,Y) (54)
and
(∇Uσ )(Y, ξ)= −σ (∇Uξ,Y)=σ (φ2U +φhU,Y)
=φσ (hU,Y)−σ (U,Y). (55) If (54) and (55) are put in (53), we reach at
[L2−κ+(ν−μφ)(κ+1)]σ (U,Y) −[(L2−κ)φ+(μ+φν)]σ (U,hY)
=0. (56)
SubstitutinghY instead ofY in (56), by virtue of Proposition2.1and (6), we can easily see that (κ+1)[(L2−κ)φ+(μ+φν)]σ (U,Y) +[L2−κ +(ν−μφ)(κ+1)]σ (U,hY)
=0. (57)
From common solutions of (56) and (57) providedκ =0, we can infer [(L2−κ)2−(κ+1)(ν2−μ2)]σ (U,Y)
+2μν(κ+1)φσ (U,Y)=0. (58)
This implies that Mis either totally geodesic or (42) is satisfied. Thus, the proof is completed.
From Theorem2.6, we have the following corollary.
Corollary 2.7 Let M be an invariant submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1(φ, η, ξ,g). M is 2-semiparallel submanifold if and only if M is totally geodesic submanifold provided
κ2−(κ+1)(ν2−μ2)=0, or μν(κ+1)=0.
Theorem 2.8 Let M be an invariant submanifold of an almost Kenmotsu(κ, μ, ν)-spaceM2n+1(φ, η, ξ,g). If M is a 2-generalized Ricci pseudoparallel submanifold ofM2n+1(φ, η, ξ,g), then M is either totally geodesic or the function L4satisfies the condition
L4= 1 2n
1∓1
k
(κ+1)(ν2−μ2)
, μ.ν.(κ+1)=0. (59)
Proof Let M be an invariant 2-generalized Ricci pseudoparallel submanifold of an almost Kenmotsu M2n+1(φ, η, ξ,g)-space. It follows that
L4Q(S,∇σ )( U,V,Z;X,Y)=(R(X,Y)·∇σ )(U,V,Z), (60) for allX,Y,Z,U,V ∈(T M), that is,
−L4{(∇(X∧SY)Uσ )(V,Z)+(∇Uσ )((X ∧SY)V,Z) +(∇Uσ )(V, (X∧SY)Z)} =R(X,Y)(∇Uσ )(V,Z)
−(∇R(X,Y)Uσ )(V,Z)−(∇Uσ )(R(X,Y)V,Z)
−(∇Uσ )(V,R(X,Y)Z).
This equality reduces forX =V =ξ
−L4{(∇(ξ∧SY)Uσ )(ξ,Z)+(∇Uσ )((ξ∧SY)ξ,Z) +(∇Uσ )(ξ, (ξ∧SY)Z)} =R(ξ,Y)(∇Uσ )(ξ,Z)
−(∇R(ξ,Y)Uσ )(ξ,Z)−(∇Uσ )(R(ξ,Y)ξ,Z)
−(∇Uσ )(ξ,R(ξ,Y)Z). (61)
Now, we will calculate these expressions separately. Firstly, making use of (6), (10), (15) and (29), we have (∇(ξ∧SY)Uσ )(ξ,Z)= −σ (∇(ξ∧SY)Uξ,Z)=σ (φ2(ξ ∧SY)U,Z)
+σ (φh(ξ∧SY)U,Z)
= −σ (S(Y,U)ξ−S(ξ,U)Y,Z) +σ (φh(S(Y,U)ξ −S(ξ,U)Y),Z)
=2nκη(U){σ (Y,Z)−φσ (hY,Z)}, (62) (∇Uσ )((ξ∧SY)ξ,Z)=(∇Uσ )(S(Y, ξ)ξ−S(ξ, ξ)Y,Z)
=2n{(∇Uσ )(κη(Y)ξ,Z)−(∇Uσ )(κY,Z)}
=2n{−σ (∇Uκη(Y)ξ,Z)−(∇Uσ )(κY,Z)}
=2n{−σ (Uη[κ(Y)]ξ +κη(Y)∇Uξ,Z)
−(∇Uσ )(κY,Z)}
=2n{κη(Y)σ (φ2U+φhU,Z)−(∇Uσ )(κY,Z)}
=2nη(Y)κ{φσ (hY,U)−σ (U,Z)}
−2n(∇Uσ )(κY,Z), (63)
(∇Uσ )(ξ,S(Y,Z)ξ −S(ξ,Z)Y)=(∇Uσ )(ξ,S(Y,Z)ξ)
−2n(∇Uσ )(κη(Z)Y, ξ)
= −2n(∇Uσ )(κη(Z)Y, ξ)
=2nκσ (∇Uξ, η(Z)Y)
= −2nκσ (φ2U +φhU, η(Z)Y)
=2nκη(Z){σ (U,Y)−σ (φhU,Y)}. (64)
Next, let’s calculate the right side of the equality.
R⊥(ξ,Y)(∇Uσ )(ξ,Z)= −R⊥(ξ,Y)σ (∇Uξ,Z)=R⊥(ξ,Y)σ (φ2U+φhU,Z)
=R⊥(ξ,Y){φσ (hU,Z)−σ (U,Z)}. (65) Furthermore, by virtue of Proposition2.1, (6) and (11), we observe
(∇R(ξ,Y)Uσ )(ξ,Z)= −σ (∇R(ξ,Y)Uξ,Z)=σ (φ2R(ξ,Y)U,Z) +σ (φh R(ξ,Y)U,Z)= −σ (R(ξ,Y)U,Z) +σ (φh R(ξ,Y)U,Z)
=η(U){κσ (Y,Z)+μσ (hY,Z)+νσ (φhU,Z)
−κσ (φhY,Z)−μσ (φh2Y,Z)−νσ (φhφhY,Z)}
=η(U){κσ (Y,Z)+μσ (hY,Z)+νσ (φhY,Z)
−κσ (φhY,Z)+μ(κ+1)φσ (Y,Z)
+ν(κ+1)σ (Y,Z)}, (66) (∇Uσ )(R(ξ,Y)ξ,Z)
=(∇Uσ )(κ[η(Y)ξ−Y] −μhY−νφhY,Z). (67) Finally,
(∇Uσ )(ξ,R(ξ,Y)Z)= −σ (∇Uξ,R(ξ,Y)Z)
=σ (φ2U+φhU,R(ξ,Y)Z)
= −σ (U,R(ξ,Y)Z)+σ (φhU,R(ξ,Y)Z)
=η(Z){σ (U,Y)+μσ (U,hY)+νφσ (U,hY)
−κφσ (hU,Y)−μφσ (hU,hY)−νφ2σ (hY,hU)}
=η(Z){κσ (U,Y)+μσ (hY,U)+νφσ (U,hY)
−κφσ (hU,Y)+μ(κ+1)φσ (U,Y)
−ν(κ+1)σ (U,Y)} (68) If (62) and (68) are put in (61), we have
2nκL4{η(U)σ (Y,Z)−η(U)φσ (hY,Z)−η(Y)σ (U,Z)+η(Y)φσ (hU,Z) +η(Z)σ (U,Y)−η(Z)φσ (hU,Y)−(∇Uσ )(Y,Z)}
=R⊥(ξ,Y){σ (U,Z)−φσ (hU,Z)} +η(U){κσ (Y,Z) +μσ (hY,Z)+νφσ (hY,Z)−κφσ (hY,Z)
+μ(κ+1)φσ (Y,Z)} +η(Z){κσ (U,Y)+μσ (U,hY) +νφσ (U,hY)−κφσ (hU,Y)+μ(κ+1)φσ (U,Y)
−ν(κ+1)σ (U,Y)}
+(∇Uσ )(κ[η(Y)ξ−Y] −μhY−νφhY,Z). (69)
TakingZ =ξ in (69), we arrive at
2nκL4{σ (U,Y)−φσ (hU,Y)−(∇Uσ )(Y, ξ)} =κσ (U,Y)+μσ (U,hY) +φσ (U,hY)−κφσ (hU,Y)+μ(κ+1)σ (U,Y)
+(∇Uσ )(κ[η(Y)ξ −Y] −μhY−νφhY, ξ)
−ν(κ+1)σ (U,Y), (70)
where
(∇Uσ )(κ[η(Y)ξ−Y] −μhY−νφhY, ξ)
= −σ (∇Uξ, κ[η(Y)ξ −Y] −μhY−νφhY)
=σ (φ2U+φhU, κ[η(Y)ξ−Y] −μhY −νφhY)
=κσ (U,Y)+μσ (U,hY)+νφσ (U,hY)−κφσ (Y,hU)
+μ(κ+1)φσ (U,Y)−ν(κ+1)σ (U,Y) (71)
and
(∇Uσ )(Y, ξ)= −σ (∇Uξ,Y)=σ (φ2U +φhU,Y)
=φσ (hU,Y)−σ (U,Y). (72) From (69), (70) and (71), we conclude
2nκL4{σ (U,Y)−φσ (hU,Y)} =κσ (U,Y)+μσ (U,hY)+νφσ (U,hY)
−κφσ (hU,Y)+μ(κ+1)φσ (U,Y)
−ν(κ+1)σ (U,Y), that is,
[2nκL4−κ+ν(κ+1)]σ (U,Y)=μ(κ+1)φσ (U,Y)+μσ (U,hY)
+[2nκL4+ν−κ]φσ (U,hY). (73) SubstitutinghY forY in (73) and again considering (6) and (11), we conclude
[2nκL4−κ+ν(κ+1)]σ (U,hY)= −(κ+1)[2nκL4+ν−κ]φσ (U,Y)
+μ(κ+1)φσ (U,hY)−μ(κ+1)σ (U,Y). (74)
From (73) and (74), we observe
[κ2(2n L4−1)2−(κ+1)(ν2−μ2)]σ (U,Y)+2μν(κ+1)φσ (U,Y)=0.
Sinceσ andφσare orthogonal, the last equality implies thatσ is either identically zero or (59) is satisfied.
Acknowledgements I would like to express my gratitude to the referees for valuable comments and suggestions.
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