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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 947640, 18 pages
http://dx.doi.org/10.1155/2012/947640
Research Article

Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Nonmetric Connection

Department of Mathematics, Kuvempu University, Shankaraghatta, Karnataka, Shimoga 577451, India

Received 30 March 2012; Accepted 2 July 2012

Academic Editor: V. R.Β Khalilov

Copyright Β© 2012 B. S. Anitha and C. S. Bagewadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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