Abstract

The object of this paper is to study invariant submanifolds 𝑀 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.

1. Semisymmetric Nonmetric Connection

The geometry of invariant submanifolds 𝑀 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)𝑋𝑤(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] 𝑋𝑌=𝑋𝑌+𝜂(𝑌)𝑋,(1.1) where 𝜂(𝑌)=𝑔(𝑌,𝜉).

The covariant differential of the 𝑝th order, 𝑝1 of a (0,𝑘)-tensor field 𝑇,𝑘1 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: 𝑋(𝑇)1,,𝑋𝑘𝑇𝑌;𝑋1,,𝑌𝑘=𝑌(𝑇)1,,𝑌𝑘𝑇𝑋;𝑋1,,𝑋𝑘,(1.2) respectively.

Consider 2𝑇𝑋1,,𝑋𝑘𝑇𝑌;𝑋,𝑌1,,𝑌𝑘=2𝑇𝑌1,,𝑌𝑘𝑇𝑋;𝑋,𝑌1,,𝑋𝑘,(1.3) where 𝑋,𝑌,𝑋1,𝑌1,,𝑋𝑘,𝑌𝑘𝑇𝑀. From (1.2) it follows that at a point 𝑥𝑀, 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 𝑇=𝑇𝜙,𝜙=𝑑(log𝑇),(1.4) respectively.

The following 2𝑇=𝑇𝜓(1.5) holds on 𝑈, where 𝑇 denotes the norm of 𝑇 and 𝑇2=𝑔(𝑇,𝑇). The tensor 𝑇 is said to be generalized 2-recurrent if 2𝑇𝑋1,,𝑋𝑘𝑋;𝑋,𝑌(𝑇𝜙)1,,𝑋𝑘𝑇𝑌;𝑋,𝑌1,,𝑌𝑘=2𝑇𝑌1,,𝑌𝑘𝑌;𝑋,𝑌(𝑇𝜙)1,,𝑌𝑘𝑇𝑋;𝑋,𝑌1,,𝑋𝑘(1.6) holds on 𝑀, where 𝜙 is a 1-form on 𝑀. From this it follows that at a point 𝑥𝑀 if the tensor 𝑇 is nonzero, then there exists a unique (0,2)-tensor 𝜓, defined on a neighborhood 𝑈 of 𝑥, such that 2𝑇=𝑇𝜙+𝑇𝜓(1.7) holds on 𝑈.

2. Isometric Immersion

Let 𝑓(𝑀,𝑔)(𝑀,̃𝑔) be an isometric immersion from an 𝑛-dimensional Riemannian manifold (𝑀,𝑔) into (𝑛+𝑑)-dimensional Riemannian manifold (𝑀,̃𝑔), 𝑛2, 𝑑1. We denote and as Levi-Civita connection of 𝑀𝑛 and 𝑀𝑛+𝑑, respectively. Then the formulas of Gauss and Weingarten are given by 𝑋𝑌=𝑋𝑌+𝜎(𝑋,𝑌),(2.1)𝑋𝑁=𝐴𝑁𝑋+𝑋𝑁,(2.2) 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 𝐴̃𝑔(𝜎(𝑋,𝑌),𝑁)=𝑔𝑁𝑋,𝑌,(2.3) for tangent vector fields 𝑋,𝑌. The first and second covariant derivatives of the second fundamental form 𝜎 are given by 𝑋𝜎(𝑌,𝑍)=𝑋(𝜎(𝑌,𝑍))𝜎𝑋𝑌,𝑍𝜎𝑌,𝑋𝑍,(2.4)2𝜎(𝑍,𝑊,𝑋,𝑌)=𝑋𝑌𝜎(𝑍,𝑊)=𝑋𝑌𝜎(𝑍,𝑊)𝑌𝜎𝑋𝑍,𝑊𝑋𝜎𝑍,𝑌𝑊𝑋𝑌𝜎(𝑍,𝑊),(2.5) 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 𝑋𝐵𝑌 of 𝜒(𝑀) by 𝑅(𝑋,𝑌)𝑍=𝑋𝑌𝑍𝑌𝑋𝑍[𝑋,𝑌]𝑍,𝑋𝐵𝑌𝑍=𝐵(𝑌,𝑍)𝑋𝐵(𝑋,𝑍)𝑌,(2.6) respectively, where 𝑋,𝑌,𝑍𝜒(𝑀) and 𝐵 is a symmetric (0,2)-tensor.

Now, for a (0,𝑘)-tensor field 𝑇,𝑘1 and a (0,2)-tensor field 𝐵 on (𝑀,𝑔), we define the tensor 𝑄(𝐵,𝑇) by 𝑄𝑋(𝐵,𝑇)1,,𝑋𝑘𝑇;𝑋,𝑌=𝑋𝐵𝑌𝑋1,,𝑋𝑘𝑋𝑇1,,𝑋𝑘1𝑋𝐵𝑌𝑋𝑘.(2.7) 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

An 𝑛-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 𝜙2=𝐼+𝜂𝜉,𝜂(𝜉)=1,𝜂𝜙=0,𝜙𝜉=0.(3.1)

Thus a manifold 𝑀 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 𝑔(𝜙𝑋,𝜙𝑌)=𝑔(𝑋,𝑌)𝜂(𝑋)𝜂(𝑌),𝑔(𝑋,𝜉)=𝜂(𝑋),(3.2) 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 Φ(𝑋,𝑌)=𝑑𝜂(𝑋,𝑌)=𝑔(𝑋,𝜙𝑌),(3.3) 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 []𝜙,𝜙(𝑋,𝑌)+2𝑑𝜂𝜉=0.(3.4)

If the contact metric structure is normal, then it is called a Sasakian structure and 𝑀 is called a Sasakian manifold. Note that an almost contact metric manifold defines Sasakian structure if and only if 𝑋𝜙𝑌=𝑔(𝑋,𝑌)𝜉𝜂(𝑌)𝑋,(3.5)𝑋𝜉=𝜙𝑋.(3.6)

Example of Sasakian Manifold
Consider the 3-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 𝐸1=𝜕𝜕𝜕𝑥2𝑦𝜕𝑧,𝐸2=𝜕𝜕𝑦,𝐸3=𝜕.𝜕𝑧(3.7) Let 𝑔 be the Riemannian metric defined by 𝑔𝐸1,𝐸2𝐸=𝑔1,𝐸3𝐸=𝑔2,𝐸3𝑔𝐸=0,1,𝐸1𝐸=𝑔2,𝐸2𝐸=𝑔3,𝐸3=1.(3.8) The (𝜙,𝜉,𝜂) is given by 𝜂=2𝑦𝑑𝑥+𝑑𝑧,𝜉=𝐸3=𝜕,𝜕𝑧𝜙𝐸1=𝐸2,𝜙𝐸2=𝐸1,𝜙𝐸3=0.(3.9) The linearity property of 𝜙 and 𝑔 yields 𝜂𝐸3=1,𝜙2𝑈=𝑈+𝜂(𝑈)𝐸3,𝑔(𝜙𝑈,𝜙𝑊)=𝑔(𝑈,𝑊)𝜂(𝑈)𝜂(𝑊),𝑔(𝑈,𝜉)=𝜂(𝑈),(3.10) for any vector fields 𝑈,𝑊 on 𝑀. By definition of Lie bracket, we have 𝐸1,𝐸2=2𝐸3.(3.11) Let be the Levi-Civita connection with respect to previously mentioned metric 𝑔 and be given by Koszula formula 2𝑔𝑋[][][]𝑌,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).(3.12) Then, we have 𝐸1𝐸1=0,𝐸1𝐸2=𝐸3,𝐸1𝐸3=𝐸2,𝐸2𝐸1=𝐸3,𝐸2𝐸2=0,𝐸2𝐸3=𝐸1,𝐸3𝐸1=𝐸2,𝐸3𝐸2=𝐸1,𝐸3𝐸3=0.(3.13) 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: 𝑅𝑅(𝑋,𝑌)𝑍={𝑔(𝑌,𝑍)𝑋𝑔(𝑋,𝑍)𝑌},(𝑋,𝑌)𝜉={𝜂(𝑌)𝑋𝜂(𝑋)𝑌},𝑅(𝜉,𝑋)𝑌={𝑔(𝑋,𝑌)𝜉𝜂(𝑌)𝑋},(3.14)𝑅(𝜉,𝑋)𝜉={𝜂(𝑋)𝜉𝑋},(3.15)𝑆(𝑋,𝜉)=(𝑛1)𝜂(𝑋),(3.16)𝑄𝜉=(𝑛1)𝜉,(3.17) 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 Admitting Semisymmetric Nonmetric Connection

If 𝑀 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 submanifold 𝑀 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 𝑥𝑀, 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 𝜎(𝑋,𝜉)=0,(4.1) for any vector 𝑋 tangent to 𝑀.

Let 𝑀 be a Sasakian manifold admitting a Semisymmetric Nonmetric connection .

Lemma 4.1. Let 𝑀 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 𝑋𝑌=𝑋𝑌+𝜂(𝑌)𝑋.(4.2) By using (2.1) in (4.2), we get 𝑋𝑌=𝑋𝑌+𝜎(𝑋,𝑌)+𝜂(𝑌)𝑋.(4.3) Now Gauss formula (2.1) with respect to Semisymmetric Nonmetric connection is given by 𝑋𝑌=𝑋𝑌+𝜎(𝑋,𝑌).(4.4) Equating (4.3) and (4.4), we get (1.1) and 𝜎(𝑋,𝑌)=𝜎(𝑋,𝑌).(4.5)

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 fields 𝑋,𝑌 tangent to 𝑀: 𝑅𝜎=0,𝑅𝜎=𝐿1𝑄(𝑔,𝜎),𝑅𝜎=𝐿2𝑄(𝑆,𝜎),(4.6) where 𝑅 denotes the curvature tensor with respect to connection . Here 𝐿1 and 𝐿2 are functions depending on 𝜎.

Lemma 4.3. Let 𝑀 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 tan𝑅(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍+𝜂𝑌𝑍𝑋+𝜂(𝑍)𝑋𝑌+𝜂(𝑍)𝜂(𝑌)𝑋𝜂𝑋𝑍𝑌𝜂(𝑍)𝑌[]𝑋𝜂(𝑍)𝜂(𝑋)𝑌𝜂(𝑍)𝑋,𝑌+tan𝑋{𝜎(𝑌,𝑍)}𝑌{𝜎(𝑋,𝑍)}𝑌𝜂(𝑍)𝑋+𝑋,𝜂(𝑍)𝑌(4.7)nor𝑅(𝑋,𝑌)𝑍=𝜎𝑋,𝑌𝑍+𝜂(𝑍)𝜎(𝑋,𝑌)𝜎𝑌,𝑋𝑍[]𝜂(𝑍)𝜎(𝑌,𝑋)𝜎(𝑋,𝑌,𝑍)+nor𝑋{𝜎(𝑌,𝑍)}Y{𝜎(𝑋,𝑍)}𝑌𝜂(𝑍)𝑋+𝑋.𝜂(𝑍)𝑌(4.8)

Proof. The Riemannian curvature tensor 𝑅 on 𝑀 with respect to Semisymmetric Nonmetric connection is given by 𝑅(𝑋,𝑌)𝑍=𝑋𝑌𝑍𝑌𝑋𝑍[𝑋,𝑌]𝑍.(4.9) Using (1.1) and (2.1) in (4.9), we get 𝑅(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍+𝜎𝑋,𝑌𝑍+𝜂𝑌𝑍𝑋+𝑋{𝜎(𝑌,𝑍)}+𝑋𝜂(𝑍)𝑌+𝜂(𝑍)𝑋𝑌+𝜂(𝑍)𝜎(𝑋,𝑌)+𝜂(𝑍)𝜂(𝑌)𝑋𝜎𝑌,𝑋𝑍𝜂𝑋𝑍𝑌𝑌{𝜎(𝑋,𝑍)}𝑌𝜂(𝑍)𝑋𝜂(𝑍)𝑌𝑋[][].𝜂(𝑍)𝜎(𝑌,𝑋)𝜂(𝑍)𝜂(𝑋)𝑌𝜎(𝑋,𝑌,𝑍)𝜂(𝑍)𝑋,𝑌(4.10) Comparing tangential and normal part of (4.10), we obtain Gauss and Weingarten formulae (4.7) and (4.8).

Lemma 4.4. Let 𝑀 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 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)𝑋𝐴𝑌𝜎(𝑈,𝑉)𝑋+𝐴𝑋𝜎(𝑈,𝑉)𝑌+𝐴𝜎(𝑈,𝑉)[]𝑋,𝑌𝜎𝑋,𝐴𝜎(𝑈,𝑉)𝑌+𝜎𝑌,𝐴𝜎(𝑈,𝑉)𝑋𝐴𝜂𝜎(𝑈,𝑉)𝑌𝐴𝑋+𝜂𝜎(𝑈,𝑉)𝑋𝑌𝜂𝑌𝑈𝜎(𝑋,𝑉)𝜂(𝑈)𝜎𝑋𝑌,𝑉𝜂(𝑈)𝜂(𝑌)𝜎(𝑋,𝑉)+𝜂𝑋𝑈𝜎(𝑌,𝑉)+𝜂(𝑈)𝜎𝑌[])𝑋,𝑉+𝜂(𝑈)𝜂(𝑋)𝜎(𝑌,𝑉)+𝜂(𝑈)𝜎(𝑋,𝑌,𝑉𝜎𝑋𝜂(𝑈)𝑌,𝑉+𝜎𝑌𝜂(𝑈)𝑋,𝑉𝜎𝑋{𝜎(𝑌,𝑈)},𝑉+𝜎𝑌𝜎{𝜎(𝑋,𝑈)},𝑉𝜎𝑋,𝑌𝑈𝜎,𝑉𝜂(𝑈)𝜎(𝜎(𝑋,𝑌),𝑉)+𝜎𝑌,𝑋𝑈[],𝑉+𝜂(𝑈)𝜎(𝜎(𝑌,𝑋),𝑉)+𝜎(𝜎(𝑋,𝑌,𝑈),𝑉)𝜂𝑌𝑉𝜎(𝑈,𝑋)𝜂(𝑉)𝜎𝑈,𝑋𝑌𝜂(𝑉)𝜂(𝑌)𝜎(𝑈,𝑋)+𝜂𝑋𝑉𝜎(𝑈,𝑌)+𝜂(𝑉)𝜎𝑈,𝑌𝑋[]+𝜂(𝑉)𝜂(𝑋)𝜎(𝑈,𝑌)+𝜂(𝑉)𝜎(𝑈,𝑋,𝑌)𝜎𝑈,𝑋𝜂(𝑉)𝑌+𝜎𝑈,𝑌𝜂(𝑉)𝑋𝜎𝑈,𝑋{𝜎(𝑌,𝑉)}+𝜎𝑈,𝑌{𝜎(𝑋,𝑉)}𝜎𝑈,𝜎𝑋,𝑌𝑉𝜂(𝑉)𝜎(𝑈,𝜎(𝑋,𝑌))+𝜎𝑈,𝜎𝑌,𝑋𝑉[]+𝜂(𝑉)𝜎(𝑈,𝜎(𝑌,𝑋))+𝜎(𝑈,𝜎(𝑋,𝑌,𝑉)),(4.11) for all vector fields 𝑋,𝑌,𝑈, and 𝑉 tangent to 𝑀, where 𝑅(𝑋,𝑌)=𝑋,𝑌[𝑋,𝑌].(4.12)

Proof. We know, from tensor algebra, that 𝑅(𝑋,𝑌)𝜎(𝑈,𝑉)=𝑅(𝑋,𝑌)𝜎(𝑈,𝑉)𝜎𝑅(𝑋,𝑌)𝑈,𝑉𝜎𝑈,𝑅(𝑋,𝑌)𝑉.(4.13) Replacing 𝑍 by 𝜎(𝑈,𝑉) in (4.9), we get 𝑅(𝑋,𝑌)𝜎(𝑈,𝑉)=𝑋𝑌𝜎(𝑈,𝑉)𝑌𝑋𝜎(𝑈,𝑉)[𝑋,𝑌]𝜎(𝑈,𝑉).(4.14) In view of (1.1), (2.1), and (2.2), we have the following equalities: 𝑋𝑌𝜎(𝑈,𝑉)=𝑋𝐴𝜎(𝑈,𝑉)𝑌+𝑌,𝜎(𝑈,𝑉)=𝑋𝐴𝜎(𝑈,𝑉)𝐴𝑌𝜂𝜎(𝑈,𝑉)𝑌𝑋𝜎𝑋,𝐴𝜎(𝑈,𝑉)𝑌𝐴𝑌𝜎(𝑈,𝑉)𝑋+𝑋𝑌𝜎(𝑈,𝑉).(4.15) Similarly 𝑌𝑋𝜎(𝑈,𝑉)=𝑌𝐴𝜎(𝑈,𝑉)𝐴𝑋𝜂𝜎(𝑈,𝑉)𝑋𝑌𝜎𝑌,𝐴𝜎(𝑈,𝑉)𝑋𝐴𝑋𝜎(𝑈,𝑉)𝑌+𝑌𝑋𝜎(𝑈,𝑉),(4.16)[𝑋,𝑌]𝜎(𝑈,𝑉)=𝐴𝜎(𝑈,𝑉)[]𝑋,𝑌+[]𝑋,𝑌𝜎(𝑈,𝑉).(4.17) Substituting (4.15), (4.16) and (4.17) into (4.14), we get 𝑅(𝑋,𝑌)𝜎(𝑈,𝑉)=𝑅(𝑋,𝑌)𝜎(𝑈,𝑉)𝑋𝐴𝜎(𝑈,𝑉)𝑌+𝑌𝐴𝜎(𝑈,𝑉)𝑋𝐴𝑌𝜎(𝑈,𝑉)𝑋+𝐴𝑋𝜎(𝑈,𝑉)𝑌+𝐴𝜎(𝑈,𝑉)[]𝑋,𝑌𝜎𝑋,𝐴𝜎(𝑈,𝑉)𝑌+𝜎𝑌,𝐴𝜎(𝑈,𝑉)𝑋𝐴𝜂𝜎(𝑈,𝑉)𝑌𝐴𝑋+𝜂𝜎(𝑈,𝑉)𝑋𝑌.(4.18) By virtue of (4.10) in 𝜎(𝑅(𝑋,𝑌)𝑈,𝑉) and 𝜎(𝑈,𝑅(𝑋,𝑌)𝑉), we get 𝜎𝑅(𝑋,𝑌)𝑈,𝑉=𝜎(𝑅(𝑋,𝑌)𝑈,𝑉)+𝜂𝑌𝑈𝜎(𝑋,𝑉)+𝜂(𝑈)𝜎𝑋𝑌,𝑉+𝜂(𝑈)𝜂(𝑌)𝜎(𝑋,𝑉)𝜂𝑋𝑈𝜎(𝑌,𝑉)𝜂(𝑈)𝜎𝑌[]𝑋,𝑉𝜂(𝑈)𝜂(𝑋)𝜎(𝑌,𝑉)𝜂(𝑈)𝜎(𝑋,𝑌,𝑉)+𝜎𝑋𝜂(𝑈)𝑌,𝑉𝜎𝑌𝜂(𝑈)𝑋,𝑉+𝜎𝑋{𝜎(𝑌,𝑈)},𝑉𝜎𝑌𝜎{𝜎(𝑋,𝑈)},𝑉+𝜎𝑋,𝑌𝑈𝜎,𝑉+𝜂(𝑈)𝜎(𝜎(𝑋,𝑌),𝑉)𝜎𝑌,𝑋𝑈[]𝜎,𝑉𝜂(𝑈)𝜎(𝜎(𝑌,𝑋),𝑉)𝜎(𝜎(𝑋,𝑌,𝑈),𝑉),(4.19)𝑈,𝑅(𝑋,𝑌)𝑉=𝜎(𝑈,𝑅(𝑋,𝑌)𝑉)+𝜂𝑌𝑉𝜎(𝑈,𝑋)+𝜂(𝑉)𝜎𝑈,𝑋𝑌+𝜂(𝑉)𝜂(𝑌)𝜎(𝑈,𝑋)𝜂𝑋𝑉𝜎(𝑈,𝑌)𝜂(𝑉)𝜎𝑈,𝑌𝑋[]𝜂(𝑉)𝜂(𝑋)𝜎(𝑈,𝑌)𝜂(𝑉)𝜎(𝑈,𝑋,𝑌)+𝜎𝑈,𝑋𝜂(𝑉)𝑌𝜎𝑈,𝑌𝜂(𝑉)𝑋+𝜎𝑈,𝑋{𝜎(𝑌,𝑉)}𝜎𝑈,𝑌{𝜎(𝑋,𝑉)}+𝜎𝑈,𝜎𝑋,𝑌𝑉+𝜂(𝑉)𝜎(𝑈,𝜎(𝑋,𝑌))𝜎𝑈,𝜎𝑌,𝑋𝑉[]𝜂(𝑉)𝜎(𝑈,𝜎(𝑌,𝑋))𝜎(𝑈,𝜎(𝑋,𝑌,𝑉)).(4.20) 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 when 𝜎 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 𝑋𝜎(𝑌,𝑍)=𝑋(𝜎(𝑌,𝑍))𝜎𝑋𝑌,𝑍𝜎𝑌,𝑋𝑍,(5.1)2𝜎(𝑍,𝑊,𝑋,𝑌)=𝑋𝑌𝜎=(𝑍,𝑊)𝑋𝑌𝜎(𝑍,𝑊)𝑌𝜎𝑋𝑍,𝑊𝑋𝜎𝑍,𝑌𝑊𝑋𝑌𝜎(𝑍,𝑊).(5.2) We prove the following theorems.

Theorem 5.1. Let 𝑀 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. Let 𝜎 be recurrent with respect to Semisymmetric Nonmetric connection; from (1.4) we get 𝑋𝜎(𝑌,𝑍)=𝜙(𝑋)𝜎(𝑌,𝑍),(5.3) where 𝜙 is a 1-form on 𝑀; in view of (5.1) and putting 𝑍=𝜉 in the above equation, we have 𝑋𝜎(𝑌,𝜉)𝜎𝑋𝑌,𝜉𝜎𝑌,𝑋𝜉=𝜙(𝑋)𝜎(𝑌,𝜉).(5.4) By virtue of (4.1) in (5.4), we get 𝜎𝑋𝑌,𝜉𝜎𝑌,𝑋𝜉=0.(5.5) Using (1.1), (3.1), (3.6), and (4.1) in (5.5), we get 𝜎(𝑌,𝜙𝑋)𝜎(𝑌,𝑋)=0.(5.6) Replacing 𝑋 by 𝜙𝑋 and by virtue of (3.1) and (4.1) in (5.6), we get 𝜎(𝑌,𝑋)𝜎(𝑌,𝜙𝑋)=0.(5.7) 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. Let 𝑀 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. Let 𝑀 have parallel third fundamental form with respect to Semisymmetric Nonmetric connection. Then we have 𝑋𝑌𝜎(𝑍,𝑊)=0.(5.8) Taking 𝑊=𝜉 and using (5.2) in the above equation, we have 𝑋𝑌𝜎(𝑍,𝜉)𝑌𝜎𝑋𝑍,𝜉𝑋𝜎𝑍,𝑌𝜉𝑋𝑌𝜎(𝑍,𝜉)=0.(5.9) In view of (4.1) and by virtue of (5.1) in (5.9), we get 0=𝑋𝜎𝑌𝑍,𝜉+𝜎𝑍,𝑌𝜉𝑌𝜎𝑋𝑍,𝜉+𝜎𝑌𝑋𝑍,𝜉+2𝜎𝑋𝑍,𝑌𝜉𝑋𝜎𝑍,𝑌𝜉+𝜎𝑍,𝑋𝑌𝜉+𝜎𝑋𝑌𝑍,𝜉+𝜎𝑍,𝑋𝑌𝜉.(5.10) Using (1.1), (3.1), (3.6), and (4.1) in (5.10), we get 0=2𝑋𝜎(𝑍,𝜙𝑌)2𝑋𝜎(𝑍,𝑌)2𝜂(𝑍)𝜎(𝑋,𝜙𝑌)+2𝜎𝑋𝑍,𝑌+2𝜂(𝑍)𝜎(𝑋,𝑌)𝜎𝑍,𝑋𝜙𝑌𝜎𝑍,𝜙𝑋𝑌𝜂(𝑌)𝜎(𝑍,𝜙𝑋)+2𝜎𝑍,𝑋𝑌+2𝜂(𝑌)𝜎(𝑍,𝑋)2𝜎𝑋.𝑍,𝜙𝑌(5.11) Putting 𝑌=𝜉 and using (3.1), (3.6), and (4.1) in (5.11), we get 0=𝜎(𝑍,𝑋)3𝜎(𝑍,𝜙𝑋).(5.12) Replacing 𝑋 by 𝜙𝑋 and by virtue of (3.1) and (4.1) in (5.12), we get 0=𝜎(𝑍,𝜙𝑋)+3𝜎(𝑍,𝑋).(5.13) 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. Let 𝑀 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. Let 𝜎 be 2-recurrent with respect to Semisymmetric Nonmetric connection; from (1.5), we have 𝑋𝑌𝜎(𝑍,𝑊)=𝜎(𝑍,𝑊)𝜙(𝑋,𝑌).(5.14) Taking 𝑊=𝜉 and using (5.2) in the above equation, we have 𝑋𝑌𝜎(𝑍,𝜉)𝑌𝜎𝑋𝑍,𝜉𝑋𝜎𝑍,𝑌𝜉𝑋𝑌𝜎(𝑍,𝜉)=𝜎(𝑍,𝜉)𝜙(𝑋,𝑌).(5.15) In view of (4.1) and by virtue of (5.1) in (5.15), we get 0=𝑋𝜎𝑌𝑍,𝜉+𝜎𝑍,𝑌𝜉𝑌𝜎𝑋𝑍,𝜉+𝜎𝑌𝑋𝑍,𝜉+2𝜎𝑋𝑍,𝑌𝜉𝑋𝜎𝑍,𝑌𝜉+𝜎𝑍,𝑋𝑌𝜉+𝜎𝑋𝑌𝑍,𝜉+𝜎𝑍,𝑋𝑌𝜉.(5.16) Using (1.1), (3.1), (3.6), and (4.1) in (5.16), we get 0=2𝑋𝜎(𝑍,𝜙𝑌)2𝑋𝜎(𝑍,𝑌)2𝜂(𝑍)𝜎(𝑋,𝜙𝑌)+2𝜎𝑋𝑍,𝑌+2𝜂(𝑍)𝜎(𝑋,𝑌)𝜎𝑍,𝑋𝜙𝑌𝜎𝑍,𝜙𝑋𝑌𝜂(𝑌)𝜎(𝑍,𝜙𝑋)+2𝜎𝑍,𝑋𝑌+2𝜂(𝑌)𝜎(𝑍,𝑋)2𝜎𝑋.𝑍,𝜙𝑌(5.17) Putting 𝑌=𝜉 and using (3.1), (3.6), (4.1) in (5.17), we get 0=𝜎(𝑍,𝑋)3𝜎(𝑍,𝜙𝑋).(5.18) Replacing 𝑋 by 𝜙𝑋 and by virtue of (3.1) and (4.1) in (5.18), we get 0=𝜎(𝑍,𝜙𝑋)+3𝜎(𝑍,𝑋).(5.19) 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. Let 𝑀 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. Letting 𝜎 be generalized 2-recurrent with respect to Semisymmetric Nonmetric connection, from (1.7), we have 𝑋𝑌𝜎(𝑍,𝑊)=𝜓(𝑋,𝑌)𝜎(𝑍,𝑊)+𝜙(𝑋)𝑌𝜎(𝑍,𝑊),(5.20) where 𝜓 and 𝜙 are 2-recurrent and 1-form, respectively. Taking 𝑊=𝜉 in (5.20) and using (4.1), we get 𝑋𝑌𝜎(𝑍,𝜉)=𝜙(𝑋)𝑌𝜎(𝑍,𝜉).(5.21) Using (4.1) and (5.2) in above equation, we get 𝑋𝑌𝜎(𝑍,𝜉)𝑌𝜎𝑋𝑍,𝜉𝑋𝜎𝑍,𝑌𝜉𝑋𝑌𝜎𝜎(𝑍,𝜉)=𝜙(𝑋)𝑌𝑍,𝜉+𝜎𝑍,𝑌𝜉.(5.22) In view of (4.1) and by virtue of (5.1) in (5.22), we get 𝑋𝜎𝑌𝑍,𝜉+𝜎𝑍,𝑌𝜉𝑌𝜎𝑋𝑍,𝜉+𝜎𝑌𝑋𝑍,𝜉+2𝜎𝑋𝑍,𝑌𝜉𝑋𝜎𝑍,𝑌𝜉+𝜎𝑍,𝑋𝑌𝜉+𝜎𝑋𝑌𝑍,𝜉+𝜎𝑍,𝑋𝑌𝜉𝜎=𝜙(𝑋)𝑌𝑍,𝜉+𝜎𝑍,𝑌𝜉.(5.23) Using (1.1), (3.1), (3.6), and (4.1) in (5.23), we get 0=2𝑋𝜎(𝑍,𝜙𝑌)2𝑋𝜎(𝑍,𝑌)2𝜂(𝑍)𝜎(𝑋,𝜙𝑌)+2𝜎𝑋𝑍,𝑌+2𝜂(𝑍)𝜎(𝑋,𝑌)𝜎𝑍,𝑋𝜙𝑌𝜎𝑍,𝜙𝑋𝑌𝜂(𝑌)𝜎(𝑍,𝜙𝑋)+2𝜎𝑍,𝑋𝑌+2𝜂(𝑌)𝜎(𝑍,𝑋)2𝜎𝑋𝑍,𝜙𝑌=𝜙(𝑋){𝜎(𝑍,𝜙𝑌)+𝜎(𝑍,𝑌)}.(5.24) Putting 𝑌=𝜉 and using (3.1), (3.6), (4.1) in (5.24), we get 0=𝜎(𝑍,𝑋)3𝜎(𝑍,𝜙𝑋).(5.25) Replacing 𝑋 by 𝜙𝑋 and by virtue of (3.1) and (4.1) in (5.25), we get 0=𝜎(𝑍,𝜙𝑋)+3𝜎(𝑍,𝑋).(5.26) 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 Admitting Semisymmetric Nonmetric Connection

We consider invariant submanifolds of Sasakian manifolds admitting Semisymmetric Nonmetric connection satisfying the conditions 𝑅𝜎=0,𝑅𝜎=𝐿1𝑄(𝑔,𝜎),𝑅𝜎=𝐿2𝑄(𝑆,𝜎).

Theorem 6.1. Let 𝑀 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. Let 𝑀 be semiparallel 𝑅𝜎=0. Putting 𝑋=𝑉=𝜉 and by virtue of (3.1), (3.6), and (4.1) in (4.11), we get 0=𝜎(𝑈,𝑅(𝜉,𝑌)𝜉)𝜎𝜉𝜂(𝑈)𝑌,𝜉+𝜎𝑌𝜂(𝑈)𝜉,𝜉𝜎𝜉𝜎(𝑌,𝑈),𝜉𝜎𝑈,𝜉𝑌+𝜎𝑈,𝑌𝜉[]+𝜎(𝑈,𝜉,𝑌)𝜎𝑈,𝜉𝑌+𝜎𝑈,𝑌𝜉+𝜎(𝑈,𝑌).(6.1) Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.1), we get 0=3𝜎(𝑈,𝑌)𝜎𝜉𝜂(𝑈)𝑌,𝜉𝜎(𝑈,𝜙𝑌)𝜎𝑈,𝜉𝑌.(6.2) 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 𝜎𝜉𝜎(𝑌,𝑈),𝜉=0.(6.3) Then from (6.2), we get 3𝜎(𝑈,𝑌)=𝜙𝜎(𝑈,𝑌)+𝜎𝑈,𝜉𝑌.(6.4) Interchanging 𝑌 and 𝑈 in (6.4), we get 3𝜎(𝑌,𝑈)=𝜙𝜎(𝑌,𝑈)+𝜎𝑈,𝜉𝑌.(6.5) Adding these tow equations, (6.4) and (6.5), we get 6=2𝜙+𝜉.(6.6)

Theorem 6.2. Let 𝑀 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/23.

Proof. Let 𝑀 be pseudoparallel 𝑅𝜎=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 𝜎(𝑈,𝑅(𝜉,𝑌)𝜉)𝜎𝜉𝜂(𝑈)𝑌,𝜉+𝜎𝑌𝜂(𝑈)𝜉,𝜉𝜎𝜉𝜎(𝑌,𝑈),𝜉𝜎𝑈,𝜉𝑌+𝜎𝑈,𝑌𝜉[]+𝜎(𝑈,𝜉,𝑌)𝜎𝑈,𝜉𝑌+𝜎𝑈,𝑌𝜉+𝜎(𝑈,𝑌)=𝐿1𝜎(𝑈,𝑌).(6.7) Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.7), we get 3𝜎(𝑈,𝑌)𝜎𝜉𝜂(𝑈)𝑌,𝜉𝜎(𝑈,𝜙𝑌)𝜎𝑈,𝜉𝑌=𝐿1𝜎(𝑈,𝑌).(6.8) Now by using (6.3) in (6.8), we get 3+𝐿1𝜎(𝑈,𝑌)=𝜙𝜎(𝑈,𝑌)+𝜎𝑈,𝜉𝑌.(6.9) Interchanging 𝑌 and 𝑈 in (6.9), we get 3+𝐿1𝜎(𝑌,𝑈)=𝜙𝜎(𝑌,𝑈)+𝜎𝑌,𝜉𝑈.(6.10) Adding (6.9) and (6.10), we get 𝐿1=𝜉𝜙+23.(6.11)

Theorem 6.3. Let 𝑀 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/(𝑛1))[𝜙+𝜉/23].

Proof. Let 𝑀 be Ricci-generalized pseudoparallel 𝑅𝜎=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 𝜎(𝑈,𝑅(𝜉,𝑌)𝜉)𝜎𝜉𝜂(𝑈)𝑌,𝜉+𝜎𝑌𝜂(𝑈)𝜉,𝜉𝜎𝜉𝜎(𝑌,𝑈),𝜉𝜎𝑈,𝜉𝑌+𝜎𝑈,𝑌𝜉[]+𝜎(𝑈,𝜉,𝑌)𝜎𝑈,𝜉𝑌+𝜎𝑈,𝑌𝜉+𝜎(𝑈,𝑌)=𝐿2(𝑛1)𝜎(𝑈,𝑌).(6.12) Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.12), we get 3𝜎(𝑈,𝑌)𝜎𝜉𝜂(𝑈)𝑌,𝜉𝜎(𝑈,𝜙𝑌)𝜎𝑈,𝜉𝑌=𝐿2(𝑛1)𝜎(𝑈,𝑌).(6.13) Now by using (6.3) in (6.13), we get 3+𝐿2(𝑛1)𝜎(𝑈,𝑌)=𝜙𝜎(𝑈,𝑌)+𝜎𝑈,𝜉𝑌.(6.14) Interchanging 𝑌 and 𝑈 in (6.14), we get 3+𝐿2(𝑛1)𝜎(𝑌,𝑈)=𝜙𝜎(𝑌,𝑈)+𝜎𝑌,𝜉𝑈.(6.15) Adding (6.14) and (6.15), we get 23+𝐿2(𝑛1)𝜎(𝑈,𝑌)=2𝜙𝜎(𝑈,𝑌)+𝜉𝜎(𝑈,𝑌).(6.16) Writting the above equation, we have 𝐿2=1(𝑛1)𝜉𝜙+23.(6.17)

Remark 6.4. Let 𝑀 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 Theorems 5.1 to 5.4 and corollary 5.3, we have the following result.

Corollary 6.5. Let 𝑀 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.