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 βˆ‡ on a manifold 𝑀 is called a metric connection if there is a Riemannian metric 𝑔 on 𝑀 if βˆ‡π‘”=0; otherwise it is Nonmetric. Further it is said to be Semisymmetric if its torsion tensor 𝑇(𝑋,π‘Œ)=0;thatis,𝑇(𝑋,π‘Œ)=𝑀(π‘Œ)π‘‹βˆ’π‘€(𝑋)π‘Œ, where 𝑀 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 βˆ‡ denotes Semisymmetric Nonmetric connection on a contact metric manifold, then it is given by [6] βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+πœ‚(π‘Œ)𝑋,(1.1) where πœ‚(π‘Œ)=𝑔(π‘Œ,πœ‰).

The covariant differential of the 𝑝th order, 𝑝β‰₯1 of a (0,π‘˜)-tensor field 𝑇,π‘˜β‰₯1 denoted by βˆ‡π‘π‘‡, defined on a Riemannian manifold (𝑀,𝑔) with the Levi-Civita connection βˆ‡. The tensor 𝑇 is said to be recurrent [7], if the following condition holds on 𝑀: 𝑋(βˆ‡π‘‡)1,…,π‘‹π‘˜ξ€Έπ‘‡ξ€·π‘Œ;𝑋1,…,π‘Œπ‘˜ξ€Έ=ξ€·π‘Œ(βˆ‡π‘‡)1,…,π‘Œπ‘˜ξ€Έπ‘‡ξ€·π‘‹;𝑋1,…,π‘‹π‘˜ξ€Έ,(1.2) respectively.

Consider ξ€·βˆ‡2𝑇𝑋1,…,π‘‹π‘˜ξ€Έπ‘‡ξ€·π‘Œ;𝑋,π‘Œ1,…,π‘Œπ‘˜ξ€Έ=ξ€·βˆ‡2π‘‡π‘Œξ€Έξ€·1,…,π‘Œπ‘˜ξ€Έπ‘‡ξ€·π‘‹;𝑋,π‘Œ1,…,π‘‹π‘˜ξ€Έ,(1.3) where 𝑋,π‘Œ,𝑋1,π‘Œ1,…,π‘‹π‘˜,π‘Œπ‘˜βˆˆπ‘‡π‘€. From (1.2) it follows that at a point π‘₯βˆˆπ‘€, if the tensor 𝑇 is nonzero, then there exists a unique 1-form πœ™, respectively, a (0,2)-tensor πœ“, defined on a neighborhood π‘ˆ of π‘₯ such that βˆ‡π‘‡=π‘‡βŠ—πœ™,πœ™=𝑑(log‖𝑇‖),(1.4) respectively.

The following βˆ‡2𝑇=π‘‡βŠ—πœ“(1.5) holds on π‘ˆ, where ‖𝑇‖ denotes the norm of 𝑇 and ‖𝑇‖2=𝑔(𝑇,𝑇). The tensor 𝑇 is said to be generalized 2-recurrent if βˆ‡ξ€·ξ€·2𝑇𝑋1,…,π‘‹π‘˜ξ€Έξ€·π‘‹;𝑋,π‘Œβˆ’(βˆ‡π‘‡βŠ—πœ™)1,…,π‘‹π‘˜π‘‡ξ€·π‘Œ;𝑋,π‘Œξ€Έξ€Έ1,…,π‘Œπ‘˜ξ€Έ=βˆ‡ξ€·ξ€·2π‘‡π‘Œξ€Έξ€·1,…,π‘Œπ‘˜ξ€Έβˆ’ξ€·π‘Œ;𝑋,π‘Œ(βˆ‡π‘‡βŠ—πœ™)1,…,π‘Œπ‘˜π‘‡ξ€·π‘‹;𝑋,π‘Œξ€Έξ€Έ1,…,π‘‹π‘˜ξ€Έ(1.6) holds on 𝑀, where πœ™ is a 1-form on 𝑀. From this it follows that at a point π‘₯βˆˆπ‘€ if the tensor 𝑇 is nonzero, then there exists a unique (0,2)-tensor πœ“, defined on a neighborhood π‘ˆ of π‘₯, such that βˆ‡2𝑇=βˆ‡π‘‡βŠ—πœ™+π‘‡βŠ—πœ“(1.7) holds on π‘ˆ.

2. Isometric Immersion

Let ξ‚‹π‘“βˆΆ(𝑀,𝑔)β†’(𝑀,̃𝑔) be an isometric immersion from an 𝑛-dimensional Riemannian manifold (𝑀,𝑔) into (𝑛+𝑑)-dimensional Riemannian manifold (𝑀,̃𝑔), 𝑛β‰₯2, 𝑑β‰₯1. We denote βˆ‡ and ξ‚βˆ‡ as Levi-Civita connection of 𝑀𝑛 and 𝑀𝑛+𝑑, respectively. Then the formulas of Gauss and Weingarten are given by ξ‚βˆ‡π‘‹π‘Œ=βˆ‡π‘‹ξ‚βˆ‡π‘Œ+𝜎(𝑋,π‘Œ),(2.1)𝑋𝑁=βˆ’π΄π‘π‘‹+βˆ‡βŸ‚π‘‹π‘,(2.2) for any tangent vector fields 𝑋,π‘Œ and the normal vector field 𝑁 on 𝑀, where 𝜎, 𝐴, and βˆ‡βŸ‚ 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,π‘Œ),𝑁)=𝑔𝑁𝑋,π‘Œ,(2.3) for tangent vector fields 𝑋,π‘Œ. The first and second covariant derivatives of the second fundamental form 𝜎 are given by ξ‚€ξ‚βˆ‡π‘‹πœŽξ‚(π‘Œ,𝑍)=βˆ‡βŸ‚π‘‹ξ€·βˆ‡(𝜎(π‘Œ,𝑍))βˆ’πœŽπ‘‹ξ€Έξ€·π‘Œ,π‘βˆ’πœŽπ‘Œ,βˆ‡π‘‹π‘ξ€Έξ‚€ξ‚βˆ‡,(2.4)2πœŽξ‚ξ‚€ξ‚βˆ‡(𝑍,π‘Š,𝑋,π‘Œ)=π‘‹ξ‚βˆ‡π‘ŒπœŽξ‚(𝑍,π‘Š)=βˆ‡βŸ‚π‘‹ξ‚βˆ‡ξ‚€ξ‚€π‘ŒπœŽξ‚ξ‚βˆ’ξ‚€ξ‚βˆ‡(𝑍,π‘Š)π‘ŒπœŽξ‚ξ€·βˆ‡π‘‹ξ€Έβˆ’ξ‚€ξ‚βˆ‡π‘,π‘Šπ‘‹πœŽξ‚ξ€·π‘,βˆ‡π‘Œπ‘Šξ€Έβˆ’ξ‚€ξ‚βˆ‡βˆ‡π‘‹π‘ŒπœŽξ‚(𝑍,π‘Š),(2.5) respectively, where ξ‚βˆ‡ is called the van der Waerden-Bortolotti connection of 𝑀 [8]. If ξ‚βˆ‡πœŽ=0, then 𝑀 is said to have parallel second fundamental form [8]. We next define endomorphisms 𝑅(𝑋,π‘Œ) and π‘‹βˆ§π΅π‘Œ of πœ’(𝑀) by 𝑅(𝑋,π‘Œ)𝑍=βˆ‡π‘‹βˆ‡π‘Œπ‘βˆ’βˆ‡π‘Œβˆ‡π‘‹π‘βˆ’βˆ‡[𝑋,π‘Œ]𝑍,π‘‹βˆ§π΅π‘Œξ€Έπ‘=𝐡(π‘Œ,𝑍)π‘‹βˆ’π΅(𝑋,𝑍)π‘Œ,(2.6) respectively, where 𝑋,π‘Œ,π‘βˆˆπœ’(𝑀) and 𝐡 is a symmetric (0,2)-tensor.

Now, for a (0,π‘˜)-tensor field 𝑇,π‘˜β‰₯1 and a (0,2)-tensor field 𝐡 on (𝑀,𝑔), we define the tensor 𝑄(𝐡,𝑇) by 𝑄𝑋(𝐡,𝑇)1,…,π‘‹π‘˜ξ€Έξ€·π‘‡ξ€·;𝑋,π‘Œ=βˆ’π‘‹βˆ§π΅π‘Œξ€Έπ‘‹1,…,π‘‹π‘˜ξ€Έξ€·π‘‹βˆ’β‹―βˆ’π‘‡1,…,π‘‹π‘˜βˆ’1ξ€·π‘‹βˆ§π΅π‘Œξ€Έπ‘‹π‘˜ξ€Έ.(2.7) Putting into consideration the previous formula β€œπ΅=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 (πœ™,πœ‰,πœ‚) if it carries a tensor field πœ™ of type (1,1), a vector field πœ‰, and 1-form πœ‚ on 𝑀, respectively, such that πœ™2=βˆ’πΌ+πœ‚βŠ—πœ‰,πœ‚(πœ‰)=1,πœ‚βˆ˜πœ™=0,πœ™πœ‰=0.(3.1)

Thus a manifold 𝑀 equipped with this structure is called an almost contact manifold and is denoted by (𝑀,πœ™,πœ‰,πœ‚). If 𝑔 is a Riemannian metric on an almost contact manifold 𝑀 such that 𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœ‚(𝑋)πœ‚(π‘Œ),𝑔(𝑋,πœ‰)=πœ‚(𝑋),(3.2) where 𝑋,π‘Œ are vector fields defined on 𝑀, then 𝑀 is said to have an almost contact metric structure (πœ™,πœ‰,πœ‚,𝑔), and 𝑀 with this structure is called an almost contact metric manifold and is denoted by (𝑀,πœ™,πœ‰,πœ‚,𝑔).

If on (𝑀,πœ™,πœ‰,πœ‚,𝑔) the exterior derivative of 1-form πœ‚ satisfies Ξ¦(𝑋,π‘Œ)=π‘‘πœ‚(𝑋,π‘Œ)=𝑔(𝑋,πœ™π‘Œ),(3.3) then (πœ™,πœ‰,πœ‚,𝑔) is said to be a contact metric structure and together with manifold 𝑀 is called contact metric manifold and Ξ¦ is a 2-form. The contact metric structure (𝑀,πœ™,πœ‰,πœ‚,𝑔) is said to be normal if []πœ™,πœ™(𝑋,π‘Œ)+2π‘‘πœ‚βŠ—πœ‰=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 ξ€·βˆ‡π‘‹πœ™ξ€Έβˆ‡π‘Œ=𝑔(𝑋,π‘Œ)πœ‰βˆ’πœ‚(π‘Œ)𝑋,(3.5)π‘‹πœ‰=βˆ’πœ™π‘‹.(3.6)

Example of Sasakian Manifold
Consider the 3-dimensional manifold 𝑀={(π‘₯,𝑦,𝑧)βˆˆπ‘…3}, where (π‘₯,𝑦,𝑧) are the standard coordinates in 𝑅3. Let {𝐸1,𝐸2,𝐸3} be linearly independent global frame field on 𝑀 given by 𝐸1=πœ•πœ•πœ•π‘₯βˆ’2π‘¦πœ•π‘§,𝐸2=πœ•πœ•π‘¦,𝐸3=πœ•.πœ•π‘§(3.7) Let 𝑔 be the Riemannian metric defined by 𝑔𝐸1,𝐸2𝐸=𝑔1,𝐸3𝐸=𝑔2,𝐸3𝑔𝐸=0,1,𝐸1𝐸=𝑔2,𝐸2𝐸=𝑔3,𝐸3ξ€Έ=1.(3.8) The (πœ™,πœ‰,πœ‚) is given by πœ‚=2𝑦𝑑π‘₯+𝑑𝑧,πœ‰=𝐸3=πœ•,πœ•π‘§πœ™πΈ1=𝐸2,πœ™πΈ2=βˆ’πΈ1,πœ™πΈ3=0.(3.9) The linearity property of πœ™ and 𝑔 yields πœ‚ξ€·πΈ3ξ€Έ=1,πœ™2π‘ˆ=βˆ’π‘ˆ+πœ‚(π‘ˆ)𝐸3,𝑔(πœ™π‘ˆ,πœ™π‘Š)=𝑔(π‘ˆ,π‘Š)βˆ’πœ‚(π‘ˆ)πœ‚(π‘Š),𝑔(π‘ˆ,πœ‰)=πœ‚(π‘ˆ),(3.10) for any vector fields π‘ˆ,π‘Š on 𝑀. By definition of Lie bracket, we have 𝐸1,𝐸2ξ€»=2𝐸3.(3.11) Let βˆ‡ be the Levi-Civita connection with respect to previously mentioned metric 𝑔 and be given by Koszula formula ξ€·βˆ‡2𝑔𝑋[][][]π‘Œ,𝑍=𝑋(𝑔(π‘Œ,𝑍))+π‘Œ(𝑔(𝑍,𝑋))βˆ’π‘(𝑔(𝑋,π‘Œ))βˆ’π‘”(𝑋,π‘Œ,𝑍)βˆ’π‘”(π‘Œ,𝑋,𝑍)+𝑔(𝑍,𝑋,π‘Œ).(3.12) Then, we have βˆ‡πΈ1𝐸1=0,βˆ‡πΈ1𝐸2=𝐸3,βˆ‡πΈ1𝐸3=βˆ’πΈ2,βˆ‡πΈ2𝐸1=βˆ’πΈ3,βˆ‡πΈ2𝐸2=0,βˆ‡πΈ2𝐸3=𝐸1,βˆ‡πΈ3𝐸1=βˆ’πΈ2,βˆ‡πΈ3𝐸2=𝐸1,βˆ‡πΈ3𝐸3=0.(3.13) The tangent vectors 𝑋 and π‘Œ to 𝑀 are expressed as linear combination of 𝐸1,𝐸2,𝐸3; that is, 𝑋=π‘Ž1𝐸1+π‘Ž2𝐸2+π‘Ž3𝐸3 and π‘Œ=𝑏1𝐸1+𝑏2𝐸2+𝑏3𝐸3, where π‘Žπ‘– and 𝑏𝑗 are scalars. Clearly (πœ™,πœ‰,πœ‚,𝑔) and 𝑋,π‘Œ satisfy (3.1), (3.2), (3.5), and (3.6). Thus 𝑀 is a Sasakian manifold. Further the following relations hold: 𝑅𝑅(𝑋,π‘Œ)𝑍={𝑔(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œ},(𝑋,π‘Œ)πœ‰={πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ},𝑅(πœ‰,𝑋)π‘Œ={𝑔(𝑋,π‘Œ)πœ‰βˆ’πœ‚(π‘Œ)𝑋},(3.14)𝑅(πœ‰,𝑋)πœ‰={πœ‚(𝑋)πœ‰βˆ’π‘‹},(3.15)𝑆(𝑋,πœ‰)=(π‘›βˆ’1)πœ‚(𝑋),(3.16)π‘„πœ‰=(π‘›βˆ’1)πœ‰,(3.17) for all vector fields, 𝑋,π‘Œ,𝑍 and where βˆ‡ denotes the operator of covariant differentiation with respect to 𝑔,πœ™ 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 (ξ‚Μƒπœ™,πœ‰,Μƒπœ‚,̃𝑔), then we know that its invariant submanifold 𝑀 has the induced Sasakian structure (πœ™,πœ‰,πœ‚,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 π‘₯βˆˆπ‘€, πœ™(𝑇π‘₯𝑀)βŠ‚π‘‡π‘₯𝑀. As a consequence, πœ‰ becomes tangent to 𝑀. For an invariant submanifold of a Sasakian manifold with a Semisymmetric Nonmetric connection we have 𝜎(𝑋,πœ‰)=0,(4.1) for any vector 𝑋 tangent to 𝑀.

Let 𝑀 be a Sasakian manifold admitting a Semisymmetric Nonmetric connection ξ‚βˆ‡.

Lemma 4.1. Let 𝑀 be an invariant submanifold of contact metric manifold 𝑀 which admits Semisymmetric Nonmetric connection ξ‚βˆ‡, 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 ξ‚βˆ‡ and ξ‚βˆ‡ are equal.

Proof. We know that the contact metric structure (ξ‚Μƒπœ™,πœ‰,Μƒπœ‚,̃𝑔) on 𝑀 induces (πœ™,πœ‰,πœ‚,𝑔) on invariant submanifold. By virtue of (1.1), we get ξ‚βˆ‡π‘‹ξ‚βˆ‡π‘Œ=π‘‹π‘Œ+πœ‚(π‘Œ)𝑋.(4.2) By using (2.1) in (4.2), we get ξ‚βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+𝜎(𝑋,π‘Œ)+πœ‚(π‘Œ)𝑋.(4.3) Now Gauss formula (2.1) with respect to Semisymmetric Nonmetric connection is given by ξ‚βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+𝜎(𝑋,π‘Œ).(4.4) Equating (4.3) and (4.4), we get (1.1) and 𝜎(𝑋,π‘Œ)=𝜎(𝑋,π‘Œ).(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,π‘Œ tangent to 𝑀: ξ‚π‘…β‹…πœŽ=0,ξ‚π‘…β‹…πœŽ=𝐿1𝑄(𝑔,𝜎),ξ‚π‘…β‹…πœŽ=𝐿2𝑄(𝑆,𝜎),(4.6) where 𝑅 denotes the curvature tensor with respect to connection ξ‚βˆ‡. 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 ξ‚΅tanξ‚ξ‚Άξ€·βˆ‡π‘…(𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)𝑍+πœ‚π‘Œπ‘ξ€Έπ‘‹+πœ‚(𝑍)βˆ‡π‘‹ξ€·βˆ‡π‘Œ+πœ‚(𝑍)πœ‚(π‘Œ)π‘‹βˆ’πœ‚π‘‹π‘ξ€Έπ‘Œβˆ’πœ‚(𝑍)βˆ‡π‘Œ[]ξ‚»π‘‹βˆ’πœ‚(𝑍)πœ‚(𝑋)π‘Œβˆ’πœ‚(𝑍)𝑋,π‘Œ+tanξ‚βˆ‡π‘‹{𝜎(π‘Œ,𝑍)}βˆ’ξ‚βˆ‡π‘Œ{𝜎(𝑋,𝑍)}βˆ’ξ‚βˆ‡π‘Œπœ‚(𝑍)𝑋+ξ‚βˆ‡π‘‹ξ‚Ό,ξ‚΅πœ‚(𝑍)π‘Œ(4.7)nor𝑅(𝑋,π‘Œ)𝑍=πœŽπ‘‹,βˆ‡π‘Œπ‘ξ€Έξ€·+πœ‚(𝑍)𝜎(𝑋,π‘Œ)βˆ’πœŽπ‘Œ,βˆ‡π‘‹π‘ξ€Έ[]ξ‚»βˆ’πœ‚(𝑍)𝜎(π‘Œ,𝑋)βˆ’πœŽ(𝑋,π‘Œ,𝑍)+norξ‚βˆ‡π‘‹{𝜎(π‘Œ,𝑍)}βˆ’ξ‚βˆ‡Y{𝜎(𝑋,𝑍)}βˆ’ξ‚βˆ‡π‘Œπœ‚(𝑍)𝑋+ξ‚βˆ‡π‘‹ξ‚Ό.πœ‚(𝑍)π‘Œ(4.8)

Proof. The Riemannian curvature tensor 𝑅 on 𝑀 with respect to Semisymmetric Nonmetric connection is given by 𝑅(𝑋,π‘Œ)𝑍=ξ‚βˆ‡π‘‹ξ‚βˆ‡π‘Œπ‘βˆ’ξ‚βˆ‡π‘Œξ‚βˆ‡π‘‹π‘βˆ’ξ‚βˆ‡[𝑋,π‘Œ]𝑍.(4.9) Using (1.1) and (2.1) in (4.9), we get 𝑅(𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)𝑍+πœŽπ‘‹,βˆ‡π‘Œπ‘ξ€Έξ€·βˆ‡+πœ‚π‘Œπ‘ξ€Έπ‘‹+ξ‚βˆ‡π‘‹{𝜎(π‘Œ,𝑍)}+ξ‚βˆ‡π‘‹πœ‚(𝑍)π‘Œ+πœ‚(𝑍)βˆ‡π‘‹ξ€·π‘Œ+πœ‚(𝑍)𝜎(𝑋,π‘Œ)+πœ‚(𝑍)πœ‚(π‘Œ)π‘‹βˆ’πœŽπ‘Œ,βˆ‡π‘‹π‘ξ€Έξ€·βˆ‡βˆ’πœ‚π‘‹π‘ξ€Έπ‘Œβˆ’ξ‚βˆ‡π‘Œ{𝜎(𝑋,𝑍)}βˆ’ξ‚βˆ‡π‘Œπœ‚(𝑍)π‘‹βˆ’πœ‚(𝑍)βˆ‡π‘Œπ‘‹[][].βˆ’πœ‚(𝑍)𝜎(π‘Œ,𝑋)βˆ’πœ‚(𝑍)πœ‚(𝑋)π‘Œβˆ’πœŽ(𝑋,π‘Œ,𝑍)βˆ’πœ‚(𝑍)𝑋,π‘Œ(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,π‘Œ)β‹…πœŽ(π‘ˆ,𝑉)=π‘…βŸ‚(𝑋,π‘Œ)𝜎(π‘ˆ,𝑉)βˆ’πœŽ(𝑅(𝑋,π‘Œ)π‘ˆ,𝑉)βˆ’πœŽ(π‘ˆ,𝑅(𝑋,π‘Œ)𝑉)βˆ’βˆ‡π‘‹π΄πœŽ(π‘ˆ,𝑉)π‘Œ+βˆ‡π‘Œπ΄πœŽ(π‘ˆ,𝑉)π‘‹βˆ’π΄βˆ‡βŸ‚π‘ŒπœŽ(π‘ˆ,𝑉)𝑋+π΄βˆ‡βŸ‚π‘‹πœŽ(π‘ˆ,𝑉)π‘Œ+𝐴𝜎(π‘ˆ,𝑉)[]𝑋,π‘Œβˆ’πœŽπ‘‹,𝐴𝜎(π‘ˆ,𝑉)π‘Œξ€Έξ€·+πœŽπ‘Œ,𝐴𝜎(π‘ˆ,𝑉)π‘‹ξ€Έξ€·π΄βˆ’πœ‚πœŽ(π‘ˆ,𝑉)π‘Œξ€Έξ€·π΄π‘‹+πœ‚πœŽ(π‘ˆ,𝑉)π‘‹ξ€Έξ€·βˆ‡π‘Œβˆ’πœ‚π‘Œπ‘ˆξ€Έξ€·βˆ‡πœŽ(𝑋,𝑉)βˆ’πœ‚(π‘ˆ)πœŽπ‘‹ξ€Έξ€·βˆ‡π‘Œ,π‘‰βˆ’πœ‚(π‘ˆ)πœ‚(π‘Œ)𝜎(𝑋,𝑉)+πœ‚π‘‹π‘ˆξ€Έξ€·βˆ‡πœŽ(π‘Œ,𝑉)+πœ‚(π‘ˆ)πœŽπ‘Œξ€Έ[])𝑋,𝑉+πœ‚(π‘ˆ)πœ‚(𝑋)𝜎(π‘Œ,𝑉)+πœ‚(π‘ˆ)𝜎(𝑋,π‘Œ,π‘‰βˆ’πœŽξ‚βˆ‡π‘‹ξ‚Άξ‚΅πœ‚(π‘ˆ)π‘Œ,𝑉+πœŽξ‚βˆ‡π‘Œξ‚Άξ‚΅πœ‚(π‘ˆ)𝑋,π‘‰βˆ’πœŽξ‚βˆ‡π‘‹ξ‚Άξ‚΅{𝜎(π‘Œ,π‘ˆ)},𝑉+πœŽξ‚βˆ‡π‘Œξ‚Άξ€·πœŽξ€·{𝜎(𝑋,π‘ˆ)},π‘‰βˆ’πœŽπ‘‹,βˆ‡π‘Œπ‘ˆξ€Έξ€Έξ€·πœŽξ€·,π‘‰βˆ’πœ‚(π‘ˆ)𝜎(𝜎(𝑋,π‘Œ),𝑉)+πœŽπ‘Œ,βˆ‡π‘‹π‘ˆξ€Έξ€Έ[]ξ€·βˆ‡,𝑉+πœ‚(π‘ˆ)𝜎(𝜎(π‘Œ,𝑋),𝑉)+𝜎(𝜎(𝑋,π‘Œ,π‘ˆ),𝑉)βˆ’πœ‚π‘Œπ‘‰ξ€Έξ€·πœŽ(π‘ˆ,𝑋)βˆ’πœ‚(𝑉)πœŽπ‘ˆ,βˆ‡π‘‹π‘Œξ€Έξ€·βˆ‡βˆ’πœ‚(𝑉)πœ‚(π‘Œ)𝜎(π‘ˆ,𝑋)+πœ‚π‘‹π‘‰ξ€Έξ€·πœŽ(π‘ˆ,π‘Œ)+πœ‚(𝑉)πœŽπ‘ˆ,βˆ‡π‘Œπ‘‹ξ€Έ[]ξ‚΅+πœ‚(𝑉)πœ‚(𝑋)𝜎(π‘ˆ,π‘Œ)+πœ‚(𝑉)𝜎(π‘ˆ,𝑋,π‘Œ)βˆ’πœŽπ‘ˆ,ξ‚βˆ‡π‘‹ξ‚Άξ‚΅πœ‚(𝑉)π‘Œ+πœŽπ‘ˆ,ξ‚βˆ‡π‘Œξ‚Άξ‚΅πœ‚(𝑉)π‘‹βˆ’πœŽπ‘ˆ,ξ‚βˆ‡π‘‹ξ‚Άξ‚΅{𝜎(π‘Œ,𝑉)}+πœŽπ‘ˆ,ξ‚βˆ‡π‘Œξ‚Άξ€·ξ€·{𝜎(𝑋,𝑉)}βˆ’πœŽπ‘ˆ,πœŽπ‘‹,βˆ‡π‘Œπ‘‰ξ€·ξ€·ξ€Έξ€Έβˆ’πœ‚(𝑉)𝜎(π‘ˆ,𝜎(𝑋,π‘Œ))+πœŽπ‘ˆ,πœŽπ‘Œ,βˆ‡π‘‹π‘‰[]ξ€Έξ€Έ+πœ‚(𝑉)𝜎(π‘ˆ,𝜎(π‘Œ,𝑋))+𝜎(π‘ˆ,𝜎(𝑋,π‘Œ,𝑉)),(4.11) for all vector fields 𝑋,π‘Œ,π‘ˆ, and 𝑉 tangent to 𝑀, where π‘…βŸ‚ξ€Ίβˆ‡(𝑋,π‘Œ)=βŸ‚π‘‹,βˆ‡βŸ‚π‘Œξ€»βˆ’βˆ‡βŸ‚[𝑋,π‘Œ].(4.12)

Proof. We know, from tensor algebra, that 𝑅(𝑋,π‘Œ)β‹…πœŽ(π‘ˆ,𝑉)=𝑅(𝑋,π‘Œ)𝜎(π‘ˆ,𝑉)βˆ’πœŽξ‚ξ‚Άξ‚΅π‘…(𝑋,π‘Œ)π‘ˆ,π‘‰βˆ’πœŽπ‘ˆ,𝑅(𝑋,π‘Œ)𝑉.(4.13) Replacing 𝑍 by 𝜎(π‘ˆ,𝑉) in (4.9), we get 𝑅(𝑋,π‘Œ)𝜎(π‘ˆ,𝑉)=ξ‚βˆ‡π‘‹ξ‚βˆ‡π‘ŒπœŽ(π‘ˆ,𝑉)βˆ’ξ‚βˆ‡π‘Œξ‚βˆ‡π‘‹πœŽ(π‘ˆ,𝑉)βˆ’ξ‚βˆ‡[𝑋,π‘Œ]𝜎(π‘ˆ,𝑉).(4.14) In view of (1.1), (2.1), and (2.2), we have the following equalities: ξ‚βˆ‡π‘‹ξ‚βˆ‡π‘ŒπœŽ(π‘ˆ,𝑉)=ξ‚βˆ‡π‘‹ξ€·βˆ’π΄πœŽ(π‘ˆ,𝑉)π‘Œ+βˆ‡βŸ‚π‘Œξ€Έ,𝜎(π‘ˆ,𝑉)=βˆ’βˆ‡π‘‹π΄πœŽ(π‘ˆ,𝑉)ξ€·π΄π‘Œβˆ’πœ‚πœŽ(π‘ˆ,𝑉)π‘Œξ€Έξ€·π‘‹βˆ’πœŽπ‘‹,𝐴𝜎(π‘ˆ,𝑉)π‘Œξ€Έβˆ’π΄βˆ‡βŸ‚π‘ŒπœŽ(π‘ˆ,𝑉)𝑋+βˆ‡βŸ‚π‘‹βˆ‡βŸ‚π‘ŒπœŽ(π‘ˆ,𝑉).(4.15) Similarly ξ‚βˆ‡π‘Œξ‚βˆ‡π‘‹πœŽ(π‘ˆ,𝑉)=βˆ’βˆ‡π‘Œπ΄πœŽ(π‘ˆ,𝑉)ξ€·π΄π‘‹βˆ’πœ‚πœŽ(π‘ˆ,𝑉)π‘‹ξ€Έξ€·π‘Œβˆ’πœŽπ‘Œ,𝐴𝜎(π‘ˆ,𝑉)π‘‹ξ€Έβˆ’π΄βˆ‡βŸ‚π‘‹πœŽ(π‘ˆ,𝑉)π‘Œ+βˆ‡βŸ‚π‘Œβˆ‡βŸ‚π‘‹πœŽ(π‘ˆ,𝑉),(4.16)ξ‚βˆ‡[𝑋,π‘Œ]𝜎(π‘ˆ,𝑉)=βˆ’π΄πœŽ(π‘ˆ,𝑉)[]𝑋,π‘Œ+βˆ‡βŸ‚[]𝑋,π‘ŒπœŽ(π‘ˆ,𝑉).(4.17) Substituting (4.15), (4.16) and (4.17) into (4.14), we get 𝑅(𝑋,π‘Œ)𝜎(π‘ˆ,𝑉)=π‘…βŸ‚(𝑋,π‘Œ)𝜎(π‘ˆ,𝑉)βˆ’βˆ‡π‘‹π΄πœŽ(π‘ˆ,𝑉)π‘Œ+βˆ‡π‘Œπ΄πœŽ(π‘ˆ,𝑉)π‘‹βˆ’π΄βˆ‡βŸ‚π‘ŒπœŽ(π‘ˆ,𝑉)𝑋+π΄βˆ‡βŸ‚π‘‹πœŽ(π‘ˆ,𝑉)π‘Œ+𝐴𝜎(π‘ˆ,𝑉)[]𝑋,π‘Œβˆ’πœŽπ‘‹,𝐴𝜎(π‘ˆ,𝑉)π‘Œξ€Έξ€·+πœŽπ‘Œ,𝐴𝜎(π‘ˆ,𝑉)π‘‹ξ€Έξ€·π΄βˆ’πœ‚πœŽ(π‘ˆ,𝑉)π‘Œξ€Έξ€·π΄π‘‹+πœ‚πœŽ(π‘ˆ,𝑉)π‘‹ξ€Έπ‘Œ.(4.18) By virtue of (4.10) in 𝜎(𝑅(𝑋,π‘Œ)π‘ˆ,𝑉) and 𝜎(π‘ˆ,𝑅(𝑋,π‘Œ)𝑉), we get πœŽξ‚΅ξ‚ξ‚Άξ€·βˆ‡π‘…(𝑋,π‘Œ)π‘ˆ,𝑉=𝜎(𝑅(𝑋,π‘Œ)π‘ˆ,𝑉)+πœ‚π‘Œπ‘ˆξ€Έξ€·βˆ‡πœŽ(𝑋,𝑉)+πœ‚(π‘ˆ)πœŽπ‘‹ξ€Έξ€·βˆ‡π‘Œ,𝑉+πœ‚(π‘ˆ)πœ‚(π‘Œ)𝜎(𝑋,𝑉)βˆ’πœ‚π‘‹π‘ˆξ€Έξ€·βˆ‡πœŽ(π‘Œ,𝑉)βˆ’πœ‚(π‘ˆ)πœŽπ‘Œξ€Έ[]𝑋,π‘‰βˆ’πœ‚(π‘ˆ)πœ‚(𝑋)𝜎(π‘Œ,𝑉)βˆ’πœ‚(π‘ˆ)𝜎(𝑋,π‘Œ,𝑉)+πœŽξ‚βˆ‡π‘‹ξ‚Άξ‚΅πœ‚(π‘ˆ)π‘Œ,π‘‰βˆ’πœŽξ‚βˆ‡π‘Œξ‚Άξ‚΅πœ‚(π‘ˆ)𝑋,𝑉+πœŽξ‚βˆ‡π‘‹ξ‚Άξ‚΅{𝜎(π‘Œ,π‘ˆ)},π‘‰βˆ’πœŽξ‚βˆ‡π‘Œξ‚Άξ€·πœŽξ€·{𝜎(𝑋,π‘ˆ)},𝑉+πœŽπ‘‹,βˆ‡π‘Œπ‘ˆξ€Έξ€Έξ€·πœŽξ€·,𝑉+πœ‚(π‘ˆ)𝜎(𝜎(𝑋,π‘Œ),𝑉)βˆ’πœŽπ‘Œ,βˆ‡π‘‹π‘ˆξ€Έξ€Έ[]πœŽξ‚΅,π‘‰βˆ’πœ‚(π‘ˆ)𝜎(𝜎(π‘Œ,𝑋),𝑉)βˆ’πœŽ(𝜎(𝑋,π‘Œ,π‘ˆ),𝑉),(4.19)π‘ˆ,ξ‚ξ‚Άξ€·βˆ‡π‘…(𝑋,π‘Œ)𝑉=𝜎(π‘ˆ,𝑅(𝑋,π‘Œ)𝑉)+πœ‚π‘Œπ‘‰ξ€Έξ€·πœŽ(π‘ˆ,𝑋)+πœ‚(𝑉)πœŽπ‘ˆ,βˆ‡π‘‹π‘Œξ€Έξ€·βˆ‡+πœ‚(𝑉)πœ‚(π‘Œ)𝜎(π‘ˆ,𝑋)βˆ’πœ‚π‘‹π‘‰ξ€Έξ€·πœŽ(π‘ˆ,π‘Œ)βˆ’πœ‚(𝑉)πœŽπ‘ˆ,βˆ‡π‘Œπ‘‹ξ€Έ[]ξ‚΅βˆ’πœ‚(𝑉)πœ‚(𝑋)𝜎(π‘ˆ,π‘Œ)βˆ’πœ‚(𝑉)𝜎(π‘ˆ,𝑋,π‘Œ)+πœŽπ‘ˆ,ξ‚βˆ‡π‘‹πœ‚ξ‚Άξ‚΅(𝑉)π‘Œβˆ’πœŽπ‘ˆ,ξ‚βˆ‡π‘Œξ‚Άξ‚΅πœ‚(𝑉)𝑋+πœŽπ‘ˆ,ξ‚βˆ‡π‘‹ξ‚Άξ‚΅{𝜎(π‘Œ,𝑉)}βˆ’πœŽπ‘ˆ,ξ‚βˆ‡π‘Œξ‚Άξ€·ξ€·{𝜎(𝑋,𝑉)}+πœŽπ‘ˆ,πœŽπ‘‹,βˆ‡π‘Œπ‘‰ξ€·ξ€·ξ€Έξ€Έ+πœ‚(𝑉)𝜎(π‘ˆ,𝜎(𝑋,π‘Œ))βˆ’πœŽπ‘ˆ,πœŽπ‘Œ,βˆ‡π‘‹π‘‰[]ξ€Έξ€Έβˆ’πœ‚(𝑉)𝜎(π‘ˆ,𝜎(π‘Œ,𝑋))βˆ’πœŽ(π‘ˆ,𝜎(𝑋,π‘Œ,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 ξ‚΅ξ‚βˆ‡π‘‹πœŽξ‚Ά(π‘Œ,𝑍)=βˆ‡βŸ‚π‘‹ξ‚€(𝜎(π‘Œ,𝑍))βˆ’πœŽβˆ‡π‘‹ξ‚ξ‚€π‘Œ,π‘βˆ’πœŽπ‘Œ,βˆ‡π‘‹π‘ξ‚ξ‚΅,(5.1)ξ‚βˆ‡2πœŽξ‚Άξ‚΅(𝑍,π‘Š,𝑋,π‘Œ)=ξ‚βˆ‡π‘‹ξ‚βˆ‡π‘ŒπœŽξ‚Ά=(𝑍,π‘Š)βˆ‡βŸ‚π‘‹ξ‚΅ξ‚΅ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚Άβˆ’ξ‚΅(𝑍,π‘Š)ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚€βˆ‡π‘‹ξ‚βˆ’ξ‚΅π‘,π‘Šξ‚βˆ‡π‘‹πœŽξ‚Άξ‚€π‘,βˆ‡π‘Œπ‘Šξ‚βˆ’ξ‚΅ξ‚βˆ‡βˆ‡π‘‹π‘ŒπœŽξ‚Ά(𝑍,π‘Š).(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 ξ‚΅ξ‚βˆ‡π‘‹πœŽξ‚Ά(π‘Œ,𝑍)=πœ™(𝑋)𝜎(π‘Œ,𝑍),(5.3) where πœ™ is a 1-form on 𝑀; in view of (5.1) and putting 𝑍=πœ‰ in the above equation, we have βˆ‡βŸ‚π‘‹ξ‚€πœŽ(π‘Œ,πœ‰)βˆ’πœŽβˆ‡π‘‹ξ‚ξ‚€π‘Œ,πœ‰βˆ’πœŽπ‘Œ,βˆ‡π‘‹πœ‰ξ‚=πœ™(𝑋)𝜎(π‘Œ,πœ‰).(5.4) By virtue of (4.1) in (5.4), we get ξ‚€βˆ’πœŽβˆ‡π‘‹ξ‚ξ‚€π‘Œ,πœ‰βˆ’πœŽπ‘Œ,βˆ‡π‘‹πœ‰ξ‚=0.(5.5) Using (1.1), (3.1), (3.6), and (4.1) in (5.5), we get 𝜎(π‘Œ,πœ™π‘‹)βˆ’πœŽ(π‘Œ,𝑋)=0.(5.6) Replacing 𝑋 by πœ™π‘‹ and by virtue of (3.1) and (4.1) in (5.6), we get βˆ’πœŽ(π‘Œ,𝑋)βˆ’πœŽ(π‘Œ,πœ™π‘‹)=0.(5.7) Adding (5.6) and (5.7), we obtain 𝜎(𝑋,π‘Œ)=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 ξ‚΅ξ‚βˆ‡π‘‹ξ‚βˆ‡π‘ŒπœŽξ‚Ά(𝑍,π‘Š)=0.(5.8) Taking π‘Š=πœ‰ and using (5.2) in the above equation, we have βˆ‡βŸ‚π‘‹ξ‚΅ξ‚΅ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚Άβˆ’ξ‚΅(𝑍,πœ‰)ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚€βˆ‡π‘‹ξ‚βˆ’ξ‚΅π‘,πœ‰ξ‚βˆ‡π‘‹πœŽξ‚Άξ‚€π‘,βˆ‡π‘Œπœ‰ξ‚βˆ’ξ‚΅ξ‚βˆ‡βˆ‡π‘‹π‘ŒπœŽξ‚Ά(𝑍,πœ‰)=0.(5.9) In view of (4.1) and by virtue of (5.1) in (5.9), we get 0=βˆ’βˆ‡βŸ‚π‘‹ξ‚†πœŽξ‚€βˆ‡π‘Œξ‚ξ‚€π‘,πœ‰+πœŽπ‘,βˆ‡π‘Œπœ‰βˆ’ξ‚ξ‚‡βˆ‡βŸ‚π‘ŒπœŽξ‚€βˆ‡π‘‹ξ‚ξ‚€π‘,πœ‰+πœŽβˆ‡π‘Œβˆ‡π‘‹ξ‚ξ‚€π‘,πœ‰+2πœŽβˆ‡π‘‹π‘,βˆ‡π‘Œπœ‰ξ‚βˆ’βˆ‡βŸ‚π‘‹πœŽξ‚€π‘,βˆ‡π‘Œπœ‰ξ‚ξ‚€+πœŽπ‘,βˆ‡π‘‹βˆ‡π‘Œπœ‰ξ‚ξ‚€+πœŽβˆ‡βˆ‡π‘‹π‘Œξ‚ξ‚€π‘,πœ‰+πœŽπ‘,βˆ‡βˆ‡π‘‹π‘Œπœ‰ξ‚.(5.10) Using (1.1), (3.1), (3.6), and (4.1) in (5.10), we get 0=2βˆ‡βŸ‚π‘‹πœŽ(𝑍,πœ™π‘Œ)βˆ’2βˆ‡βŸ‚π‘‹ξ€·βˆ‡πœŽ(𝑍,π‘Œ)βˆ’2πœ‚(𝑍)𝜎(𝑋,πœ™π‘Œ)+2πœŽπ‘‹ξ€Έξ€·π‘,π‘Œ+2πœ‚(𝑍)𝜎(𝑋,π‘Œ)βˆ’πœŽπ‘,βˆ‡π‘‹ξ€Έξ€·πœ™π‘Œβˆ’πœŽπ‘,πœ™βˆ‡π‘‹π‘Œξ€Έξ€·βˆ’πœ‚(π‘Œ)𝜎(𝑍,πœ™π‘‹)+2πœŽπ‘,βˆ‡π‘‹π‘Œξ€Έξ€·βˆ‡+2πœ‚(π‘Œ)𝜎(𝑍,𝑋)βˆ’2πœŽπ‘‹ξ€Έ.𝑍,πœ™π‘Œ(5.11) Putting π‘Œ=πœ‰ and using (3.1), (3.6), and (4.1) in (5.11), we get 0=𝜎(𝑍,𝑋)βˆ’3𝜎(𝑍,πœ™π‘‹).(5.12) Replacing 𝑋 by πœ™π‘‹ and by virtue of (3.1) and (4.1) in (5.12), we get 0=𝜎(𝑍,πœ™π‘‹)+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 ξ‚΅ξ‚βˆ‡π‘‹ξ‚βˆ‡π‘ŒπœŽξ‚Ά(𝑍,π‘Š)=𝜎(𝑍,π‘Š)πœ™(𝑋,π‘Œ).(5.14) Taking π‘Š=πœ‰ and using (5.2) in the above equation, we have βˆ‡βŸ‚π‘‹ξ‚΅ξ‚΅ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚Άβˆ’ξ‚΅(𝑍,πœ‰)ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚€βˆ‡π‘‹ξ‚βˆ’ξ‚΅π‘,πœ‰ξ‚βˆ‡π‘‹πœŽξ‚Άξ‚€π‘,βˆ‡π‘Œπœ‰ξ‚βˆ’ξ‚΅ξ‚βˆ‡βˆ‡π‘‹π‘ŒπœŽξ‚Ά(𝑍,πœ‰)=𝜎(𝑍,πœ‰)πœ™(𝑋,π‘Œ).(5.15) In view of (4.1) and by virtue of (5.1) in (5.15), we get 0=βˆ’βˆ‡βŸ‚π‘‹ξ‚†πœŽξ‚€βˆ‡π‘Œξ‚ξ‚€π‘,πœ‰+πœŽπ‘,βˆ‡π‘Œπœ‰βˆ’ξ‚ξ‚‡βˆ‡βŸ‚π‘ŒπœŽξ‚€βˆ‡π‘‹ξ‚ξ‚€π‘,πœ‰+πœŽβˆ‡π‘Œβˆ‡π‘‹ξ‚ξ‚€π‘,πœ‰+2πœŽβˆ‡π‘‹π‘,βˆ‡π‘Œπœ‰ξ‚βˆ’βˆ‡βŸ‚π‘‹πœŽξ‚€π‘,βˆ‡π‘Œπœ‰ξ‚ξ‚€+πœŽπ‘,βˆ‡π‘‹βˆ‡π‘Œπœ‰ξ‚ξ‚€+πœŽβˆ‡βˆ‡π‘‹π‘Œξ‚ξ‚€π‘,πœ‰+πœŽπ‘,βˆ‡βˆ‡π‘‹π‘Œπœ‰ξ‚.(5.16) Using (1.1), (3.1), (3.6), and (4.1) in (5.16), we get 0=2βˆ‡βŸ‚π‘‹πœŽ(𝑍,πœ™π‘Œ)βˆ’2βˆ‡βŸ‚π‘‹ξ€·βˆ‡πœŽ(𝑍,π‘Œ)βˆ’2πœ‚(𝑍)𝜎(𝑋,πœ™π‘Œ)+2πœŽπ‘‹ξ€Έξ€·π‘,π‘Œ+2πœ‚(𝑍)𝜎(𝑋,π‘Œ)βˆ’πœŽπ‘,βˆ‡π‘‹ξ€Έξ€·πœ™π‘Œβˆ’πœŽπ‘,πœ™βˆ‡π‘‹π‘Œξ€Έξ€·βˆ’πœ‚(π‘Œ)𝜎(𝑍,πœ™π‘‹)+2πœŽπ‘,βˆ‡π‘‹π‘Œξ€Έξ€·βˆ‡+2πœ‚(π‘Œ)𝜎(𝑍,𝑋)βˆ’2πœŽπ‘‹ξ€Έ.𝑍,πœ™π‘Œ(5.17) Putting π‘Œ=πœ‰ and using (3.1), (3.6), (4.1) in (5.17), we get 0=𝜎(𝑍,𝑋)βˆ’3𝜎(𝑍,πœ™π‘‹).(5.18) Replacing 𝑋 by πœ™π‘‹ and by virtue of (3.1) and (4.1) in (5.18), we get 0=𝜎(𝑍,πœ™π‘‹)+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 ξ‚΅ξ‚βˆ‡π‘‹ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚΅(𝑍,π‘Š)=πœ“(𝑋,π‘Œ)𝜎(𝑍,π‘Š)+πœ™(𝑋)ξ‚βˆ‡π‘ŒπœŽξ‚Ά(𝑍,π‘Š),(5.20) where πœ“ and πœ™ are 2-recurrent and 1-form, respectively. Taking π‘Š=πœ‰ in (5.20) and using (4.1), we get ξ‚΅ξ‚βˆ‡π‘‹ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚΅(𝑍,πœ‰)=πœ™(𝑋)ξ‚βˆ‡π‘ŒπœŽξ‚Ά(𝑍,πœ‰).(5.21) Using (4.1) and (5.2) in above equation, we get βˆ‡βŸ‚π‘‹ξ‚΅ξ‚΅ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚Άβˆ’ξ‚΅(𝑍,πœ‰)ξ‚βˆ‡π‘ŒπœŽξ‚Άξ‚€βˆ‡π‘‹ξ‚βˆ’ξ‚΅π‘,πœ‰ξ‚βˆ‡π‘‹πœŽξ‚Άξ‚€π‘,βˆ‡π‘Œπœ‰ξ‚βˆ’ξ‚΅ξ‚βˆ‡βˆ‡π‘‹π‘ŒπœŽξ‚Άξ‚†πœŽξ‚€(𝑍,πœ‰)=βˆ’πœ™(𝑋)βˆ‡π‘Œξ‚ξ‚€π‘,πœ‰+πœŽπ‘,βˆ‡π‘Œπœ‰.(5.22) In view of (4.1) and by virtue of (5.1) in (5.22), we get βˆ’βˆ‡βŸ‚π‘‹ξ‚†πœŽξ‚€βˆ‡π‘Œξ‚ξ‚€π‘,πœ‰+πœŽπ‘,βˆ‡π‘Œπœ‰βˆ’ξ‚ξ‚‡βˆ‡βŸ‚π‘ŒπœŽξ‚€βˆ‡π‘‹ξ‚ξ‚€π‘,πœ‰+πœŽβˆ‡π‘Œβˆ‡π‘‹ξ‚ξ‚€π‘,πœ‰+2πœŽβˆ‡π‘‹π‘,βˆ‡π‘Œπœ‰ξ‚βˆ’βˆ‡βŸ‚π‘‹πœŽξ‚€π‘,βˆ‡π‘Œπœ‰ξ‚ξ‚€+πœŽπ‘,βˆ‡π‘‹βˆ‡π‘Œπœ‰ξ‚ξ‚€+πœŽβˆ‡βˆ‡π‘‹π‘Œξ‚ξ‚€π‘,πœ‰+πœŽπ‘,βˆ‡βˆ‡π‘‹π‘Œπœ‰ξ‚ξ‚†πœŽξ‚€=βˆ’πœ™(𝑋)βˆ‡π‘Œξ‚ξ‚€π‘,πœ‰+πœŽπ‘,βˆ‡π‘Œπœ‰.(5.23) Using (1.1), (3.1), (3.6), and (4.1) in (5.23), we get 0=2βˆ‡βŸ‚π‘‹πœŽ(𝑍,πœ™π‘Œ)βˆ’2βˆ‡βŸ‚π‘‹ξ€·βˆ‡πœŽ(𝑍,π‘Œ)βˆ’2πœ‚(𝑍)𝜎(𝑋,πœ™π‘Œ)+2πœŽπ‘‹ξ€Έξ€·π‘,π‘Œ+2πœ‚(𝑍)𝜎(𝑋,π‘Œ)βˆ’πœŽπ‘,βˆ‡π‘‹ξ€Έξ€·πœ™π‘Œβˆ’πœŽπ‘,πœ™βˆ‡π‘‹π‘Œξ€Έξ€·βˆ’πœ‚(π‘Œ)𝜎(𝑍,πœ™π‘‹)+2πœŽπ‘,βˆ‡π‘‹π‘Œξ€Έξ€·βˆ‡+2πœ‚(π‘Œ)𝜎(𝑍,𝑋)βˆ’2πœŽπ‘‹ξ€Έπ‘,πœ™π‘Œ=βˆ’πœ™(𝑋){βˆ’πœŽ(𝑍,πœ™π‘Œ)+𝜎(𝑍,π‘Œ)}.(5.24) Putting π‘Œ=πœ‰ and using (3.1), (3.6), (4.1) in (5.24), we get 0=𝜎(𝑍,𝑋)βˆ’3𝜎(𝑍,πœ™π‘‹).(5.25) Replacing 𝑋 by πœ™π‘‹ and by virtue of (3.1) and (4.1) in (5.25), we get 0=𝜎(𝑍,πœ™π‘‹)+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 ξ‚π‘…β‹…πœŽ=0,ξ‚π‘…β‹…πœŽ=𝐿1𝑄(𝑔,𝜎),ξ‚π‘…β‹…πœŽ=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=2πœ™+πœ‰.

Proof. Let 𝑀 be semiparallel ξ‚π‘…β‹…πœŽ=0. Putting 𝑋=𝑉=πœ‰ and by virtue of (3.1), (3.6), and (4.1) in (4.11), we get ξ‚΅0=βˆ’πœŽ(π‘ˆ,𝑅(πœ‰,π‘Œ)πœ‰)βˆ’πœŽξ‚βˆ‡πœ‰ξ‚Άξ‚΅πœ‚(π‘ˆ)π‘Œ,πœ‰+πœŽξ‚βˆ‡π‘Œξ‚Άξ‚΅πœ‚(π‘ˆ)πœ‰,πœ‰βˆ’πœŽξ‚βˆ‡πœ‰ξ‚Άξ€·πœŽ(π‘Œ,π‘ˆ),πœ‰βˆ’πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έξ€·+πœŽπ‘ˆ,βˆ‡π‘Œπœ‰ξ€Έ[]ξ‚΅+𝜎(π‘ˆ,πœ‰,π‘Œ)βˆ’πœŽπ‘ˆ,ξ‚βˆ‡πœ‰π‘Œξ‚Άξ‚΅+πœŽπ‘ˆ,ξ‚βˆ‡π‘Œπœ‰ξ‚Ά+𝜎(π‘ˆ,π‘Œ).(6.1) Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.1), we get ξ‚΅0=3𝜎(π‘ˆ,π‘Œ)βˆ’πœŽξ‚βˆ‡πœ‰ξ‚Άξ€·πœ‚(π‘ˆ)π‘Œ,πœ‰βˆ’πœŽ(π‘ˆ,πœ™π‘Œ)βˆ’πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έ.(6.2) By definition 𝜎 is a vector-valued covariant tensor, and so 𝜎(π‘ˆ,π‘Œ) is a vector. Therefore ξ‚βˆ‡πœ‰πœŽ(π‘Œ,π‘ˆ) is a vector, and hence by (4.1), we have πœŽξ‚΅ξ‚βˆ‡πœ‰ξ‚ΆπœŽ(π‘Œ,π‘ˆ),πœ‰=0.(6.3) Then from (6.2), we get 3𝜎(π‘ˆ,π‘Œ)=ξ€·πœ™πœŽ(π‘ˆ,π‘Œ)+πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έ.(6.4) Interchanging π‘Œ and π‘ˆ in (6.4), we get 3𝜎(π‘Œ,π‘ˆ)=ξ€·πœ™πœŽ(π‘Œ,π‘ˆ)+πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έ.(6.5) Adding these tow equations, (6.4) and (6.5), we get 6=2πœ™+πœ‰.(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=πœ™+πœ‰/2βˆ’3.

Proof. Let 𝑀 be pseudoparallel ξ‚π‘…β‹…πœŽ=𝐿1𝑄(𝑔,𝜎). Putting 𝑋=𝑉=πœ‰ and by virtue of (3.1), (3.6), and (4.1) in (2.7), (4.11), we get ξ‚΅βˆ’πœŽ(π‘ˆ,𝑅(πœ‰,π‘Œ)πœ‰)βˆ’πœŽξ‚βˆ‡πœ‰ξ‚Άξ‚΅πœ‚(π‘ˆ)π‘Œ,πœ‰+πœŽξ‚βˆ‡π‘Œξ‚Άξ‚΅πœ‚(π‘ˆ)πœ‰,πœ‰βˆ’πœŽξ‚βˆ‡πœ‰ξ‚Άξ€·πœŽ(π‘Œ,π‘ˆ),πœ‰βˆ’πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έξ€·+πœŽπ‘ˆ,βˆ‡π‘Œπœ‰ξ€Έ[]ξ‚΅+𝜎(π‘ˆ,πœ‰,π‘Œ)βˆ’πœŽπ‘ˆ,ξ‚βˆ‡πœ‰π‘Œξ‚Άξ‚΅+πœŽπ‘ˆ,ξ‚βˆ‡π‘Œπœ‰ξ‚Ά+𝜎(π‘ˆ,π‘Œ)=βˆ’πΏ1𝜎(π‘ˆ,π‘Œ).(6.7) Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.7), we get ξ‚΅3𝜎(π‘ˆ,π‘Œ)βˆ’πœŽξ‚βˆ‡πœ‰ξ‚Άξ€·πœ‚(π‘ˆ)π‘Œ,πœ‰βˆ’πœŽ(π‘ˆ,πœ™π‘Œ)βˆ’πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έ=βˆ’πΏ1𝜎(π‘ˆ,π‘Œ).(6.8) Now by using (6.3) in (6.8), we get ξ€·3+𝐿1ξ€ΈπœŽ(π‘ˆ,π‘Œ)=ξ€·πœ™πœŽ(π‘ˆ,π‘Œ)+πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έ.(6.9) Interchanging π‘Œ and π‘ˆ in (6.9), we get ξ€·3+𝐿1ξ€ΈπœŽ(π‘Œ,π‘ˆ)=ξ€·πœ™πœŽ(π‘Œ,π‘ˆ)+πœŽπ‘Œ,βˆ‡πœ‰π‘ˆξ€Έ.(6.10) Adding (6.9) and (6.10), we get 𝐿1=πœ‰πœ™+2βˆ’3.(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/(π‘›βˆ’1))[πœ™+πœ‰/2βˆ’3].

Proof. Let 𝑀 be Ricci-generalized pseudoparallel ξ‚π‘…β‹…πœŽ=𝐿2𝑄(𝑆,𝜎). Putting 𝑋=𝑉=πœ‰ and by virtue of (3.1), (3.6), (3.16), and (4.1) in (2.7), (4.11), we get ξ‚΅βˆ’πœŽ(π‘ˆ,𝑅(πœ‰,π‘Œ)πœ‰)βˆ’πœŽξ‚βˆ‡πœ‰ξ‚Άξ‚΅πœ‚(π‘ˆ)π‘Œ,πœ‰+πœŽξ‚βˆ‡π‘Œξ‚Άξ‚΅πœ‚(π‘ˆ)πœ‰,πœ‰βˆ’πœŽξ‚βˆ‡πœ‰ξ‚Άξ€·πœŽ(π‘Œ,π‘ˆ),πœ‰βˆ’πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έξ€·+πœŽπ‘ˆ,βˆ‡π‘Œπœ‰ξ€Έ[]ξ‚΅+𝜎(π‘ˆ,πœ‰,π‘Œ)βˆ’πœŽπ‘ˆ,ξ‚βˆ‡πœ‰π‘Œξ‚Άξ‚΅+πœŽπ‘ˆ,ξ‚βˆ‡π‘Œπœ‰ξ‚Ά+𝜎(π‘ˆ,π‘Œ)=βˆ’πΏ2(π‘›βˆ’1)𝜎(π‘ˆ,π‘Œ).(6.12) Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.12), we get ξ‚΅3𝜎(π‘ˆ,π‘Œ)βˆ’πœŽξ‚βˆ‡πœ‰ξ‚Άξ€·πœ‚(π‘ˆ)π‘Œ,πœ‰βˆ’πœŽ(π‘ˆ,πœ™π‘Œ)βˆ’πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έ=βˆ’πΏ2(π‘›βˆ’1)𝜎(π‘ˆ,π‘Œ).(6.13) Now by using (6.3) in (6.13), we get ξ€·3+𝐿2ξ€Έ(π‘›βˆ’1)𝜎(π‘ˆ,π‘Œ)=ξ€·πœ™πœŽ(π‘ˆ,π‘Œ)+πœŽπ‘ˆ,βˆ‡πœ‰π‘Œξ€Έ.(6.14) Interchanging π‘Œ and π‘ˆ in (6.14), we get ξ€·3+𝐿2ξ€Έ(π‘›βˆ’1)𝜎(π‘Œ,π‘ˆ)=ξ€·πœ™πœŽ(π‘Œ,π‘ˆ)+πœŽπ‘Œ,βˆ‡πœ‰π‘ˆξ€Έ.(6.15) Adding (6.14) and (6.15), we get 2ξ€·3+𝐿2ξ€Έ(π‘›βˆ’1)𝜎(π‘ˆ,π‘Œ)=2πœ™πœŽ(π‘ˆ,π‘Œ)+βˆ‡πœ‰πœŽ(π‘ˆ,π‘Œ).(6.16) Writting the above equation, we have 𝐿2=1ξ‚Έ(π‘›βˆ’1)πœ‰πœ™+2ξ‚Ήβˆ’3.(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 πœ™, πœ‰, 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.