We consider a nearly hyperbolic Kenmotsu manifold admitting a semi-symmetric metric connection and study semi-invariant submanifolds of a nearly hyperbolic Kenmotsu manifold with semi-symmetric metric connection. We also find the integrability conditions of some distributions on nearly hyperbolic Kenmotsu manifold andstudy parallel distributions on nearly hyperbolic Kenmotsu manifold.

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On Semi-Invariant Submanifolds of a Nearly Hyperbolic Kenmotsu Manifold with Semi-symmetric Metric Connection TOUKEER KHAN1, SHADAB AHMAD KHAN2 AND MOBIN AHMAD3 1,2 Department of Mathematics,Integral University, Kursi Road, Lucknow-226026, India. 3Department of Mathematics,Faculty of Science, Jazan University, Jazan-2069, Saudi Arabia. Abstract We consider a nearly hyperbolic Kenmotsu manifold admitting a semi-symmetric metric connection and study semi-invariant submanifolds of a nearly hyperbolic Kenmotsu manifold with semi-symmetric metric connection. We also find the integrability conditions of some distributions on nearly hyperbolic Kenmotsu manifold andstudy parallel distributions on nearly hyperbolic Kenmotsu manifold. Key Words and Phrases: Semi-invariant submanifolds, Nearly hyperbolic Kenmotsu manifold, Semi-symmetric metric connection, Parallel distribution, Integrability condition. 2000 AMS Mathematics Subject Classification: 53D05, 53D25, 53D12.
I. Introduction
Let ∇ be a linear connection in an n-dimensional differentiable manifoldM . The torsion tensor T and curvature tensor R of ∇ given respectively by 푇 푋,푌 =훻푋푌−훻푌푋− 푋,푌 , 푅 푋,푌 푍=∇푋∇푌푍−∇푌∇푋푍−∇[푋,푌]푍. The connection ∇ is symmetric if its torsion tensor T vanishes, otherwise it is non-symmetric. The connection ∇ is metric connection if there is a Riemannian metric g in 푀 such that 훻푔=0, otherwise it is non-metric. It is well known that a linear connection is symmetric and metric if and only if it is the Levi-Civita connection. A. Friedmann and J. A. Schouten introduced the idea of a semi- symmetric connection. A linear connection is said to be a semi-symmetric connection if its torsion tensor T is of the form: 푇 푋,푌 =휂 푋 푌−휂 푌 푋. Many geometers (see,[20], [21]) have studied some properties of semi-symmetric metric connection.
The study of CR-submanifolds of Kaehler manifold as generalization of invariant and anti- invariant submanifolds was initiated by A. Bejancu in [8]. A semi-invariant submanifold is the extension of a CR-submanifold of a Kaehler manifold to submanifolds of almost contact manifolds. The study of semi-invariant submanifolds of Sasakian manifolds was initiated by Bejancu-Papaghuic in [10]. The same concept was studied under the name contact CR-submanifold by Yano-Kon in [19] and K. Matsumoto in [16].The study of semi-invariant submanifolds in almost contact manifold was enriched by several geometers (see, [2], [4], [5], [6], [7], [8], [12], [13], [17]). On the otherhand, almost hyperbolic 푓,휉,휂,푔 -structure was defined and studied by Upadhyay and Dube in [18].Joshi and Dube studied semi-invariant submanifolds of an almost r-contact hyperbolic metric manifold in [15]. Ahmad M., K. Ali, studied semi-invariant submanifolds of a nearly hyperbolic Kenmotsu manifold with a semi-symmetric non-metric connection in [2]. In this paper, we study semi- invariant submanifolds of a nearly hyperbolic Kenmotsu manifold admitting a semi-symmetric metric connection.
RESEARCH ARTICLE OPEN ACCESS

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This paper is organized as follows. In section 2, we give a brief description of nearly hyperbolic Kenmotsu manifold admitting a semi- symmetric metric connection. In section 3, we study some properties of semi-invariant submanifolds of a nearly hyperbolic Kenmotsu manifold with a semi- symmetric metric connection. In section 4,we discuss the integrability conditions of some distributions on nearly hyperbolic Kenmotsu manifold with a semi- symmetric metric connection. In section 5, we study parallel horizontal distribution on nearly hyperbolic Kenmotsu manifold with a semi-symmetric metric connection.
II. Preliminaries
Let 푀 be an 푛-dimensional almost hyperbolic contact metric manifold with the almost hyperbolic contact metric structure ∅,휉,휂,푔 , where a tensor ∅ of type 1,1 , a vector field 휉, called structure vector field and 휂, the dual 1-form of 휉and the associated Riemannian metric 푔satisfying the following ∅2푋=푋+휂 푋 휉, 푔 푋,휉 =휂 푋 (2.1) 휂 휉 =−1, ∅ 휉 =0, 휂표∅=0, (2.2) 푔 ∅푋,∅푌 =−푔 푋,푌 −휂 푋 휂 푌 (2.3) for any 푋,푌 tangent to 푀 [19]. In this case 푔 ∅푋,푌 =−푔 ∅푌,푋 . (2.4) An almost hyperbolic contact metric structure ∅,휉,휂,푔 on 푀 is called hyperbolic Kenmotsu manifold [7] if and only if ∇푋∅ 푌=푔 ∅푋,푌 − 휂(푌)∅푋 (2.5) for all 푋,푌 tangent to 푀 . On a hyperbolic Kenmotsu manifold푀 , we have ∇푋휉=푋+ 휂(푋)휉 (2.6) for a Riemannian metric g and Riemannian connection∇. Further, an almost hyperbolic contact metric manifold 푀 on ∅,휉,휂,푔 is called a nearly hyperbolic Kenmotsu manifold [7], if ∇푋∅ 푌+ ∇푌∅ 푋=−휂 푋 ∅푌− 휂 푌 ∅푋, (2.7) where∇ is Riemannian connection 푀 . Now, we define a semi-symmetric metric connection ∇ 푋푌=∇푋푌+휂 푌 푋−푔(푋,푌)휉 (2.8) Such that(∇ 푋푔) 푌,푍 =0.From(2.7) and(2.8), replacing 푌by ∅푌, we have ∇ 푋∅ 푌+ ∇ 푌∅ 푋=−2휂 푋 ∅푌−2휂 푌 ∅푋 (2.9) ∇ 푋휉=0. (2.10) An almost hyperbolic contact metric manifold with almost hyperbolic contact structure ∅,휉,휂,푔 is called nearly hyperbolic Kenmotsu manifold with semi-symmetric metric connection if it is satisfied (2.9) and (2.10). Let Riemannian metric symbol 푔 and ∇ be induced Levi-Civita connection on 푀 then the Guass formula and Weingarten formula is given by ∇ 푋푌=∇푋푌+푕 푋,푌 , (2.11) ∇ 푋푁=−퐴푁푋+∇푋 ⊥푁 (2.12)

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for any 푋,푌 ∈푇푀 and 푁∈푇푀, where ∇ is a connection on the normal bundle 푇푀, 푕is the second fundamental form and 퐴푁 is the Weingarten map associated with 푁 as 푔 푕 푋,푌 ,푁 =푔 퐴푁푋,푌 . (2.13) Any vector 푋 tangent to 푀 is given as 푋=푃푋+푄푋+휂 푋 휉, (2.14) where푃푋∈퐷 푎푛푑 푄푋∈퐷⊥. For any 푁 normal to 푀, we have ∅푁=퐵푁+퐶푁, (2.15) where퐵푁 푟푒푠푝.퐶푁 is the tangential component 푟푒푠푝.푛표푟푚푎푙 푐표푚푝표푛푒푛푡 of ∅푁. For almost contact structure the Nijenhuis tensor 푁 푋,푌 is expressed as 푁 푋,푌 = 훻 ∅푋∅ 푌− 훻 ∅푌∅ 푋−∅ 훻 푋∅ 푌+∅ 훻 푌∅ 푋 (2.16) Now from (2.9),replacing 푋 by ∅푋, we have ∇ ∅푋∅ 푌=− 휂 푌 푋−휂 푋 휂 푌 휉− ∇ 푌∅ ∅푋. (2.17) Differentiating (2.1) conveniently along the vector and using (2.10), we have ∇ 푌∅ ∅푋= ∇ 푌휂 푋 휉−∅ ∇ 푌∅ 푋. (2.18) From (2.17) and (2.18), we have ∇ ∅푋∅ 푌=−휂 푌 푋−휂 푋 휂 푌 휉− ∇ 푌휂 푋 휉+∅ ∇ 푌∅ 푋. (2.19) Interchanging 푋and 푌, we have ∇ ∅푌∅ 푋=− 휂 푋 푌−휂 푋 휂 푌 휉− ∇ 푋휂 푌 휉+∅ ∇ 푋∅ 푌. (2.20) Using equation (2.19), (2.20) and (2.9) in (2.16), we have 푁 푋,푌 =휂 푋 푌− 휂 푌 푋+2푑휂 푋,푌 휉+2∅ ∇ 푌∅ 푋−2∅ ∇ 푋∅ 푌, (2.21) 푁 푋,푌 =2푔 ∅푋,푌 휉+5휂 푋 푌+3휂 푌 푋+8휂 푋 휂 푌 휉+4∅ ∇ 푌∅ 푋. (2.22) As we know that, ∇ 푌∅ 푋=∇ 푌∅푋−∅ ∇ 푌푋 .UsingGuass formula (2.11), we have ∅ ∇ 푌∅ 푋=∅ ∇푌∅푋 +∅푕 푌,∅푋 −∇푌푋−휂 ∇푌푋 휉−푕 푌,푋 . Using this equation in (2.22), we have 푁 푋,푌 =2푔 ∅푋,푌 휉+5휂 푋 푌+3휂 푌 푋+8휂 푋 휂 푌 휉+4∅ ∇푌∅푋 +4∅푕 푌,∅푋 −4 ∇푌푋 −4휂 ∇푌푋 휉−4푕 푌,푋 . (2.23)
III. Semi-invariant Submanifold
Let 푀 be submanifold immersed in 푀 , we assume that the vector 휉 is tangent to 푀, denoted by {휉} the 1- dimentional distribution spanned by 휉 on 푀, then 푀 is called a semi-invariant submanifold [9] of 푀 if there exist two differentiable distribution 퐷 &퐷 on 푀 satisfying (i) 푇푀= 퐷⨁퐷⨁휉 , where 퐷 ,퐷& 휉 are mutually orthogonal to each other, (ii) the distribution퐷 is invariant under ∅that is ∅퐷푥= 퐷푥 for all 푥 ∈ 푀, (iii) the distribution퐷is anti-invariant under ∅, that is ∅퐷푥 ⊂푇푀 for all푥 ∈ 푀,where 푇푀 & 푇푀 be the Lie algebra of vector fields tangential & normal to 푀 respectively. Theorem 3.1.The connection induced on semi-invariant submanifold of nearly hyperbolic Kenmotsu manifold with semi-symmetric metric connection also semi-symmetric metric.

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Proof:Let ∇be induced connection with respect to the normal 푁 on semi-invariant submanifold of nearly hyperbolic Kenmotsu manifold with semi-symmetric metricconnection ∇ , then ∇ 푋푌=∇푋푌+푚 푋,푌 , (3.1) where m is the tensor filed of type (0,2) on semi-invariant submanifold 푀. If ∇∗ be the induced connection on semi-invariant submanifold from Riemannian connection ∇ ,then ∇ 푋푌=∇푋∗ 푌+푕 푋,푌 , (3.2) where푕 is second fundamental tensor and we know that semi-symmetric metric connection on nearly hyperbolic Kenmotsu manifold ∇ 푋푌=∇ 푋푌+ 푌 푋−푔 푋,푌 휉. (3.3) Using (3.1), (3.2) in (3.3), we have ∇푋푌+푚 푋,푌 =∇푋∗ 푌+푕 푋,푌 +휂 푌 푋−푔 푋,푌 휉. (3.4) Comparing tangent and normal parts, we have ∇푋푌=∇푋∗ 푌+휂 푌 푋−푔 푋,푌 휉, 푚 푋,푌 =푕 푋,푌 . Thus ∇ is also semi-symmetric metric connection. Lemma 3.2. Let 푀 be a semi-invariant submanifold of a nearly hyperbolic Kenmotsu manifold 푀 with semi- symmetric metric connection, then 2 ∇ 푋∅ 푌=∇푋∅푌−∇푌∅푋+푕 푋,∅푌 −푕 푌,∅푋 −∅ 푋,푌 for each 푋,푌∈퐷. Proof.By Gauss formula (2.11), we have ∇ 푋∅푌−∇ 푌∅푋=∇푋∅푌−∇푌∅푋+푕 푋,∅푌 −푕 푌,∅푋 . (3.5) Also, by covariantly differentiation, we know that ∇ 푋∅푌−∇ 푌∅푋= ∇ 푋∅ 푌− ∇ 푌∅ 푋+∅ 푋,푌 . (3.6) From (3.5) and (3.6), we have ∇ 푋∅ 푌− ∇ 푌∅ 푋=∇푋∅푌−∇푌∅푋+푕 푋,∅푌 −푕 푌,∅푋 −∅ 푋,푌 . (3.7) Adding (2.9) and (3.7), we obtain 2 ∇ 푋∅ 푌=∇푋∅푌−∇푌∅푋+푕 푋,∅푌 −푕 푌,∅푋 −∅ 푋,푌 for each 푋,푌∈퐷. Lemma 3.3. Let 푀 be a semi-invariant submanifold of a nearly hyperbolic Kenmotsu manifold 푀 with semi- symmetric metric connection, then 2 ∇ 푌∅ 푋=∇푌∅푋−∇푋∅푌+푕 푌,∅푋 −푕 푋,∅푌 +∅ 푋,푌 for each 푋,푌∈퐷. Theorem 3.4. Let 푀 be a semi-invariant submanifold of a nearly hyperbolic Kenmotsu manifold 푀 with semi- symmetric metric connection, then 2 ∇ 푋∅ 푌=−퐴∅푌푋+∇푋 ⊥∅푌−∇푌∅푋−푕 푌,∅푋 −∅ 푋,푌 for all X∈퐷 푎푛푑 푌∈퐷⊥. Proof.By Gauss formulas (2.11), we have ∇ 푌∅푋=∇푌∅푋+푕 푌,∅푋 .

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Also, by Weingarten formula (2.12), we have ∇ 푋∅푌=−퐴∅푌푋+∇푋 ⊥∅푌. From abovetwo equations, we have ∇ 푋∅푌−∇ 푌∅푋=−퐴∅푌푋+∇푋 ⊥∅푌−∇푌∅푋−푕 푌,∅푋 . (3.8) Comparing equation (3.6) and (3.8), we have ∇ 푋∅ 푌− ∇ 푌∅ 푋=−퐴∅푌푋+∇푋 ⊥∅푌−∇푌∅푋−푕 푌,∅푋 −∅ 푋,푌 . (3.9) Adding equation (2.9) in (3.9), we have 2 ∇ 푋∅ 푌=−퐴∅푌푋+∇푋 ⊥∅푌−휂 푋 푌−∇푌∅푋−푕 푌,∅푋 −∅ 푋,푌 −2휂 푋 ∅푌−2휂 푌 ∅푋 2 ∇ 푋∅ 푌=−퐴∅푌푋+∇푋 ⊥∅푌−∇푌∅푋−푕 푌,∅푋 −∅ 푋,푌 for all 푋∈퐷 and 푌∈퐷⊥. Lemma 3.5. Let 푀 be a semi-invariant submanifold of a nearly hyperbolic Kenmotsu manifold 푀 with semi- symmetric metric connection, then 2 ∇ 푌∅ 푋=퐴∅푌푋−∇푋 ⊥∅푌+∇푌∅푋+푕 푌,∅푋 +∅ 푋,푌 (3.10) for all푋∈퐷 and 푌∈퐷⊥. Lemma 3.6. Let 푀 be a semi-invariant submanifold of a nearly hyperbolic Kenmotsu manifold 푀 with semi- symmetric metric connection, then 2 ∇ 푋∅ 푌=퐴∅푋푌−퐴∅푌푋+∇푋 ⊥∅푌−∇푌 ⊥∅푋−∅ 푋,푌 (3.11) for all 푋,푌∈퐷⊥. Proof.Using Weingarten formula (2.12), we have ∇ 푋∅푌−∇ 푌∅푋=퐴∅푋푌−퐴∅푌푋+∇푋 ⊥∅푌−∇푌 ⊥∅푋. Using equation (3.6) in above, we have ∇ 푋∅ 푌− ∇ 푌∅ 푋=퐴∅푋푌−퐴∅푌푋+∇푋 ⊥∅푌−∇푌 ⊥∅푋−∅ 푋,푌 . (3.12) Adding (2.9) in above, we have 2 ∇ 푋∅ 푌=퐴∅푋푌−퐴∅푌푋+∇푋 ⊥∅푌−∇푌 ⊥∅푋−∅ 푋,푌 for all 푋,푌∈퐷⊥. Lemma 3.7. Let 푀 be a semi-invariant submanifold of a nearly hyperbolic Kenmotsu manifold 푀 with semi- symmetric metric connection, then 2 ∇ 푌∅ 푋=퐴∅푌푋−퐴∅푋푌+∇푌 ⊥∅푋−∇푋 ⊥∅푌+∅ 푋,푌 for all푋,푌∈퐷⊥ Lemma 3.8. Let 푀 be a semi-invariant submanifold of a nearly hyperbolic Kenmotsu manifold 푀 with semi- symmetric metricconnection.Then 푃 ∇푋∅푃푌 +푃 ∇푌∅푃푋 −푃퐴∅푄푌푋−푃퐴∅푄푋푌=∅푃 ∇푋푌 +∅푃 ∇푌푋 −2휂 푋 ∅푃푌−2휂 푌 ∅푃푋, (3.13) 푄 ∇푋∅푃푌 +푄 ∇푌∅푃푋 −푄퐴∅푄푌푋−푄퐴∅푄푋푌=2퐵푕 푋,푌 , (3.14) 푕 푌,∅푃푋 +푕 푋,∅푃푌 +∇푋 ⊥∅푄푌+∇푌 ⊥∅푄푋=∅푄 ∇푋푌 +∅푄 ∇푌푋 +2퐶푕 푋,푌 −2휂 푋 ∅푄푌−2휂 푌 ∅푄푋, (3.15) 휂 ∇푋∅푃푌 +휂 ∇푌∅푃푋 −휂 퐴∅푄푌푋 −휂 퐴∅푄푋푌 =0 (3.16) for all 푋,푌∈푇푀.

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Proof.Differentiating covariantlyequation (2.14) and using equation (2.11) and(2.12), we have ∇ 푋∅ 푌+∅(∇푋푌)+∅푕 푋,푌 =∇푋∅푃푌+푕 푋,∅푃푌 −퐴∅푄푌푋+∇푋 ⊥∅푄푌. Interchanging 푋and 푌, we have ∇ 푌∅ 푋+∅(∇푌푋)+∅푕 푌,푋 =∇푌∅푃푋+푕 푌,∅푃푋 −퐴∅푄푋푌+∇푌 ⊥∅푄푋. Adding above two equations, we have ∇ 푋∅ 푌+ ∇ 푌∅ 푋+∅(∇푋푌)+∅(∇푌푋)+2∅푕 푋,푌 =∇푋∅푃푌+∇푌∅푃푋 +푕 푋,∅푃푌 +푕 푌,∅푃푋 −퐴∅푄푌푋−퐴∅푄푋푌+∇푋 ⊥∅푄푌+∇푌 ⊥∅푄푋. Using equation (2.9) in above, we have −2휂 푋 ∅푌–2휂 푌 ∅푋+∅(∇푋푌)+∅(∇푌푋)+2∅푕 푋,푌 =∇푋∅푃푌+∇푌∅푃푋 +푕 푋,∅푃푌 +푕 푌,∅푃푋 −퐴∅푄푌푋−퐴∅푄푋푌+∇푋 ⊥∅푄푌+∇푌 ⊥∅푄푋. Using equations (2.14), (2.15)&(2.2), we have −2휂 푋 ∅푃푌−2휂 푋 ∅푄푌−2휂 푌 ∅푃푋−2휂 푌 ∅푄푋+∅푃 ∇푋푌 +∅푄 ∇푋푌 +∅푃 ∇푌푋 +∅푄 ∇푌푋 +2퐵푕 푋,푌 +2퐶푕 푋,푌 =푃 ∇푋∅푃푌 +푄 ∇푋∅푃푌 +휂 ∇푋∅푃푌 휉+푃 ∇푌∅푃푋 +푄 ∇푌∅푃푋 +휂 ∇푌∅푃푋 휉+푕 푋,∅푃푌 +푕 푌,∅푃푋 −푃퐴∅푄푌푋−푄퐴∅푄푌푋−휂 퐴∅푄푌푋 휉−푃퐴∅푄푋푌−푄퐴∅푄푋푌 −휂 퐴∅푄푋푌 휉+∇푋 ⊥∅푄푌+∇푌 ⊥∅푄푋. Comparing horizontal, vertical and normal components we get desired results. ⊡
IV. Integrability of Distributions
Theorem4.1. Let 푀 be a semi-invariant submanifold of a nearly hyperbolic Kenmotsu manifold 푀 with semi- symmetric metric connection, then the distribution 퐷⨁ 휉 is integrable if the following conditions are satisfied 푆 푋,푌 ∈ 퐷⨁ 휉 (4.1) 푕 푋,∅푌 =푕 ∅푋,푌 (4.2) for each 푋,푌∈ 퐷⨁ 휉 . Proof.The torsion tensor 푆 푋,푌 of an almost hyperbolic contact manifold is given by 푆 푋,푌 =푁 푋,푌 +2푑휂 푋,푌 휉, where 푁 푋,푌 is Neijenhuis tensor, 푆 푋,푌 = ∅푋,∅푌 −∅ ∅푋,푌 −∅ 푋,∅푌 +2푑휂 푋,푌 휉. (4.3) Suppose that 퐷⨁ 휉 is integrable, then푁 푋,푌 =0for any 푋,푌∈ 퐷⨁ 휉 . Therefore, 푆 푋,푌 =2푑휂 푋,푌 휉∈ 퐷⨁ 휉 . From (4.3), (2.23) and comparing normal part, we have ∅푄 ∇푌∅푋 +퐶푕 푌,∅푋 −푕 푋,푌 =0 for 푋,푌∈ 퐷⨁ 휉 . Replacing푌 푏푦 ∅푍,where 푍∈퐷, ∅푄 ∇∅푍∅푋 +퐶푕 ∅푍,∅푋 −푕 푋,∅푍 =0. (4.4) Interchanging 푋 and 푍, we have ∅푄 ∇∅푋∅푍 +퐶푕 ∅푋,∅푍 −푕 푍,∅푋 =0. (4.5)

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Subtracting (4.4) from (4.5), we obtain ∅푄 ∅푋,∅푍 +푕 푋,∅푍 −푕 푍,∅푋 =0. Since 퐷⨁ 휉 is integrable so that ∅푋,∅푍 ∈ 퐷⨁ 휉 for 푋,푌∈퐷. Consequently, above equation gives 푕 푋,∅푍 =푕 ∅푋,푍 . ⊡ Proposition 4.2.Let 푀 be a semi-invariant submanifold of a nearly hyperbolicKenmotsu manifold 푀 with semi- symmetric metricconnection, then 퐴∅푌푍−퐴∅푍푌= 13∅푃 푌,푍 for each 푌,푍∈퐷⊥. Proof.Let 푌,푍∈퐷⊥and 푋∈푇푀, from (2.13), we have
2푔 퐴∅푍푌,푋 =푔 푕 푌,푋 ,∅푍 +푔 푕 푋,푌 ,∅푍 . (4.6) Using (2.11) and (2.9) in (4.6), we have 2 푔 퐴∅푍푌,푋 =−푔 ∇푋∅푌,푍 −푔 ∇푌∅푋,푍)−푔(푕 푋,∅푌 +푕 푌,∅푋 ,푍 −2휂 푋 푔 ∅푌,푍 −2휂 푌 푔 ∅푋,푍 . (4.7) From (2.12) and (4.7), we have 2푔 퐴∅푍푌,푋 =−푔 ∅∇푌푍,푋 +푔 퐴∅푌푍,푋 . Transvecting푋 from both sides, we obtain 2퐴∅푍푌=−∅∇푌푍+퐴∅푌푍. (4.8) Interchanging푌 and 푍, we have 2퐴∅푌푍=−∅∇푍푌+퐴∅푍푌. (4.9) Subtracting(4.8) from (4.9), we have 퐴∅푌푍−퐴∅푍푌 = 13∅푃 푌,푍 . (4.10) Theorem4.3.Let 푀 be a semi-invariant submanifold of a nearly hyperbolic Kenmotsu manifold 푀 with semi- symmetric metric connection, then the distribution is integrable if and only if 퐴∅푌푍−퐴∅푍푌=0 (4.11) for all 푌,푍∈퐷⊥. Proof.Suppose that the distribution 퐷⊥ is integrable, that is 푌,푍 ∈퐷⊥. Therefore, 푃 푌,푍 =0 for any 푌,푍∈ 퐷⊥. Consequently, from (4.10) we have 퐴∅푌푍−퐴∅푍푌=0. Conversely, let (4.11) holds. Then by virtue of (4.10), we have either 푃 푌,푍 =0 or 푃 푌,푍 =푘휉.But푃 푌,푍 = 푘휉is not possible as 푃 being a projection operator on 퐷.So,푃 푌,푍 =0.This implies that 푌,푍 ∈퐷⊥for all 푌,푍∈퐷⊥. Hence 퐷⊥ is integrable.
V. Parallel Distribution

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Definition 5.1.The horizontal (resp., vertical) distribution 퐷(푟푒푠푝.,퐷⊥) is said to be parallel [7] with respect to the connection on 푀 푖푓 ∇푋푌∈퐷 (푟푒푠푝.,∇푍푊∈퐷⊥) for any vector field. Proposition 5.2.Let 푀 be a semi-invariant submanifold of a nearly hyperbolicKenmotsu manifold 푀 with semi- symmetric metric connection. If the horizontal distribution 퐷 is parallel then푕 푋,∅푌 =푕 푌,∅푋 for all푋,푌∈ 퐷. Proof.Let푋,푌∈퐷, as퐷 is parallel distribution.So that ∇푋∅푌∈퐷and∇푌∅푋∈퐷. Now, from (3.14) and (3.15), we have 푄 ∇푋∅푃푌 +푄 ∇푌∅푃푋 −푄퐴∅푄푌푋−푄퐴∅푄푋푌+푕 푌,∅푃푋 +푕 푋,∅푃푌 +∇푋 ⊥∅푄푌+∇푌 ⊥∅푄푋=2퐵푕 푋,푌 +∅푄 ∇푋푌 +∅푄 ∇푌푋 +2퐶푕 푋,푌 −2휂 푋 ∅푄푌−2휂 푌 ∅푄푋. As푄 being a projection operator on 퐷⊥, then we have 푕 푋,∅푃푌 +푕 푌,∅푃푋 =2퐵푕 푋,푌 +2퐶푕 푋,푌 . Using (2.14) and (2.15) in above, we have 푕 푋,∅푌 +푕 푌,∅푋 =2∅푕 푋,푌 . (5.1) Replacing푋 푏푦 ∅푋 in (5.1) and using (2.1), we have 푕 ∅푋,∅푌 +푕 푌,푋 =2∅푕 ∅푋,푌 . (5.2) Replacing푌 푏푦 ∅푌 in (5.1) and using (2.1), we have 푕 푋,푌 +푕 ∅푌,∅푋 =2∅푕 푋,∅푌 . (5.3) Comparing (5.2) and (5.3), we get 푕 푋,∅푌 =푕 푌,∅푋 for all푋,푌∈퐷. Definition 5.3.A Semi-invariant submanifold is said to be mixed totally geodesic if 푕 푋,푌 =0 for all푋∈ 퐷 and 푌∈퐷⊥. Proposition 5.4.Let 푀 be a semi-invariant submanifold of a nearly hyperbolicKenmotsu manifold 푀 with semi- symmetric metric connection. Then 푀 is a mixed totally geodesic if and only if 퐴푁푋∈퐷 푓표푟 푎푙푙 푋∈퐷. Proof.Let 퐴푁푋∈퐷 for all푋∈퐷. Now, 푔 푕 푋,푌 ,푁 =푔 퐴푁푋,푌 =0 푓표푟 푌∈퐷⊥, which is equivalent to 푕 푋,푌 =0. Hence 푀 is totally mixed geodesic. Conversely, let 푀 is totally mixed geodesic.That is 푕 푋,푌 =0 푓표푟 푋∈퐷 and 푌∈퐷⊥. Now, from푔 푕 푋,푌 ,푁 =푔 퐴푁푋,푌 , we get 푔 퐴푁푋,푌 =0. That is퐴푁푋∈퐷 for all 푌∈퐷⊥. References [1] Agashe N. S. and Chafle M. R., A semi-symmetric non-metric connection in a Riemannian manifold, Indian J. Pure Applied Math., 23 (1992), 399-409. [2] M. Ahmad, K. Ali, Semi-invariant submanifolds of a nearly hyperbolic Kenmotsu manifold with semi- symmetric non-metric connection, International J. of Math. Sci. &Engg.Appls.(IJMSEA), Vol. 7 No.IV(July, 2013), pp. 107-119. [3] Ahmad M. and Jun J. B., On semi-invariant submanifolds of a nearly Kenmotsu Manifold with semi- symmetric non-metric connection, Journal of the Chungcheong Math. Soc., 23(2) (June 2010), 257-266.

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