# Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Nonmetric Connection

Authors: B. S. Anitha; C. S. Bagewadi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

The object of this paper is to study invariant submanifoldsM of Sasakian manifolds M˜ admitting a semisymmetric nonmetric connection, and it is shown that M admits semisymmetric nonmetric connection. Further it is proved that the second fundamental forms σ and σ¯ with respect to Levi-Civita connection and semi-symmetric nonmetric connection coincide. It is shown that if the second fundamental form σ is recurrent, 2-recurrent, generalized 2-recurrent, semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel and  M has parallel third fundamental form with respect to semisymmetric nonmetric connection, then  M is totally geodesic with respect to Levi-Civita connection.


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## Body

## 1.  Semisymmetric Nonmetric Connection

The geometry of invariant submanifoldsM of Sasakian manifolds M~ is carried out from 1970’s by M. Kon [1], D. Chinea [2], K. Yano and M. Kon [3] and B.S. Anitha and C.S. Bagewadi [4]. The aurthor [1] has proved that invariant submanifold of Sasakian structure also carries Sasakian structure. In this paper we extend the results to invariant submanifolds M of Sasakian manifolds admitting Semisymmetric Nonmetric connection.We know that a connection∇ on a manifold M is called a metric connection if there is a Riemannian metric g on M if ∇g=0; otherwise it is Nonmetric. Further it is said to be Semisymmetric if its torsion tensor T(X,Y)=0;thatis,T(X,Y)=w(Y)X-w(X)Y, where w is a 1-form. A study of Semisymmetric connection on a Riemannian manifold was initiated by Yano [5]. In 1992, Agashe and Chafle [6] introduced the notion of Semisymmetric Nonmetric connection. If ∇¯ denotes Semisymmetric Nonmetric connection on a contact metric manifold, then it is given by [6]
(1.1)∇¯XY=∇XY+η(Y)X,
where η(Y)=g(Y,ξ).The covariant differential of thepth order, p≥1 of a (0,k)-tensor field T,k≥1 denoted by ∇pT, defined on a Riemannian manifold (M,g) with the Levi-Civita connection ∇. The tensor T is said to be recurrent [7], if the following condition holds on M:
(1.2)(∇T)(X1,…,Xk;X)T(Y1,…,Yk)=(∇T)(Y1,…,Yk;X)T(X1,…,Xk),
respectively.Consider(1.3)(∇2T)(X1,…,Xk;X,Y)T(Y1,…,Yk)=(∇2T)(Y1,…,Yk;X,Y)T(X1,…,Xk),
where X,Y,X1,Y1,…,Xk,Yk∈TM. From (1.2) it follows that at a point x∈M, if the tensor T is nonzero, then there exists a unique 1-form ϕ, respectively, a (0,2)-tensor ψ, defined on a neighborhood U of x such that
(1.4)∇T=T⊗ϕ,ϕ=d(log⁡∥T∥),
respectively.The following(1.5)∇2T=T⊗ψ
holds on U, where ∥T∥ denotes the norm of T and ∥T∥2=g(T,T). The tensor T is said to be generalized2-recurrent if
(1.6)((∇2T)(X1,…,Xk;X,Y)-(∇T⊗ϕ)(X1,…,Xk;X,Y))T(Y1,…,Yk)=((∇2T)(Y1,…,Yk;X,Y)-(∇T⊗ϕ)(Y1,…,Yk;X,Y))T(X1,…,Xk)
holds on M, where ϕ is a 1-form on M. From this it follows that at a point x∈M if the tensor T is nonzero, then there exists a unique (0,2)-tensor ψ, defined on a neighborhood U of x, such that
(1.7)∇2T=∇T⊗ϕ+T⊗ψ
holds on U.


## 2. Isometric Immersion

Letf:(M,g)→(M~,g~) be an isometric immersion from an n-dimensional Riemannian manifold (M,g) into (n+d)-dimensional Riemannian manifold (M~,g~), n≥2, d≥1. We denote ∇ and ∇~ as Levi-Civita connection of Mn and M~n+d, respectively. Then the formulas of Gauss and Weingarten are given by
(2.1)∇~XY=∇XY+σ(X,Y),(2.2)∇~XN=-ANX+∇X⊥N,
for any tangent vector fields X,Y and the normal vector field N on M, where σ, A, and ∇⊥ are the second fundamental form, the shape operator, and the normal connection, respectively. If the second fundamental form σ is identically zero, then the manifold is said to be totally geodesic. The second fundamental form σ and AN  is related by
(2.3)g~(σ(X,Y),N)=g(ANX,Y),
for tangent vector fields X,Y. The first and second covariant derivatives of the second fundamental form σ are given by
(2.4)(∇~Xσ)(Y,Z)=∇X⊥(σ(Y,Z))-σ(∇XY,Z)-σ(Y,∇XZ),(2.5)(∇~2σ)(Z,W,X,Y)=(∇~X∇~Yσ)(Z,W)=∇X⊥((∇~Yσ)(Z,W))-(∇~Yσ)(∇XZ,W)-(∇~Xσ)(Z,∇YW)-(∇~∇XYσ)(Z,W),
respectively, where ∇~ is called the van der Waerden-Bortolotti connection of M [8]. If ∇~σ=0, then M is said to have parallel second fundamental form [8]. We next define endomorphisms R(X,Y) and X∧BY of χ(M) by
(2.6)R(X,Y)Z=∇X∇YZ-∇Y∇XZ-∇[X,Y]Z,(X∧BY)Z=B(Y,Z)X-B(X,Z)Y,
respectively, where X,Y,Z∈χ(M) and B is a symmetric (0,2)-tensor.Now, for a(0,k)-tensor field T,k≥1 and a (0,2)-tensor field B on (M,g), we define the tensor Q(B,T) by
(2.7)Q(B,T)(X1,…,Xk;X,Y)=-(T(X∧BY)X1,…,Xk)-⋯-T(X1,…,Xk-1(X∧BY)Xk).
Putting into consideration the previous formula “B=g,S and T=σ,” we obtain the tensors Q(g,σ) and Q(S,σ).


## 3. Sasakian Manifolds

Ann-dimensional differential manifold M is said to have an almost contact structure (ϕ,ξ,η) if it carries a tensor field ϕ of type (1,1), a vector field ξ, and 1-form η on M, respectively, such that
(3.1)ϕ2=-I+η⊗ξ,η(ξ)=1,η∘ϕ=0,ϕξ=0.Thus a manifoldM equipped with this structure is called an almost contact manifold and is denoted by (M,ϕ,ξ,η). If g is a Riemannian metric on an almost contact manifold M such that
(3.2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),g(X,ξ)=η(X),
where X,Y are vector fields defined on M, then M is said to have an almost contact metric structure (ϕ,ξ,η,g), and M with this structure is called an almost contact metric manifold and is denoted by (M,ϕ,ξ,η,g).If on(M,ϕ,ξ,η,g) the exterior derivative of 1-form η satisfies
(3.3)Φ(X,Y)=dη(X,Y)=g(X,ϕY),
then (ϕ,ξ,η,g) is said to be a contact metric structure and together with manifold M is called contact metric manifold and Φ is a 2-form. The contact metric structure (M,ϕ,ξ,η,g) is said to be normal if
(3.4)[ϕ,ϕ](X,Y)+2dη⊗ξ=0.If the contact metric structure is normal, then it is called a Sasakian structure andM is called a Sasakian manifold. Note that an almost contact metric manifold defines Sasakian structure if and only if
(3.5)(∇Xϕ)Y=g(X,Y)ξ-η(Y)X,(3.6)∇Xξ=-ϕX.Example of Sasakian Manifold
Consider the3-dimensional manifold M={(x,y,z)∈R3}, where (x,y,z) are the standard coordinates in R3. Let {E1,E2,E3} be linearly independent global frame field on M given by
(3.7)E1=∂∂x-2y∂∂z,E2=∂∂y,E3=∂∂z.
Let g be the Riemannian metric defined by
(3.8)g(E1,E2)=g(E1,E3)=g(E2,E3)=0,g(E1,E1)=g(E2,E2)=g(E3,E3)=1.
The (ϕ,ξ,η) is given by
(3.9)η=2ydx+dz,ξ=E3=∂∂z,ϕE1=E2,ϕE2=-E1,ϕE3=0.
The linearity property of ϕ and g yields
(3.10)η(E3)=1,ϕ2U=-U+η(U)E3,g(ϕU,ϕW)=g(U,W)-η(U)η(W),g(U,ξ)=η(U),
for any vector fields U,W on M. By definition of Lie bracket, we have
(3.11)[E1,E2]=2E3.
Let ∇ be the Levi-Civita connection with respect to previously mentioned metric g and be given by Koszula formula
(3.12)2g(∇XY,Z)=X(g(Y,Z))+Y(g(Z,X))-Z(g(X,Y))-g(X,[Y,Z])-g(Y,[X,Z])+g(Z,[X,Y]).
Then, we have
(3.13)∇E1E1=0,∇E1E2=E3,∇E1E3=-E2,∇E2E1=-E3,∇E2E2=0,∇E2E3=E1,∇E3E1=-E2,∇E3E2=E1,∇E3E3=0.
The tangent vectors X and Y to M are expressed as linear combination of E1,E2,E3; that is, X=a1E1+a2E2+a3E3 and Y=b1E1+b2E2+b3E3, where ai and bj are scalars. Clearly (ϕ,ξ,η,g) and X,Y satisfy (3.1), (3.2), (3.5), and (3.6). Thus M is a Sasakian manifold. Further the following relations hold:
(3.14)R(X,Y)Z={g(Y,Z)X-g(X,Z)Y},R(X,Y)ξ={η(Y)X-η(X)Y},R(ξ,X)Y={g(X,Y)ξ-η(Y)X},(3.15)R(ξ,X)ξ={η(X)ξ-X},(3.16)S(X,ξ)=(n-1)η(X),(3.17)Qξ=(n-1)ξ,
for all vector fields, X,Y,Z and where ∇ denotes the operator of covariant differentiation with respect to g,ϕ is a (1,1) tensor field, S is the Ricci tensor of type (0,2), and R is the Riemannian curvature tensor of the manifold.


## 4. Invariant Submanifolds of Sasakian Manifolds AdmittingSemisymmetric Nonmetric Connection

IfM~ is a Sasakian manifold with structure tensors (ϕ~,ξ~,η~,g~), then we know that its invariant submanifold M has the induced Sasakian structure (ϕ,ξ,η,g).A submanifoldM of a Sasakian manifold M~ with a Semisymmetric Nonmetric connection is called an invariant submanifold of M~ with a Semisymmetric Nonmetric connection, if for each x∈M, ϕ(TxM)⊂TxM. As a consequence, ξ becomes tangent to M. For an invariant submanifold of a Sasakian manifold with a Semisymmetric Nonmetric connection we have
(4.1)σ(X,ξ)=0,
for any vector X tangent to M.LetM~ be a Sasakian manifold admitting a Semisymmetric Nonmetric connection ∇~.Lemma 4.1.
LetM be an invariant submanifold of contact metric manifold M~ which admits Semisymmetric Nonmetric connection ∇~¯, and let σ and σ¯ be the second fundamental forms with respect to Levi-Civita connection and Semisymmetric Nonmetric connection; then (1) M admits Semisymmetric Nonmetric connection and  (2) the second fundamental forms with respect to ∇~ and ∇~¯ are equal.Proof.
We know that the contact metric structure(ϕ~,ξ~,η~,g~) on M~ induces (ϕ,ξ,η,g) on invariant submanifold. By virtue of (1.1), we get
(4.2)∇~¯XY=∇~XY+η(Y)X.
By using (2.1) in (4.2), we get
(4.3)∇~¯XY=∇XY+σ(X,Y)+η(Y)X.
Now Gauss formula (2.1) with respect to Semisymmetric Nonmetric connection is given by
(4.4)∇~¯XY=∇¯XY+σ¯(X,Y).
Equating (4.3) and (4.4), we get (1.1) and
(4.5)σ¯(X,Y)=σ(X,Y).Now we introduce the definitions of semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel with respect to Semisymmetric Nonmetric connection.Definition 4.2.
An immersion is said to be semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel with respect to Semisymmetric Nonmetric connection, respectively, if the following conditions hold for all vector fieldsX,Y tangent to M:
(4.6)R~¯⋅σ=0,R~¯⋅σ=L1Q(g,σ),R~¯⋅σ=L2Q(S,σ),
where R~¯ denotes the curvature tensor with respect to connection ∇~¯. Here L1 and L2 are functions depending on σ.Lemma 4.3.
LetM be an invariant submanifold of contact manifold M~ which admits Semisymmetric Nonmetric connection. Then Gauss and Weingarten formulae with respect to Semisymmetric Nonmetric connection are given by
(4.7)tan(R~¯(X,Y)Z)=R(X,Y)Z+η(∇YZ)X+η(Z)∇XY+η(Z)η(Y)X-η(∇XZ)Y-η(Z)∇YX-η(Z)η(X)Y-η(Z)[X,Y]+tan{∇~¯X{σ(Y,Z)}-∇~¯Y{σ(X,Z)}-∇~¯Yη(Z)X+∇~¯Xη(Z)Y},(4.8)nor(R~¯(X,Y)Z)=σ(X,∇YZ)+η(Z)σ(X,Y)-σ(Y,∇XZ)-η(Z)σ(Y,X)-σ([X,Y],Z)+nor{∇~¯X{σ(Y,Z)}-∇~¯Y{σ(X,Z)}-∇~¯Yη(Z)X+∇~¯Xη(Z)Y}.Proof.
The Riemannian curvature tensorR~ on M~ with respect to Semisymmetric Nonmetric connection is given by
(4.9)R~¯(X,Y)Z=∇~¯X∇~¯YZ-∇~¯Y∇~¯XZ-∇~¯[X,Y]Z.
Using (1.1) and (2.1) in (4.9), we get
(4.10)R~¯(X,Y)Z=R(X,Y)Z+σ(X,∇YZ)+η(∇YZ)X+∇~¯X{σ(Y,Z)}+∇~¯Xη(Z)Y+η(Z)∇XY+η(Z)σ(X,Y)+η(Z)η(Y)X-σ(Y,∇XZ)-η(∇XZ)Y-∇~¯Y{σ(X,Z)}-∇~¯Yη(Z)X-η(Z)∇YX-η(Z)σ(Y,X)-η(Z)η(X)Y-σ([X,Y],Z)-η(Z)[X,Y].
Comparing tangential and normal part of (4.10), we obtain Gauss and Weingarten formulae (4.7) and (4.8).Lemma 4.4.
LetM be an invariant submanifold of contact manifold M~ which admits Semisymmetric Nonmetric connection. If σ is semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel with respect to Semisymmetric Nonmetric connection, then we have
(4.11)(R~¯(X,Y)⋅σ)(U,V)=R⊥(X,Y)σ(U,V)-σ(R(X,Y)U,V)-σ(U,R(X,Y)V)-∇XAσ(U,V)Y+∇YAσ(U,V)X-A∇Y⊥σ(U,V)X+A∇X⊥σ(U,V)Y+Aσ(U,V)[X,Y]-σ(X,Aσ(U,V)Y)+σ(Y,Aσ(U,V)X)-η(Aσ(U,V)Y)X+η(Aσ(U,V)X)Y-η(∇YU)σ(X,V)-η(U)σ(∇XY,V)-η(U)η(Y)σ(X,V)+η(∇XU)σ(Y,V)+η(U)σ(∇YX,V)+η(U)η(X)σ(Y,V)+η(U)σ([X,Y],V)-σ(∇~¯Xη(U)Y,V)+σ(∇~¯Yη(U)X,V)-σ(∇~¯X{σ(Y,U)},V)+σ(∇~¯Y{σ(X,U)},V)-σ(σ(X,∇YU),V)-η(U)σ(σ(X,Y),V)+σ(σ(Y,∇XU),V)+η(U)σ(σ(Y,X),V)+σ(σ([X,Y],U),V)-η(∇YV)σ(U,X)-η(V)σ(U,∇XY)-η(V)η(Y)σ(U,X)+η(∇XV)σ(U,Y)+η(V)σ(U,∇YX)+η(V)η(X)σ(U,Y)+η(V)σ(U,[X,Y])-σ(U,∇~¯Xη(V)Y)+σ(U,∇~¯Yη(V)X)-σ(U,∇~¯X{σ(Y,V)})+σ(U,∇~¯Y{σ(X,V)})-σ(U,σ(X,∇YV))-η(V)σ(U,σ(X,Y))+σ(U,σ(Y,∇XV))+η(V)σ(U,σ(Y,X))+σ(U,σ([X,Y],V)),
for all vector fields X,Y,U, and V tangent to M, where
(4.12)R⊥(X,Y)=[∇X⊥,∇Y⊥]-∇[X,Y]⊥.Proof.
We know, from tensor algebra, that(4.13)(R~¯(X,Y)⋅σ)(U,V)=R~¯(X,Y)σ(U,V)-σ(R~¯(X,Y)U,V)-σ(U,R~¯(X,Y)V).
Replacing Z by σ(U,V) in (4.9), we get
(4.14)R~¯(X,Y)σ(U,V)=∇~¯X∇~¯Yσ(U,V)-∇~¯Y∇~¯Xσ(U,V)-∇~¯[X,Y]σ(U,V).
In view of (1.1), (2.1), and  (2.2), we have the following equalities:
(4.15)∇~¯X∇~¯Yσ(U,V)=∇~¯X(-Aσ(U,V)Y+∇Y⊥σ(U,V)),=-∇XAσ(U,V)Y-η(Aσ(U,V)Y)X-σ(X,Aσ(U,V)Y)-A∇Y⊥σ(U,V)X+∇X⊥∇Y⊥σ(U,V).
Similarly
(4.16)∇~¯Y∇~¯Xσ(U,V)=-∇YAσ(U,V)X-η(Aσ(U,V)X)Y-σ(Y,Aσ(U,V)X)-A∇X⊥σ(U,V)Y+∇Y⊥∇X⊥σ(U,V),(4.17)∇~¯[X,Y]σ(U,V)=-Aσ(U,V)[X,Y]+∇[X,Y]⊥σ(U,V).
Substituting (4.15), (4.16) and (4.17) into (4.14), we get
(4.18)R~¯(X,Y)σ(U,V)=R⊥(X,Y)σ(U,V)-∇XAσ(U,V)Y+∇YAσ(U,V)X-A∇Y⊥σ(U,V)X+A∇X⊥σ(U,V)Y+Aσ(U,V)[X,Y]-σ(X,Aσ(U,V)Y)+σ(Y,Aσ(U,V)X)-η(Aσ(U,V)Y)X+η(Aσ(U,V)X)Y.
By virtue of (4.10) in σ(R~¯(X,Y)U,V) and σ(U,R~¯(X,Y)V), we get
(4.19)σ(R~¯(X,Y)U,V)=σ(R(X,Y)U,V)+η(∇YU)σ(X,V)+η(U)σ(∇XY,V)+η(U)η(Y)σ(X,V)-η(∇XU)σ(Y,V)-η(U)σ(∇YX,V)-η(U)η(X)σ(Y,V)-η(U)σ([X,Y],V)+σ(∇~¯Xη(U)Y,V)-σ(∇~¯Yη(U)X,V)+σ(∇~¯X{σ(Y,U)},V)-σ(∇~¯Y{σ(X,U)},V)+σ(σ(X,∇YU),V)+η(U)σ(σ(X,Y),V)-σ(σ(Y,∇XU),V)-η(U)σ(σ(Y,X),V)-σ(σ([X,Y],U),V),(4.20)σ(U,R~¯(X,Y)V)=σ(U,R(X,Y)V)+η(∇YV)σ(U,X)+η(V)σ(U,∇XY)+η(V)η(Y)σ(U,X)-η(∇XV)σ(U,Y)-η(V)σ(U,∇YX)-η(V)η(X)σ(U,Y)-η(V)σ(U,[X,Y])+σ(U,∇~¯Xη(V)Y)-σ(U,∇~¯Yη(V)X)+σ(U,∇~¯X{σ(Y,V)})-σ(U,∇~¯Y{σ(X,V)})+σ(U,σ(X,∇YV))+η(V)σ(U,σ(X,Y))-σ(U,σ(Y,∇XV))-η(V)σ(U,σ(Y,X))-σ(U,σ([X,Y],V)).
Substituting (4.18), (4.19) and (4.20) into (4.13), we get (4.11).


## 5. Recurrent Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Nonmetric Connection

We consider invariant submanifolds of a Sasakian manifold whenσ is recurrent, 2-recurrent, and generalized 2-recurrent and M has parallel third fundamental form with respect to Semisymmetric Nonmetric connection. We write (2.4) and (2.5) with respect to Semisymmetric Nonmetric connection, and they are given by
(5.1)(∇~¯Xσ)(Y,Z)=∇¯X⊥(σ(Y,Z))-σ(∇¯XY,Z)-σ(Y,∇¯XZ),(5.2)(∇~¯2σ)(Z,W,X,Y)=(∇~¯X∇~¯Yσ)(Z,W)=∇¯X⊥((∇~¯Yσ)(Z,W))-(∇~¯Yσ)(∇¯XZ,W)-(∇~¯Xσ)(Z,∇¯YW)-(∇~¯∇¯XYσ)(Z,W).
We prove the following theorems.Theorem 5.1.
LetM be an invariant submanifold of a Sasakian manifold M~ admitting a Semisymmetric Nonmetric connection. Then σ is recurrent with respect to Semisymmetric Nonmetric connection if and only if it is totally geodesic with respect to Levi-Civita connection.Proof.
Letσ be recurrent with respect to Semisymmetric Nonmetric connection; from (1.4) we get
(5.3)(∇~¯Xσ)(Y,Z)=ϕ(X)σ(Y,Z),
where ϕ is a 1-form on M; in view of (5.1) and putting Z=ξ in the above equation, we have
(5.4)∇¯X⊥σ(Y,ξ)-σ(∇¯XY,ξ)-σ(Y,∇¯Xξ)=ϕ(X)σ(Y,ξ).
By virtue of (4.1) in (5.4), we get
(5.5)-σ(∇¯XY,ξ)-σ(Y,∇¯Xξ)=0.
Using (1.1), (3.1), (3.6), and (4.1) in (5.5), we get
(5.6)σ(Y,ϕX)-σ(Y,X)=0.
Replacing X by ϕX and by virtue of (3.1) and (4.1) in (5.6), we get
(5.7)-σ(Y,X)-σ(Y,ϕX)=0.
Adding (5.6) and (5.7), we obtain σ(X,Y)=0. Thus M is totally geodesic. The converse statement is trivial. This proves the theorem.Theorem 5.2.
LetM be an invariant submanifold of a Sasakian manifold M~ admitting a Semisymmetric Nonmetric connection. Then M has parallel third fundamental form with respect to Semisymmetric Nonmetric connection if and only if it is totally geodesic with respect to Levi-Civita connection.Proof.
LetM have parallel third fundamental form with respect to Semisymmetric Nonmetric connection. Then we have
(5.8)(∇~¯X∇~¯Yσ)(Z,W)=0.
Taking W=ξ and using (5.2) in the above equation, we have
(5.9)∇¯X⊥((∇~¯Yσ)(Z,ξ))-(∇~¯Yσ)(∇¯XZ,ξ)-(∇~¯Xσ)(Z,∇¯Yξ)-(∇~¯∇¯XYσ)(Z,ξ)=0.
In view of (4.1) and by virtue of (5.1) in (5.9), we get
(5.10)0=-∇¯X⊥{σ(∇¯YZ,ξ)+σ(Z,∇¯Yξ)}-∇¯Y⊥σ(∇¯XZ,ξ)+σ(∇¯Y∇¯XZ,ξ)+2σ(∇¯XZ,∇¯Yξ)-∇¯X⊥σ(Z,∇¯Yξ)+σ(Z,∇¯X∇¯Yξ)+σ(∇¯∇¯XYZ,ξ)+σ(Z,∇¯∇¯XYξ).
Using (1.1), (3.1), (3.6), and (4.1) in (5.10), we get
(5.11)0=2∇¯X⊥σ(Z,ϕY)-2∇¯X⊥σ(Z,Y)-2η(Z)σ(X,ϕY)+2σ(∇XZ,Y)+2η(Z)σ(X,Y)-σ(Z,∇XϕY)-σ(Z,ϕ∇XY)-η(Y)σ(Z,ϕX)+2σ(Z,∇XY)+2η(Y)σ(Z,X)-2σ(∇XZ,ϕY).
Putting  Y=ξ and using (3.1), (3.6), and (4.1) in (5.11), we get
(5.12)0=σ(Z,X)-3σ(Z,ϕX).
Replacing X by ϕX and by virtue of (3.1) and (4.1) in (5.12), we get
(5.13)0=σ(Z,ϕX)+3σ(Z,X).
Multiplying (5.12) by 1 and (5.13) by 3 and adding these two equations, we obtain σ(X,Z)=0. Thus M is totally geodesic. The converse statement is trivial. This proves the theorem.Corollary 5.3.
LetM be an invariant submanifold of a Sasakian manifold M~ admitting a Semisymmetric Nonmetric connection. Then σ is 2-recurrent with respect to Semisymmetric Nonmetric connection if and only if it is totally geodesic with respect to Levi-Civita connection.Proof.
Letσ be 2-recurrent with respect to Semisymmetric Nonmetric connection; from (1.5), we have
(5.14)(∇~¯X∇~¯Yσ)(Z,W)=σ(Z,W)ϕ(X,Y).
Taking W=ξ and using (5.2) in the above equation, we have
(5.15)∇¯X⊥((∇~¯Yσ)(Z,ξ))-(∇~¯Yσ)(∇¯XZ,ξ)-(∇~¯Xσ)(Z,∇¯Yξ)-(∇~¯∇¯XYσ)(Z,ξ)=σ(Z,ξ)ϕ(X,Y).
In view of (4.1) and by virtue of (5.1) in (5.15), we get
(5.16)0=-∇¯X⊥{σ(∇¯YZ,ξ)+σ(Z,∇¯Yξ)}-∇¯Y⊥σ(∇¯XZ,ξ)+σ(∇¯Y∇¯XZ,ξ)+2σ(∇¯XZ,∇¯Yξ)-∇¯X⊥σ(Z,∇¯Yξ)+σ(Z,∇¯X∇¯Yξ)+σ(∇¯∇¯XYZ,ξ)+σ(Z,∇¯∇¯XYξ).
Using (1.1), (3.1), (3.6), and (4.1) in (5.16), we get
(5.17)0=2∇¯X⊥σ(Z,ϕY)-2∇¯X⊥σ(Z,Y)-2η(Z)σ(X,ϕY)+2σ(∇XZ,Y)+2η(Z)σ(X,Y)-σ(Z,∇XϕY)-σ(Z,ϕ∇XY)-η(Y)σ(Z,ϕX)+2σ(Z,∇XY)+2η(Y)σ(Z,X)-2σ(∇XZ,ϕY).
Putting Y=ξ and using (3.1), (3.6), (4.1) in (5.17), we get
(5.18)0=σ(Z,X)-3σ(Z,ϕX).
Replacing X by ϕX and by virtue of (3.1) and (4.1) in (5.18), we get
(5.19)0=σ(Z,ϕX)+3σ(Z,X).
Multiplying (5.18) by 1 and (5.19) by 3 and adding these two equations, we obtain σ(X,Z)=0. Thus M is totally geodesic. The converse statement is trivial. This proves the theorem.Theorem 5.4.
LetM be an invariant submanifold of a Sasakian manifold M~ admitting a Semisymmetric Nonmetric connection. Then σ is generalized 2-recurrent with respect to Semisymmetric Nonmetric connection if and only if it is totally geodesic with respect to Levi-Civita connection.Proof.
Lettingσ be generalized 2-recurrent with respect to Semisymmetric Nonmetric connection, from (1.7), we have
(5.20)(∇~¯X∇~¯Yσ)(Z,W)=ψ(X,Y)σ(Z,W)+ϕ(X)(∇~¯Yσ)(Z,W),
where ψ and ϕ are 2-recurrent and 1-form, respectively. Taking W=ξ in (5.20) and using (4.1), we get
(5.21)(∇~¯X∇~¯Yσ)(Z,ξ)=ϕ(X)(∇~¯Yσ)(Z,ξ).
Using (4.1) and (5.2) in above equation, we get
(5.22)∇¯X⊥((∇~¯Yσ)(Z,ξ))-(∇~¯Yσ)(∇¯XZ,ξ)-(∇~¯Xσ)(Z,∇¯Yξ)-(∇~¯∇¯XYσ)(Z,ξ)=-ϕ(X){σ(∇¯YZ,ξ)+σ(Z,∇¯Yξ)}.
In view of (4.1) and by virtue of (5.1) in (5.22), we get
(5.23)-∇¯X⊥{σ(∇¯YZ,ξ)+σ(Z,∇¯Yξ)}-∇¯Y⊥σ(∇¯XZ,ξ)+σ(∇¯Y∇¯XZ,ξ)+2σ(∇¯XZ,∇¯Yξ)-∇¯X⊥σ(Z,∇¯Yξ)+σ(Z,∇¯X∇¯Yξ)+σ(∇¯∇¯XYZ,ξ)+σ(Z,∇¯∇¯XYξ)=-ϕ(X){σ(∇¯YZ,ξ)+σ(Z,∇¯Yξ)}.
Using (1.1), (3.1), (3.6), and (4.1) in (5.23), we get
(5.24)0=2∇¯X⊥σ(Z,ϕY)-2∇¯X⊥σ(Z,Y)-2η(Z)σ(X,ϕY)+2σ(∇XZ,Y)+2η(Z)σ(X,Y)-σ(Z,∇XϕY)-σ(Z,ϕ∇XY)-η(Y)σ(Z,ϕX)+2σ(Z,∇XY)+2η(Y)σ(Z,X)-2σ(∇XZ,ϕY)=-ϕ(X){-σ(Z,ϕY)+σ(Z,Y)}.
Putting Y=ξ and using (3.1), (3.6), (4.1) in (5.24), we get
(5.25)0=σ(Z,X)-3σ(Z,ϕX).
Replacing  X by ϕX and by virtue of (3.1) and (4.1) in (5.25), we get
(5.26)0=σ(Z,ϕX)+3σ(Z,X).
Multiplying (5.25) by 1 and (5.26) by 3 and adding these two equations, we obtain σ(X,Z)=0. Thus M is totally geodesic. The converse statement is trivial. This proves the theorem.


## 6. Semiparallel, Pseudoparallel, and Ricci-Generalized Pseudoparallel Invariant Submanifolds of Sasakian Manifolds AdmittingSemisymmetric Nonmetric Connection

We consider invariant submanifolds of Sasakian manifolds admitting Semisymmetric Nonmetric connection satisfying the conditionsR~¯·σ=0,R~¯·σ=L1Q(g,σ),R~¯·σ=L2Q(S,σ).Theorem 6.1.
LetM be an invariant submanifold of a Sasakian manifold M~ admitting a Semisymmetric Nonmetric connection. Then we prove that M is semiparallel with respect to Semisymmetric Nonmetric connection if and only if 6=2ϕ¯+ξ.Proof.
LetM be semiparallel R~¯·σ=0. Putting X=V=ξ and by virtue of (3.1), (3.6), and (4.1) in (4.11), we get
(6.1)0=-σ(U,R(ξ,Y)ξ)-σ(∇~¯ξη(U)Y,ξ)+σ(∇~¯Yη(U)ξ,ξ)-σ(∇~¯ξσ(Y,U),ξ)-σ(U,∇ξY)+σ(U,∇Yξ)+σ(U,[ξ,Y])-σ(U,∇~¯ξY)+σ(U,∇~¯Yξ)+σ(U,Y).
Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.1), we get
(6.2)0=3σ(U,Y)-σ(∇~¯ξη(U)Y,ξ)-σ(U,ϕY)-σ(U,∇ξY).
By definition σ is a vector-valued covariant tensor, and so σ(U,Y) is a vector. Therefore ∇~¯ξσ(Y,U) is a vector, and hence by (4.1), we have
(6.3)σ(∇~¯ξσ(Y,U),ξ)=0.
Then from (6.2), we get
(6.4)3σ(U,Y)=ϕ¯σ(U,Y)+σ(U,∇ξY).
Interchanging Y and U in (6.4), we get
(6.5)3σ(Y,U)=ϕ¯σ(Y,U)+σ(U,∇ξY).
Adding these tow equations, (6.4) and (6.5), we get
(6.6)6=2ϕ¯+ξ.Theorem 6.2.
LetM be an invariant submanifold of a Sasakian manifold M~ admitting a Semisymmetric Nonmetric connection. Then we prove that M is pseudoparallel with respect to Semisymmetric Nonmetric connection if and only if L1=ϕ¯+ξ/2-3.Proof.
LetM be pseudoparallel R~¯·σ=L1Q(g,σ). Putting X=V=ξ and by virtue of (3.1), (3.6), and (4.1) in (2.7), (4.11), we get
(6.7)-σ(U,R(ξ,Y)ξ)-σ(∇~¯ξη(U)Y,ξ)+σ(∇~¯Yη(U)ξ,ξ)-σ(∇~¯ξσ(Y,U),ξ)-σ(U,∇ξY)+σ(U,∇Yξ)+σ(U,[ξ,Y])-σ(U,∇~¯ξY)+σ(U,∇~¯Yξ)+σ(U,Y)=-L1σ(U,Y).
Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.7), we get
(6.8)3σ(U,Y)-σ(∇~¯ξη(U)Y,ξ)-σ(U,ϕY)-σ(U,∇ξY)=-L1σ(U,Y).
Now by using (6.3) in (6.8), we get
(6.9)(3+L1)σ(U,Y)=ϕ¯σ(U,Y)+σ(U,∇ξY).
Interchanging Y and U in (6.9), we get
(6.10)(3+L1)σ(Y,U)=ϕ¯σ(Y,U)+σ(Y,∇ξU).
Adding (6.9)  and (6.10), we get
(6.11)L1=ϕ¯+ξ2-3.Theorem 6.3.
LetM be an invariant submanifold of a Sasakian manifold M~ admitting a Semisymmetric Nonmetric connection. Then we prove that M is Ricci-generalized pseudoparallel with respect to Semisymmetric Nonmetric connection if and only if L2=(1/(n-1))[ϕ¯+ξ/2-3].Proof.
LetM be Ricci-generalized pseudoparallel R~¯·σ=L2Q(S,σ). Putting X=V=ξ and by virtue of (3.1), (3.6), (3.16), and (4.1) in (2.7), (4.11), we get
(6.12)-σ(U,R(ξ,Y)ξ)-σ(∇~¯ξη(U)Y,ξ)+σ(∇~¯Yη(U)ξ,ξ)-σ(∇~¯ξσ(Y,U),ξ)-σ(U,∇ξY)+σ(U,∇Yξ)+σ(U,[ξ,Y])-σ(U,∇~¯ξY)+σ(U,∇~¯Yξ)+σ(U,Y)=-L2(n-1)σ(U,Y).
Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.12), we get
(6.13)3σ(U,Y)-σ(∇~¯ξη(U)Y,ξ)-σ(U,ϕY)-σ(U,∇ξY)=-L2(n-1)σ(U,Y).
Now by using (6.3) in (6.13), we get
(6.14)(3+L2(n-1))σ(U,Y)=ϕ¯σ(U,Y)+σ(U,∇ξY).
Interchanging Y and U in (6.14), we get
(6.15)(3+L2(n-1))σ(Y,U)=ϕ¯σ(Y,U)+σ(Y,∇ξU).
Adding (6.14) and (6.15), we get
(6.16)2(3+L2(n-1))σ(U,Y)=2ϕ¯σ(U,Y)+∇ξσ(U,Y).
Writting the above equation, we have
(6.17)L2=1(n-1)[ϕ¯+ξ2-3].Remark 6.4.
LetM be an invariant submanifold of a Sasakian manifold which admits Semisymmetric Nonmetric connection. If M is semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel, then we have obtained conditions connecting ϕ, ξ, L1, and L2. These conditions need further investigation and are to be interpreted geometrically.Using Theorems5.1 to 5.4 and corollary 5.3, we have the following result.Corollary 6.5.
LetM be an invariant submanifold of a Sasakian manifold M~ admitting a Semisymmetric Nonmetric connection. Then the following statements are equivalent: (1)
σis recurrent,(2)
σ is 2-recurrent,(3)
σ is generalized 2-recurrent,(4)
M has parallel third fundamental form.



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*Source: 947640-2012-08-09.xml*