INVARIANT SUB-MANIFOLDS OF LP-SASAKIAN MANIFOLDS WITH SEMI-SYMMETRIC METRIC CONNECTIONS G

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ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 12 Issue 2(2020), Pages 35-44.

INVARIANT SUB-MANIFOLDS OF LP-SASAKIAN MANIFOLDS WITH SEMI-SYMMETRIC METRIC CONNECTIONS

G. SOMASHEKHARA, N. PAVANI AND P. SIVA KOTA REDDY¹

Abstract. The object of the present paper is to study some results on invariant sub-manifolds of LP-Sasakian manifolds endowed with semi-symmetric metric conection. Further, we have shown that theLP-Sasakian manifold is totally geodesic.

1. Introduction

In [11], Matsumoto defined the notion of Lorentzian para-Sasakian manifold.

The same notion was independently defined by Mihai and Rosca [13] and obtained several results. In the modern analysis, the geometry of sub-manifold has turned into a subject of growing interest for its significant applications in applied mathematics and theoretical physics. Submanifold theory is a very active vast research field which plays an important role in the development of modern differential geometry (See ). For instance, the notion of invariant submanifold is used to study the properties of non-linear autonomous system (See [9]). Also the notion of geodesics plays an important role in the theory of relativity (See [12]). Later on several authors studied Lorentzian almost paracontact manifolds, their different classes, but Lorentzian para-Sasakian manifolds and their sub-manifolds were initiated in [5, 6].

In this paper, we have notice that the invariant sub-manifolds of a LP-Sasakian manifolds satisfying the conditions:

P(X1, X2).∇h = 0,W(X1, X2).∇h = 0, P(X1, X2).∇h = f Q(g, h), P(X1, X2).∇h=f Q(S, h),W(X1, X2).∇h=f Q(S, h),W(X1, X2).∇h=f Q(g, h) , whereP denotes the pseudo-projective curvature tensor andW is the Weyl curvature tensor.

2. Preliminaries

Let (M, g) be ann-dimensional Riemannian sub-manifold of an (2n+1)-dimensional Riemannian manifold ( ˜M ,˜g) endowed with an almost contact metric structure

1Corresponding author: [email protected]

2000Mathematics Subject Classification. 53C15, 53C40, 53C50.

Key words and phrases. Sub-manifold, Invariant sub-manifold,LP-Sasakian manifold, Semi- Symmetric Metric, Pseudo-Projective Curvature tensor, Weyl curvature tensor, Totally geodesic.

c

2020 Universiteti i Prishtinës, Prishtinë, Kosovë.

Submitted December 27, 2019. Published May 25, 2020.

Communicated by U. C. De.

35

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(φ, ξ, η, g), where φ is a (1,1) tensor field, ξ is a vector field , η is a one-form andg is a compatible Riemannian metric on ¯M. That is,

φ²(X1) =X1+η(X1)ξ, η(ξ) =−1,

˜

g(φX1, φX2) = ˜g(X1, X2) +η(X1)η(X2), (2.1) φξ= 0, η(φX1) = 0, ˜g(X1, φX2) = ˜g(φX1, X2), (2.2) for all vector fieldsX1, X2. The manifold with the structure (φ, ξ, η,g) is called a˜ Lorentzian almost paracontact manifold.

In the Lorentzian almost paracontact manifold ¯M the following relation hold:

φξ = 0, η(φX1) = 0,˜g(X1, φX2) = ˜g(φX1, X2). (2.3) We denote by ∇ and ∇ the Levi-Cevita connections of M and M, respectively.

Then for any vector fields X1, X2 ∈ Γ(T M), the second fundamental form h is given by:

∇_X₁X₂=∇_X₁X₂+h(X₁, X₂).

Furthermore, for any sectionN of normal bundleT^⊥M we have

∇X₁N =−ANX1+∇^⊥_X

1N,

where ∇^⊥ denotes the normal bundle connection ofM. The second fundamental formhand shape operatorAN are related by

g(ANX1, X2) =g(h(X1, X2), N).

A sub-manifoldM is said to be totally geodesic ifh= 0. For Riemannian manifold M, we have:

Q(E, T)(Y₁, Y₂, ..., Y_k;X₁, X₂) =−T((X₁∧_EX₂)Y₁, Y₂,· · · , Y_k)

−T(X1,(X1∧EX2)Y2,· · · , Yk)

· · ·T(Y1, Y2,· · · , Y_k−1,(X1∧EX2)Yk), (2.4)

where (X1∧EX2)X3=E(X2, X3)X1−E(X1, X3)X2. We haveM is said to be pseudo-parallel if

R(X1, X2).h=f Q(g, h).

ForLP-Sasakian manifold, the following relations hold (See [11]):

(∇_X₁φ)X₂= ˜g(φX₁, φX₂)ξ+η(X₂)φ²X₁,

where∇denotes the operator of covariant differentiation with respect to the Lorentzian metric ˜g. Further, the LP-Sasakian manifold ¯M with the structure (φ, ξ, η, g), we have

∇˜X₁ξ=φX1, (2.5)

R(ξ, X˜ 1)X2= ˜g(X1, X2)ξ−η(X2)X1, (2.6) R(X˜ ₁, X₂)ξ=η(X₂)X₁−η(X₁)X₂, (2.7) S(X˜ ₁, ξ) = (n−1)η(X₁), (2.8)

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for all vector fieldsX1, X2 on ¯M [14], where ¯S denotes the Ricci tensor of ¯M and R¯ is the curvature tensor of ¯M.

A sub-manifoldM of aLP-Sasakian manifold ¯M is called an invariant submanifold of ¯M ifφ(T M)⊂T M. An invariant sub-manifold of aLP-Sasakian manifold is given by

h(X1, ξ) = 0, (2.9)

h(X₁, φX₂) =φh(X₁, X₂) =h(φX₁, X₂) (2.10) (∇X1φ)X2=g(X1, X2)ξ+η(X2)X1+ 2η(X1)η(X2)ξ, (2.11) for any vector fieldX1 tangent toM. For the second fundamental form, the first derivatives ofhis defined by

(∇_X₁)h(X₂, X₃) =∇^⊥_X

1h(X₂, X₃)−h(∇_X₁X₂, X₃)−h(X₂,∇_X₁X₃) (2.12) for any vector fieldX₁, X₂, X₃ tangent toM. Then∇his a normal bundle valued tensor type (0,3).

If ∇h = 0 then M is said to have parallel second fundamental form or the sub-manifoldM is said to be parallel. An immersion is said to be semi parallel if

R(X₁, X₂).h= (∇X1∇X2− ∇X2∇X1− ∇[X₁,X₂])h= 0 (2.13) for all vector fieldX1, X2tangent toM. The notionRdenotes the curvature tensor of the connection∇.

In [2, 3], the authors have studied the semi-parallel immersion and Arslan et al.

[1] defined sub-manifolds, which satisfies the condition:

R(X₁, X₂).∇h= 0 (2.14)

for all vector fields X1, X2 tangent to M and such sub-manifolds are called 2- semiparallel. We now have:

(R(X1, X2).∇h)(X3, U, V) =R^⊥(X1, X2)(∇h)(X3, U, V)−

(∇h)R(X₁, X₂)X₃, U, V)−(∇h)(X₃, R(X₁, X₂)U, V)−

(∇h)(X3, U, R(X1, X2)V). (2.15)

Let (M, g) be an (2n+ 1)-dimensional Riemannian manifold then the Pseudo- projective curvature tensor and Weyl curvature tensor respectively are defined by

P(X1, X2)X3=aR(X1, X2)X3+b[S(X2, X3)X1−S(X1, X3)X2]

−( r

2n+ 1)(( a

2n) +b)[g(X2, X3)X1−g(X1, X3)X2], (2.16) W(X₁, X₂)X₃=R(X₁, X₂)X₃−( 1

2n)[S(X₂, X₃)X₁−S(X₁, X₃)X₂], (2.17) whereaandb are constants andS is the Ricci tensor.

The notion of recurrent tensor was introduced by Roter [14]. Further, the sub- manifolds with recurrent tensors was studied by Sular and Ozgur [16].

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A semi-symmetric connection∇e is called semi-symmetric metric connection if it satisfies

∇ge = 0.

The relation between semi-symmetric metric connection ∇˜ and the Remannian connection∇e of a LP-Sasakian manifoldM⁽²ⁿ⁺¹⁾ is given by

∇e_X₁X₂=∇_X₁X₂+η(X₂)X₁−g(X₁, X₂)ξ (2.18) If R and Re are the Remannian curvature tensor of LP-Sasakian manifolds with respect to Levi-Civita connection and semi-symmetric connection then

R(Xe ₁, X₂)X₃=R(X₁, X₂)X₃− (2.19) α(X2, X3)X1+α(X1, X3)X2−g(X2, X3)LX1+g(X1, X3)LX2,

whereαis a (0,2) tensor field and is given by:

α(X₁, X₂) = (∇eX1η)X₂+1

2g(X₁, X₂) (2.20)

LX₁=∇eX1ξ+1

2X₁ (2.21)

g(LX1, X2) =α(X1, X2) (2.22)

S(Xe 1, X2) =S(X1, X2)−(2n−1)α(X1, X2)−ag(X1, X2) (2.23)

er(X₁, X₂) =r−4nc (2.24)

wherec= trace(α),Se anderare the Ricci tensor and scalar curvature with respect to semi-symmetric metric connection. In fact, the notion semi-symmetric metric connection in a Riemannian manifold was introduced by Yano [18]. It was sepa- rately also introduced by Golab [8]. In [8], the author also introduced the idea of a quarter symmetric linear connection in differentiable manifolds.

With respect to Levi-Civita connection, the following relations hold:

R(Xe ₁, X₂)ξ=1

2[η(X₂)X₁−η(X₁)X₂] +η(X₂)LX₁+η(X₁)LX₂ (2.25) R(ξ, Xe ₁)X₂=1

2[3g(X₁, X₂)ξ−η(X₂)X₁]−α(X₁, X₂)ξ+η(X₂)LX₁ (2.26) S(Xe ₁, ξ) =−(1

2 +a)η(X₁) (2.27)

QXe ₁=−(1

2 +a)X₁ (2.28)

for all arbitrary vector fieldsX₁, X₂andX₃onM.

3. Invariant Sub-manifolds of LP-Sasakian Manifolds Satisfying P(X1, X2).∇h=f[Q(g,∇h)]

In view of the equation (2.15), the pseudo-projective curvature tensor in (2.16) becomes:

(P(X₁, X₂).∇h) =f[Q(g, h)], (3.1)

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R^⊥(X1, X2)(∇h)(X3, U, V)−(∇h)P(X1, X2)X3, U, V)−(∇h)(X3, P(X1, X2)U, V)

−(∇h)(X3, U, P(X₁, X₂)V) =−f[g(X₂, X₃)∇h(X1, U, V)−g(X₁, X₃)∇h(X2, U, V) +g(X2, U)∇h(X3, X1, V)−g(X1, U)∇h(X3, X2, V) +g(X2, V)∇h(X3, U, X1)

−g(X1, V)∇h(X3, U, X2)],

(3.2) Since (X₁∧EX₂)X₃ = E(X₂, X₃)X₁−E(X₁, X₃)X₂. Now put X₁ = U = ξ in (3.2), we obtain

R^⊥(ξ, X2)(∇h)(X3, ξ, V)−(∇h)P(ξ, X2)X3, ξ, V)−(∇h)(X3, P(ξ, X2)ξ, V)−

(∇h)(X3, ξ, P(ξ, X2)V) =−f[g(X2, X3)∇h(X1, ξ, V)−g(ξ, X3)∇h(X2, ξ, V)+

g(X₂, ξ)∇h(X3, ξ, V)−g(ξ, ξ)∇h(X3, X₂, V) +g(X₂, V)∇h(X3, ξ, ξ)−

g(ξ, V)∇h(X3, ξ, X2)].

(3.3) With reference to the equations (2.16), (2.25), (2.26) and (2.27), the equation (3.3) becomes:

−R^⊥(ξ, X2)h(φX3, V)−a

2η(X3)h(φX2, V) +aη(X3)h(LX2, φV)−b(n−1) ηX₃h(φX₂, V) +b(2n−1)

2 η(X₃)h(φX₂, V) +abη(X₃)h(φX₂, V)− ∇^⊥_X

3

[(a−b(1

2 +a)) + r (2n+ 1)( a

2n+b)h(X₂, V)−ah(φX₂, V)]−a[h(X₂,∇_X₃V)−

h(φX2,∇X₃V)] + [b(1

2 +a) + r (2n+ 1)( a

2n+b)]h(X2,∇X₃V)−aη(V)h(φX3, φX2)+

ah(φX₃, X₂)η(V) =−f[∇_X₃h(X₂, V)].

(3.4) Now putV =ξ in above equation, we get,

−f−2a+b 1

2 +a

+ r

2n+ 1 a

2n +b

h(X2, φX3) = 0. (3.5) Thus in the equation (3.5),h= 0 if and only if

f 6=

r 2n+ 1

a 2n +b

−2a+b 1

2 +a

(3.6) Conversely, ifM⁽²ⁿ⁺¹⁾ be totally geodesic, then we getM⁽²ⁿ⁺¹⁾satisfies

(P(X₁, X₂).∇h) =f[Q(g, h)].

Hence, we can state the following result:

Theorem 3.1. Let M⁽²ⁿ⁺¹⁾ be an invariant sub-manifold of a LP-Sasakian manifold of M¯ with semi-symmetric metric connection. Then M⁽²ⁿ⁺¹⁾ satisfies (P(X1, X2).∇h) =f[Q(g, h)] if and only ifM⁽²ⁿ⁺¹⁾ is totally geodesic provided,

f 6=

r 2n+ 1

a 2n+b

−2a+b 1

2 +a

).

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4. Invariant Sub-manifolds of LP-Sasakian Manifolds satisfying P(X1, X2).∇h=f[Q(S,∇h)]

LetM⁽²ⁿ⁺¹⁾be an invariant sub-manifold of aLP-Sasakian manifold with semi- symmetric metric connection, satisfying

P(X1, X2).∇h=f[Q(S,∇h)] (4.1) for all vector fieldsX₁, X₂ tangent toM, wheref denotes the real valued function onM⁽²ⁿ⁺¹⁾(4.1) can be written as

R^⊥(X1, X2)(∇h)(X3, U, V)−(∇h)P(X1, X2)X3, U, V)

−(∇h)(X3, P(X1, X2)U, V)−(∇h)(X3, U, P(X1, X2)V)

=−f[S(Xe ₂, X₃)∇h(X1, U, V)−S(Xe ₁, X₃)∇h(X2, U, V) +S(Xe 2, U)∇h(X3, X1, V)−S(Xe 1, U)∇h(X3, X2, V)

+S(Xe 2, V)∇h(X3, U, X1)−S(Xe 1, V)∇h(X3, U, X2)]. (4.2) Since (X₁∧_EX₂)X₃=E(X₂, X₃)X₁−E(X₁, X₃)X₂.

Now putX₁=U =ξin (4.2) we get,

R^⊥(ξ, X2)(∇h)(X3, ξ, V)−(∇h)P(ξ, X2)X3, ξ, V)

−(∇h)(X3, P(ξ, X2)ξ, V)−(∇h)(X3, ξ, P(ξ, X2)V)

=−f[S(Xe 2, X3)∇h(ξ, ξ, V)−S(ξ, Xe 3)∇h(X2, ξ, V)

−S(ξ, ξ)∇h(Xe ₃, X₂, V) +S(Xe ₂, ξ)∇h(X₃, ξ, V))

+S(Xe 2, V)∇h(X3, ξ, ξ−S(ξ, Ve )∇h(X3, ξ, X2)]. (4.3) Now with the reference of (2.15), (2.16), (2.19), (2.25),(2.26) and (2.27) in (4.3), we get:

−R^⊥(ξ, X2)h(φX3, V)−a

2η(X3)h(φX2, V) +aη(X3)h(LX2, φV)−

b(n−1)ηX₃h(φX₂, V) +b(2n−1)

2 η(X₃)h(φX₂, V) +abη(X₃)h(φX₂, V)−

∇^⊥_X₃[(a−b(1

2 +a)) + r (2n+ 1)( a

2n+b)h(X₂, V)−ah(φX₂, V)]−a[h(X₂,∇X3V)−

h(φX2,∇X₃V)] + [b(1

2 +a) + r (2n+ 1)( a

2n+b)]h(X2,∇X₃V)−aη(V)h(φX3, φX2)+

ah(φX3, X2)η(V) =−f[S(ξ, Xe 3)h(φX2, V)−S(Xe 2, ξ)h(φX3, V)

−S(ξ, ξ)[∇e ^⊥h(X₂, V)−h(∇_X₃X₂, V)−h(X₂,∇_X₃V)] +S(ξ, Ve )h(φX₃, X₂)]

[(2f+b)(1

2+a) + r (2n+ 1)( a

2n+b)]h(X2, φX3) = 0. (4.5) Therefore by using (4.4) we get,

f 6=−b

2 − r

(2n+ 1)(1 + 2a) a

2n+b

. (4.6)

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Theorem 4.1. Let M⁽²ⁿ⁺¹⁾ be an invariant sub-manifold of a LP-Sasakian manifold of M¯ with semi-symmetric metric connection. Then M⁽²ⁿ⁺¹⁾ satisfies (P(X1, X2).∇h) =f[Q(S, h)]if and only if M⁽²ⁿ⁺¹⁾ is totally geodesic provided,

f 6=−b

2 − r

(2n+ 1)(1 + 2a) a

2n+b .

5. Invariant sub-manifolds of LP-Sasakian manifolds satisfying P(X1, X2).∇h= 0

SinceP(X₁, X₂).∇h= 0 and using (2.16), (2.18), (3.2), we have [b(1

2 +a) + r (2n+ 1)( a

2n +b)−c)]φh(X2, X3) = 0 (5.1) we haveh= 0 if and only if

r6= −bn(2n+ 1)(1 + 2a)

a+ 2nb (5.2)

ThereforeM is totally geodesic if and only ifMis pseudo-projectively 2-semiparallel.

Theorem 5.1. Let M⁽²ⁿ⁺¹⁾ be an invariant sub-manifold of a LP-Sasakian manifold ofM¯ if and only ifM⁽²ⁿ⁺¹⁾ is pseudo-projectively 2-semiparallel provided

r6= −bn(2n+ 1)(1 + 2a)

a+ 2nb .

6. Invariant sub-manifolds of LP-Sasakian manifolds satisfying W(X₁, X₂).∇h=f[Q(g,∇h)]

Using (2.15), the Weyl curvature tensor in (2.17) becomes:

(W(X1, X2).∇h) =f[Q(g, h)] (6.1)

R^⊥(X1, X2)(∇h)(X3, U, W)−(∇h)W(X1, X2)X3, U, V)−(∇h)(X3, W(X1, X2)U, V)

−(∇h)(X₃, U, W(X₁, X₂)V) =−f[g(X₂, X₃)∇h(X₁, U, V)−g(X₁, X₃)∇h(X₂, U, V) +g(X2, U)∇h(X3, X1, V)−g(X1, U)∇h(X3, X2, V) +g(X2, V)∇h(X3, U, X1)

−g(X1, V)∇h(X3, U, X2)]. (6.2)

Since (X1∧EX2)X3=E(X2, X3)X1−E(X1, X3)X2. Now putX1=U =ξin above equation we get:

R^⊥(ξ, X₂)(∇h)(X3, ξ, V)−(∇h)W(ξ, X₂)X₃, ξ, V)−(∇h)(X3, W(ξ, X₂)ξ, V)

−(∇h)(X3, ξ, W(ξ, X2)V) =−f[g(X2, X3)∇h(X, ξ, V)−g(ξ, X3)∇h(X2, ξ, V) +g(X2, ξ)∇h(X3, ξ, V)−g(ξ, ξ)∇h(X3, X2, V)

+g(X₂, V)∇h(X₃, ξ, ξ)−g(ξ, V)∇h(X₃, ξ, X₂)]. (6.3)

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Now with the reference of (2.16), (2.25), (2.26) and (2.27) in (3.3) we get:

−R^⊥(ξ, X2)h(φX3, V)−1

2η(X3)h(φX2, V)−aη(X3)h(LX2, V)

−(n−1)

2n η(X₃)h(φX₂, V) +2n−1

4n η(X₃)h(φX₂, V) + a

2nη(X3)h(φX2, V)−1

2h(X2,∇V)−h(LX2,∇V) +(1)

2n[−(n−1)h(X2, ξ, X3) +(2n−1)

2n h(X2, ξ, X3) +ah(X2, ξ, X3)]

−η(V)h(φX3, X2) +η(V)h(φX3, φX2)− 1 2n(1

2+aη(V)h(φX3, X2)

= 2f φh(X2, X3). (6.4)

Now putV =ξ in above equation, we get,

(2f+ 1)φh(X2, X3) = 0. (6.5)

Therefore in the above equation we haveh= 0 if and only if f 6=

−1 2

(6.6) Conversely, if M⁽²ⁿ⁺¹⁾ be totally geodesic, then we get M⁽²ⁿ⁺¹⁾ satisfies (W(X1, X2).∇h) =f[Q(g, h)]. Thus we can state that

Theorem 6.1. Let M⁽²ⁿ⁺¹⁾ be an invariant sub-manifold of a LP-Sasakian manifold of M¯ with semi-symmetric metric connection. Then M⁽²ⁿ⁺¹⁾ satisfies (W(X₁, X₂).∇h) =f[Q(g, h)]if and only if M⁽²ⁿ⁺¹⁾ is totally geodesic provided,

f 6=

−1 2

.

7. Invariant sub-manifolds of LP-Sasakian manifolds satisfying P(X1, X2).∇h=f[Q(S,∇h)]

LetM⁽²ⁿ⁺¹⁾be an invariant sub-manifold of aLP-Sasakian manifold with semi- symmetric metric connection, satisfying

(W(X1, X2).∇h) =f[Q(S, h)] (7.1)

R^⊥(X1, X2)(∇h)(X3, U, V)−(∇h)W((X1, X2)X3, U, V)−

(∇h)(X₃, W(X₁, X₂)U, V)−(∇h)(X₃, U, W(X₁, X₂)V) =

−f[S(Xe 2, X3)∇h(X1, U, V)−S(Xe 1, X3)∇h(X2, U, V) +S(Xe 2, U)∇h(X3, X1, V)−S(Xe 1, U)∇h(X3, X2, V)+

S(Xe ₂, V)∇h(X₃, U, X₁)−S(Xe ₁, V)∇h(X₃, U, X₂)]. (7.2)

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Since (X1∧EX2)X3=E(X2, X3)X1−E(X1, X3)X2, now putX1=U =ξ in the above equation, we get

R^⊥(ξ, X2)(∇h)(X3, ξ, V)−(∇h)W(ξ, X2)X3, ξ, V)−(∇h)(X3, W(ξ, X2)ξ, V)−

(∇h)(X3, ξ, W(ξ, X2)V) =−f[S(Xe 2, X3)∇h(ξ, ξ, V)−eS(ξ, X3)∇h(X2, ξ, V)+

S(Xe ₂, ξ)∇h(X₃, ξ, V)−S(ξ, ξ)∇h(Xe ₃, X₂, V) +S(Xe ₂, V)∇h(X₃, ξ, ξ)−

S(ξ, Ve )∇h(X3, ξ, X2)]. (7.3)

In view of (2.15), (2.16), (2.19), (2.25), (2.26) and (2.27), the above equation becomes:

−R^⊥(ξ, X₂)h(φX₃, V)−1

2η(X₃)h(φX₂, V)−aη(X₃)h(LX₂, V)−(n−1) 2n η(X₃) h(φX₂, V) +2n−1

4n η(X₃)h(φX₂, V) + a

2nη(X₃)h(φX₂, V)−1

2h(X₂,∇V)−

h(LX2,∇V) +(1)

2n[−(n−1)h(X2, ξ, X3) +(2n−1)

2n h(X2, ξ, X3) +ah(X2, ξ, X3)]−

η(V)h(φX3, X2) +η(V)h(φX3, φX2)− 1 2n(1

2 +aη(V)h(φX3, X2) =

−2f(1

2 +a)φh(X2, X3).

−1 + (2f(¹₂+a)

φh(X2, X3) = 0.

Therefore by using the above equation we get, f 6= 1

1 + 2a.

Theorem 7.1. Let M⁽²ⁿ⁺¹⁾ be an invariant sub-manifold of a LP-Sasakian manifold of M¯ with semi-symmetric metric connection. Then M⁽²ⁿ⁺¹⁾ satisfies (W(X₁, X₂).∇h) =f[Q(S, h)]if and only ifM⁽²ⁿ⁺¹⁾is totally geodesic provided,

f 6= 1 1 + 2a.

8. Invariant sub-manifolds of LP-Sasakian manifolds satisfying W(X1, X2).∇h= 0

SinceW(X₁, X₂).∇h= 0 and using (2.16), (2.18), we have

−2n−a+b

2n = 0. (8.1)

Thenh= 0 if and only if

n6= (a−b

2 ). (8.2)

ThereforeM is totally geodesic if and only ifMis pseudo-projectively 2-semiparallel.

Theorem 8.1. Let M⁽²ⁿ⁺¹⁾ be an invariant sub-manifold of a LP-Sasakian manifold of M¯ with semi-symmetric connection. Then M⁽²ⁿ⁺¹⁾ satisfies W(X1, X2).∇h= 0 if and only ifM⁽²ⁿ⁺¹⁾ is totally geodesic provided

n6= (a−b

2 ). (8.3)

(10)

Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this manuscript.

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G. Somashekhara

Department of Mathematics, Ramaiah University of Applied Sciences, Bengaluru – 560 054, India

E-mail address:[email protected]

N. Pavani

Department of Mathematics, Sri Krishna Institute of Technology, Bengaluru – 560 090, India

P. Siva Kota Reddy

Department of Mathematics, Sri Jayachamarajendra College of Engineering, JSS Science and Technology University, Mysuru – 570 006, India