On pseudo-slant submanifolds of nearly quasi-Sasakian manifolds Текст научной статьи по специальности «Физика»

www.volsu.ru

D0I: https://doi.Org/10.15688/mpcm.jvolsu.2019.4.3

UDC 004.89 Submitted: 04.06.2019

LBC 55.6 Accepted: 03.10.2019

ON PSEUDO-SLANT SUBMANIFOLDS OF NEARLY QUASI-SASAKIAN MANIFOLDS

Shamsur Rahman

PhD, Assistant Professor, Department of Mathematics, Maulana Azad National Urdu University shamsur@rediffmail.com

Polytechnic Satellite Campus Darbhanga,846002 Bihar, India

Mohd Sadiq Khan

PhD, Department of Mathematics, Shibli National College msadiqkhan.snc@gmail.com Azamgarh, Uttar Pradesh 276001, India

Aboo Horaira

Department of Mathematics,

Shibli National College

aboohoraria@gmail.com

Azamgarh, Uttar Pradesh 276001, India

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Abstract. The geometry of pseudo-slant submanifolds of nearly quasi Sasakian manifold is studied. It is proved that totally umbilical proper-slant sub-manifold of nearly quasi Sasakian manifold admits totally geodesic if the mean curvature vector H G \x . The integrability conditions of the distributions of pseudo-slant submanifolds of nearly quasi Sasakian manifold are also obtained.

~ Key words: Nearly quasi-Sasakian manifold, slant submanifold, proper

§ slant submanifold, pseudo-slant submanifold.

Introduction

E The notion of a slant submanifolds as natural generallization of both holomorphic and totally

real immersions was given by B.Y. Chen [6]. Latter many research articles have been @ appeared on the existence of these submanifolds in various know spaces. The properties

of slant submanifolds of an almost contact metric manifolds were studied by A. Lotta [10]. L. Cabrerizo et.al [8] were defined slant submanifolds of Sasakian manifolds. N. Papagiuc [12], introduced and studied the notion of semi-slant submanifolds of an almost Hermitian manifold. A. Carrizo [5; 7] defined hemi-slant submanifolds. The contact version of pseudo-slant submanifold in a Sasakian manifold have been studied by V.A. Khan et.al [11]. In [13] the authors studied nearly quasi Sasakian manifold. The purpose of the paper is to study the notion of pseudo-slant submanifold of nearly quasi Sasakian manifold. In section 1 we recall some results and formula later use. In section 2 we define pseudo-slant submanifold of nearly quasi Sasakian manifold. In section 3 it is concern with the integrability of the distribution on the pseudo slant submanifolds of nearly quasi Sasakian manifold and obtain some characterizations. In section 4 we obtain a classification theorem for totally umbilical pseudo-slant submanifold M of nearly quasi Sasakian manifold M.

1. Preliminaries

Let M be a real (2n + 1) dimensional differentiable manifold endowed with an almost contact metric structure (/, £,,r|,g), where / is a tensor field of type (1,1), £, vector field, r| is a 1-form and g is Riemannian metric on M such that

(a) /2 = -J + n®£„ (6) r|(£,) = 1, (c) tio/ = 0,

(d) /(£,)= 0, (e) t!(X)=£(X£,), (1)

(/) g( fX, fY) = g(X, Y) - n(X)n(y)

for any vector field A", Y tangent to M, where I is the identity on the tangent bundle TM of M. An almost contact metric structure (/, £,,r|, g) on M is called quasi-Sasakian manifold if

(V y/)Y' = ti(Y)AX - g(AX, Y)E,, (2)

where A a symmetric linear transformation field, V denotes the Riemannian connection of g on M. If in a addition to above relations

(yxf)Y + (yYf)X = n(Y)AX +r\(X)AY - 2 g(AX, ¥)£,, (3)

then, M is called a nearly quasi-Sasakian manifold. We have also on nearly quasi-Sasakian manifold M

= fAX. (4)

Now, let M be a submanifold immersed in M. The Riemannian metric induced on M is denoted by the same symbol g. Let PM and PLM be the Lie algebras of vector fields tangential to M and normal to M respectively and N be the induced Levi-Civita connection on M, then the Gauss and Weingarten formulas are given by

VXY = VxY + h(X,Y), (5)

VXV = -AvX + VjtV (6)

for any X, Y e PM and V £ PLM, where V^ is the connection on the normal bundle PLM, h is the second fundamental form and Ay is the Weingarten map associated with V as

g(AvX,Y) = g(h(X,Y),V). (7)

For any X G PM and V E P±M, we write

fX = PX + VX (PX e PM and VX e P±M), (8)

fV = tV + nV (tV e PM and nV e P±M). (9)

The submanifold M is invariant if V is identically zero. On the other hand, M is anti-invariant if P is identically zero. From (1) and (8), we have

g(X,PY) = -g(PX,Y) (10)

for any X, Y e PM. If we put Q = P2 we have

(VxQ)Y = VXQY - QVXY, (11)

(VA:P)Y = VA-PY - PVxY., (12)

(VXV)Y = V^VY-VVXY (13)

for any X, Y e PM. In view of (5), (8), and (4) it follows that

Val = PAX, (14)

h(X, £,) = VAX. (15)

The mean curvature vector H of M is given by

1 1 n

H = — trace (h) = — } h(ei, e«), (16)

n n ^^

i=i

where n is the dimension of M and e±, e2, •••, en is a local orthonormal frame of M. A submanifold M of an contact metric manifold M is said to be totally umbilical if

h{X.Y) = g{X.Y)H, (17)

where H is the mean curvature vector. A submanifold M is said to be totally geodesic if h(X, Y) = 0, for each X, Y G T(PM) and M is said to be minimal if H = 0 .

2. Pseudo-slant submanifolds of nearly quasi-Sasakian manifolds

The purpose of this section is study the existence of pseudo-slant submanifolds of nearly quasi-Sasakian manifolds

Definition 1. Let M be a submanifold of a nearly quasi-Sasakian manifold M . For each non-zero vector X tangent to M at x, the angle 9(x) G [0, tt/2] , between f X and PX is called the slant angle or the Wirtinger angle of M . If the slant angle is constant for each X e T(PM) and X e M, then the submanifold is also called the slant submanifold. If 0 = 0 the submanifold is invariant submanifold. If 0 = n/2 then it is called anti-invariant submanifold. If 0(x) e [0, tt/2] , then it is called proper-slant submanifold.

Now, we will give the definition of pseudo-slant submanifold which are a generalization of the slant submanifolds.

Definition 2. We say that M is a pseudo-slant submanifold of nearly quasi Sasakian manifold M if there exist two orthogonal distributions DL and De on M such that

1) PM admits the orthogonal direct decomposition PM = DL © De , £, = T(D).

2) The distribution D± is anti-invariant i.e., f(DL) C P±M.

3) The distribution Dg is a slant with slant angle 0^0, that is, the angle between f(De) and Dq is a constant.

From the definition, it is clear that if 0 = 0 , then the pseudo-slant submanifold is a semi invariant submanifold. On the other hand, if 0 = n/2, submanifold becomes an anti-invariant.

On the other hand we suppose that M is a pseudo-slant submanifold of nearly quasi Sasakian manifold M and we denote the dimensions of distributions DL and Dg by d\ and d2, respectively, then we have the following cases:

1) If d2 = 0 then M is an anti-invariant submanifold.

2) If d\ = 0 and 0 = 0, then M is an invariant submanifold.

3) If d\ = 0 and 0^0, then M is a proper slant submanifold with slant angle 0.

4) If dx.d2 ^ 0 and 0 e [0, n/2] then M is a proper pseudo-slant submanifold.

Theorem 1. Let M be a submanifold of a nearly quasi-Sasakian manifold M such that e PM. Then M is slant if and only if there exists a constant A e [0,1] such that

Furthermore, in such a case if 0 is the slant angle of M then A = cos2Q. Corollary 1. Let M be a slant submanifold of nearly quasi-Sasakian manifold M with slant angle 0. Then for any X, Y e T(PM) we have

Let M a proper pseudo slant submanifold of a contact metric manifold M and the projections on Dl and De by Pi and P2, respectively, then for any vector field X e T(PM), we can write

P2 = -Л{/ - л (g> 1}

(18)

д(РХ, PY) = cos2d(g(X, Y) - ц(Х)л(У)), g(VX, VY) = sin2e(g{X, Y) - ц(Х)ф')).

(19)

(20)

X = PlX + P2X+ n(X)f, Now applying / on both sides of equation (3.4), we obtain

(21)

fX = j'P]X + fP2X

that is,

PX + VX = VPxX + PP2X + VP2X

(22)

We can easily to see

PX = PP2X, VX = VPi X + VP2X and

(23)

fPiX = VPiX, TF\X = 0, fP2X = TP2X + VP2X, TP2X e T(De)

(24)

If we denote the orthogonal complementary of fPM in DLM by |i, then the normal bundle PLM can be decomposed as follows

P±M = V(D±) © V(DQ) © |lx, (26)

where (x is an invariant sub bundle of PLM as N(D±) and N(De) are orthogonal distribution on M. Indeed, g{Z,X) = 0 for each Z G T(JD±) and X G T(De). Thus, by equation (1) and (25), we can write

g(VZ, VX) = g(fZ, fX) = g(Z, X) = 0, (27)

that is, the distributions V(D±) and V(De) are mutually perpendicular. In fact, the decomposition (26) is an orthogonal direct decomposition.

3. Integrability of Distributions

In this section we shall discuss the integrability of involved distributions. Theorem 2. Let M be a pseudo-slant submanifold of nearly quasi Sasakian manifold M. Then for all X, Y G DL we have

AfyX - AfXY = WX(PY) + h(X, PY) - AVVX + V{-(VY) --P(VXY) - V(VXY) - V(h(X, Y)).

(28)

Proof. In view of (7), we get

g(AfYX, Z) = g(h(X, Z), fY) = -g(fh(X, Z), Y). (29)

From (5) and (29), we get

g(AfYX, Z) = -g(fVzX, Y) + g(f\/zX, Y) = -g(fVzX,Y) since JXx.\ ( I'M (30)

= g((Vzf)X,Y)-g(VzfX,Y).

Now, for X G £>i, fX G P±M. Hence, from (6) we have

X zfX = —AfXZ + XzfX. (31)

Combining (30) and (31), we obtain

g(AfYX, Z) = g((Xzf)X, Y) + g(AfXZ, Y). (32)

Since h{X,Y) = h{Y,X), if follows from (7) that

g(AfXZ,Y)=g(AfXY,Z).

Hence, from (32) we obtain, with the help of (3),

g(AfYX, Z) - g(AfXY, Z) = л(Z)g(AX, Y) + Л(X)g(AZ, Y) --2g(AZ, Х)ц(У) + g{(Vxf)Y., Z) = = Л (Z)g(AX, Y) + Л (X)g(AZ, Y) - 2g{AZ, Х)л(У) + +g(Xx(PY) + V v{PY) - f(yxY) - f(h(X, Y)), Z) = (33)

= л (Z)g(AX, Y) + л (X)g(AZ, Y) - 2Л(Y)g(AZ, X) +

+g{Vx{PY) + h(X, PY) - Avy(X) + -

-P(XxY) - V(XxY) - P(h(X, Y) - V{h(X, Y))), Z).

Since X.Y.Z e Dl an orthonormal distribution to the distribution (£,) it follows that r|(X) = r|(F) = 0. Therefore, the above equation reduces to

AfyX - AfXY = WX{PY) + h{X, PY) - AvyX + V^(VF) --P(yxY) - V(WXY) - V(h(X, Y)).

Theorem 3. In a pseudo-slant submanifold of a nearly quasi Sasakian manifold is given by

(VXP)Y = AyyX + AyxY + th(X, Y) + r}{Y)AX + ^X)AY -

- 2g(AX, Y)E, + Vy {PX) + P(VyX) + P(h(Y, X)).

Proof. Let X, Y e PM, we have

Vx/F = (Vxf)Y + f(VxY)

From (8) and (9), we obtain

WXPY + VxVF = (yxf)Y + f(yxY + h(X, Y))

Also from (8) and (9), we obtain

VxPY + VxVV = (Vxf)Y + PiyxY) + V(VXY) + th(X, Y) + nh(X, Y)

Using (5) and (6) from above, we obtain

VXPY + h(X, PY) - AVYX + Vj^VY = ri(Y)AX +1](Y)AX -- 2g(AX, Y)E, + P(yxY) + V(yxY) + th(X, Y) + nh(X, Y) --VyPX - h{Y, PX) + - V^l/X + PVyX +

+VVYX + P(h(Y, X)) + V(h(Y, X)).

Comparing tangential and normal parts, we obtain

VXPY - AyyX = t\(Y)AX + y\(X)AY - 2g{AX, Y)E, + P{VXY) + +th{X, Y) + Vy (PX) + ,lrvV + P(VyX) + P(/i(Y, X)).

That is,

(VA P)Y = AyyX + AVXY + i/i(X, Y) +r\{Y)AX + n{X)AY -

- 2g(AX, Y)l + Vy (PX) + P(VyX) + P(/i(F, X)).

(35)

(36)

(37)

Theorem 4. Let M be a pseudo-slant of a nearly quasi Sasakian manifold M. Then the anti-invariant distribution DL is integrable if and only if for any Z, W G r(D^)

Ay\yZ + + 2TXZW + 2th(W, Z) = -t\(W)AZ - r\(Z)AW + 2g(AZ, W)E,. (38) ^

Proof. Let Z, W E Г (В1) and using (3), we obtain

(Vzf)W + (Vwf)Z = r\{W)AZ + r\(Z)AW - 2g(AZ, W)t,,

which is equivalent to

VzfW - fVzW + VwfW - fWwZ = л(W)AZ + л(Z)AW - 2g(AZ, W)l.

By using (5), (6), (8) and (9), we obtain

Л {W)AZ + л {Z)AW - 2g(AZ, W)E = VZNW - TS7ZW - VVZW - th(W., Z) -—nh(W, Z) + VwVZ - TVWZ - VVWZ - th(W, Z) - nh(W, Z).

So we have

Л{W)AZ + л{Z)AW - 2g(AZ, W)l = -AVWZ + V^VW - TVZW - VVZW --2th(W, Z) - AVZW + V^VZ - TWwZ - VWwZ - 2nh{W, Z).

Corresponding the tangent components of the last equation, we conclude

-Л (W)AZ - л (Z)AW + 2g(AZ, W)l = AVWZ + TVZW + 2th(W, Z) + AVZW + TVWZ.

From the above equation, we can infer

-Л{W)AZ - л{Z)AW + 2g(AZ, W)E, = + AVZW + 2TVZW

-T(VZW - VWZ) + 2th(W, Z) T[Z, W] = Avwz + AvzW + 2TWzW + 2th(W, Z) +Л{W)A + л{Z)AW - 2g(AZ, W)E,

Thus [Z, W] E Г(D1) if and only if (38) is satisfied.

Theorem 5. Let M be a pseudo-slant submanifold of a nearly quasi Sasakian manifold M. Then the slant distribution Bq is integrable if and only if for any X, Y e Г(Bq)

Pi{VxPY - PVyX + {S7yP)X - AVXY - AVVX - 2ЩХ, Y) -

-Л {Y)AX - л {X)AY + 2g(AX, Y)E} = 0. '

Proof. For any X, Y e T(Be) and we denote the projections on BL and Bq by Pi and P2 , respectively, then for any vector fields X.Y e T(Be), by using equation (4), we obtain

(Xxf)Y + (Vyf)X = л (Y) AX + л (X)AY - 2 g(AX, Y) £,

or

XxfY - fXxY + VyfX - fXyX = л {Y) AX + л {X)AY - 2 g{AX, Y) £,. By using equations (5), (6), (8), and (9), we can write

VXPY + XxVY - f(X\Y + h{X, Y)) + VyPX + VyVX --fiXyX + h(X, Y)) = л (Y)AX + л(X)AY - 2g(AX, Y)E VXPY + h(X, PY) - AvyX + V^.l/F - PVXY -

- ЩХ, Y) - nh(X, Y) + XyPX + h(Y, PX) - ^ Uj

-AnxY + V^l/x - PXyX - VXyX - th{X, Y) -- nh(X, Y) = л {Y) AX + лРО AY - 2 g(AX, Y)E

From tangential components of (40) reach

_ A„YV — A,,V Y — O+h ( Y VI =

(41)

VA PY - PVXY + (VYP)X - AVXY - AVYX - 2th{X, Y) = = л {Y) AX + л (X)AY - 2g(AX, Y)i,

P[X, Y] = VaPY - PXxY + (yYP)X - AVXY --AVYX - 2 ЩХ, Y) - л {Y) AX - л {X)AY + 2 g(AX, Y)l,.

(42)

Applying Pl to (42), we get (39).

Theorem 6. Let M be a pseudo-slant submanifold of a nearly quasi Sasakian manifold M. Then the distribution DL © is integrable if and only if for any Z, W G © £,)

AfZ\V - AfWZ = -[n{AZ)W - r[{Z)A\V + r[{\V)AZ - n(AW)Z]

O

Proof. For any Z, W G T(D± © £,) and U G T(PM) , by using (7), we can write

2g(AfZW, U) = g(h(U, W), fZ) + g(h(U, W), fZ)

By using (5), we have

2 g(AfZW, U) = g(ywU, fZ) + g(VuW, fZ) = = -g(fVwU,Z)-g(fVuW,Z)

So we have

2g(AfZW, U) = g((Vwf)U + (Vuf)W, Z) --g(ywfU,Z)-g(VufW,Z)

By using equation (3), we obtain

2g(AfZW, U) = g(r\(U)AW +x](W)AU - 2g(AW, U)£,, Z) - g(XwfU, Z) - g(XufW, Z) = = g(r\(U)AW + r\(W)AU - 2g(A\V, U)E,, Z) - g(XWZ, fU) - g(-AfWU, Z) =

= g(y\(AW)Z + ri(W)AZ - 2n(AW)Z, U) - g{fXWZ, U) + g(AfWU, Z) = = g(y\(AW)Z + n{W)AZ - 2rj{AW)Z - PXWZ - th(Z, W), U) + g(AfWZ, U)

which is equivalent to

2AfZW = ri{W)AZ - n(AW)Z + AfWZ - PXWZ - th(Z, W). (43)

Take Z = W in (43), we infer

2AfWZ = ti(Z)AW - ti{AZ)W + AfZW - PXZ\V - th(W., Z) (44)

By using equation (43) and (45), we obtain

3{AfZW - AfwZ) = P[Z, W] - ti{Z)AW + r\(AZ)W +y\(W)AZ - n(AW)Z, (45)

thus the distribution DL © Í, is integrable if and only if P[Z, W] = 0 which proves our assertion.

4. Totally umbilical pseudo-slant submanifolds

In this section we shall consider M as a totally umbilical pseudo-slant submanifold of nearly quasi Sasakian manifold M. We have the following preparatory results. Theorem 7. Let M be a totally umbilical pseudo-slant submanifold of a nearly quasi Sasakian manifold M. Then at least one of the following statements is true,

1) dim(D±) = 1,

2) He r(n),

3) M is proper pseudo-slant submanifold. Proof. Let Z e and using (3), we obtain

(\7zf)Z=r](Z)AZ-g(AZ,Z)i,

WzVZ - f(VzZ + h(Z, Z)) = t\(Z)AZ - g(AZ, Z)i,. From the last equation, we have

-AvzZ + V^VZ - NVzZ - th(Z, Z) - nh(Z, Z) = t](Z)AZ - g(AZ, Z% (46)

From (12) and from the tangential components of (46), we obtain

+ th(Z, Z) = -ti(Z)AZ + g(AZ, Z)P£,. (47)

Taking the product by W e T(D±), we obtain

g(AvzZ + th(Z, Z) + n(Z)AZ - g(AZ, Z)PE, W) = 0.

It implies that

g(h(Z, W), NZ) + g(th(Z, Z), W) + T](Z)g(AZ, W) - g(AZ, Z)g(PE„ W) = 0. (48) Since M is totally umbilical submanifold, we obtain

g(Z, W)g(H. VZ) + g(Z, Z)g(tH, W) + T](Z)g(AZ, W) - g(AZ, Z)g(PE,, W) = 0, (49) that is

-g(tH, Z)W + g(tH, W)Z + g(AZ, W)f, - g{PE, W)AZ = 0. (50)

Here tH is either zero or Z and W are linearly dependent vector fields. If tH ^ 0, then

dimT(D±) = 1.

Otherwise H e r([x). Since De ^ 0, M is pseudo-slant submanifold. Since 9^0 and di.d-2 ^ 0, M is proper pseudo-slant submanifold.

Theorem 8. Let M be totally umbilical proper pseudo-slant submanifold of nearly quasi Sasakian manifold M . Then M is an either totally geodesic submanifold or it is an anti-invariant if H, V y H G r(|i).

Proof. Since the ambient space is nearly quasi Sasakian manifold, for any X e T(PM), by using 3, we have

(Vxf)X = ri(X)AX - g{AX,X)lS7xfX - fXxX = ^X)AX - g(AX,X)£,. (51) Using (5), (7), (8) and (12) in (51) and we get

V vPX - g(X, PX)H - AvxX + Vj^VX = = fXxX + g(X, X)fH + ц(Х)АХ - g(AX, X)£,.

giXxfH, VX) = -sin2e{g(X,X)\\H\\2}, g(XxVH, fX) = sin2e{g(X,X)\\H\\2}.

Thus, (54) and (57) imply

g(X,X)\\H\\2 = згп2в{д(Х,Х)\\Н\\2}, cos2eg{x,x)\\H\\2 = o.

(52)

Applying product fH to the above equation we get

g(VjtVX, fH) = g(VXxX, fH) + g(X, X)\\H\\2 - g(AX, X)g(V£,, fH) (53) taking into account (6), we get

g(\7jcVXJH) = g(X,X)\\H\\2-g(AX,X)g(ViJH). (54)

Now, for any X G T(PM) , we obtain

XxfH = (Vxf)H + fX xH (55)

In view of (6), (8), (9), (17) and (55) we obtain

-AfHX + VjtfH = (Vxf)H - PAHX - VAHX + tX^H + nX^H. (56) Applying product VX to the above equation we get

giXjcfH, VX) = g{(Xxf)H, VX) - g(VAHX. VX),

g(XxfH, VX) = g{(Xxn)H + h(tH, X) + VAHX, VX) - g(VAHX, VX). By using (7), (17) and (20), we have

g(XxfH, VX) = -sin2e{g(X, X)\\H\\2 - g(h(X, £,), H)ri(X)} From (15), we obtain

(57)

(58)

From (58), we conclude that g(X,X)\\H\\2 = 0, for any X E T(PM). Since M proper pseudo slant submanifold of nearly quasi Sasakian manifold we obtain H = 0. This tells us that M is totally geodesic in M.

Theorem 9. Let M be totally umbilical proper pseudo-slant submanifold of nearly quasi Sasakian manifold M. Then at least one of the following statements is true.

1) н e\i.

2) g(VPXE,X) = 0.

3) t\((VxP)X) = 0.

4) М is a anti-invariant submanifold.

5) If M proper slant submanifold then, dim(M) > 3, for any X e Г(РМ). Proof. From equation (3) and M is nearly quasi Sasakian manifold, we have

XxfX - f\/xX = л(X)AX - g(AX, X)l.

By using (5), (6), (8) and (9), we have

VxPX + h(X. PX) - AVXX + Vj^VX - PVxX --VVxX - th(X, X) - nh(X, X) = r\(X)AX - g(AX, X)t, 1 j

tangential components of (59), we obtain

VxPX - PVxX - ЩХ, X) - AvxX = л {X)AX. (60)

Since M is a totally umbilical pseudo-slant submanifold, by using (7) and (17), we can write

g(AvxX, X) = g(h(X, X), VX) = g(H, VX)g(X, X) = g(g(H, VX)X, X) = 0. (61)

If H e Г(ц), then from (60), we obtain

VxPX - PVxX = л(X)AX.

Taking the product of (61) by £,, we obtain

giVxPX, £,) - л (PVxX) = л(Х)л(AX)g(VxPX1 £,) = 0. (62)

Interchanging X by PX in (62), we derive

g(VPXP2X, 0 = 0 ^ g{VpxL P2X) = 0

by using (18), we have

g{VpxK ~cos2d(X - л(Х)£,) = 0 cos2Qg{VpxL (-X - ц{Х)1) = 0.

Since, M is a proper pseudo-slant submanifold, we have

g(Vpxt, (X — л(А")£,) = 0.

From which

g(VpXl, X) = л(X)g(Vpxt, £,)• (63)

Now, we have g(E,, £,) = 1. Taking the covariant derivative of above equation with respect to PX for any X G Г(PM), we obtain g(VРХ1Л) + g{KVpxl) = 0 which implies giVpxK 0 = 0 and then (63) gives

g(VPX£,,X) = 0. (64)

This proves 2) of theorem.

Now, Inter changing X by PX in the equation (64), we derive

g(\7P2Xt, TX) = £(Vcos26(_x+^xjQt, PX) = 0,

cos2eg{X{-x+^x)i)t,PX) = 0, -cos2Qg(Xxl, PX) + -cos2dr[(X)g(XPX) = 0. Since V^f, = 0, we obtain

cos2dg(Xxt,PX) = 0. (65)

From (65) if cosQ = 0, 0 = 7r/2 then M is an anti-invariant submanifold. On the other hand, g{XxKPX) = 0, that is VxE = 0. This implies that is a the Killing vector field on M. If the vector field £, is not Killing, then we can take at least two linearly independent vectors A" and PX to span De, that is, the dim(M) > 3.

Example 1. Suppose M be a submanifold of R' with coordinates (xi, x2, x3, yi, y2l y3, w), defined by

x\ = V3u sinh ot, x2 = —v cosh ot, x3 = s sinh z,

yi = v cosh ot, y2 = 2v cosh a, y3 = —s sinh z.t = w,

where u, v, s and z denote the arbitrary parameters. The tangent bundal space of M is spanned by tangent vectors.

/— d 0 d d

ei = v 3 smh ot-—. e2 = cosh ot---cosh ot—--h 2 cosh ot-—.

ox 1 dy 1 ox 2 oy2

e3 = sinh^---sinh^-—. e4 = s coshz—--scoshx; ——.

ox 3 dy 3 dx3 dy 3

For the almost contact metric structure cj) of R', choosing and £, = J^, r| = dt. For any vector field W = u(-+ Vj-J^ + A^ G T(R7), then we have

4>Z = ^ " (t,Z) = ^ + 4

and

for any i,j = 1,2,3. It follows that g($Z,4>Z) = g{Z,Z) -T]2{Z). Thus (cf>, f,,r|, is an almost contact metric structure on R7. Thus we have

r , 0 0 0 0 4>ei = v 3 smh ot——, 4>e2 = — cosh ot—--cosh a—--2 cosh ot

dyi dx i dy2 dx2 X^

i?

0 0 0 0 4>e3 = sinn £ —--h sinn ——. cj>e4 = s cosh z —--1- s cosh z -

<>!/ ■, ' dx3' ' dy3 ' dx3'

By direct calculations, we can infer Dq = span{ei, e2} is a slant distribution with slant angle 0 = cos_1(4g) . Since

g((\)e3, ei) = g(4>e3, e2) = g{4>e3, e4) = g(4>e3, e5) = 0, 5,(ct)e4i ei) = 5,(<t>e4, e2) = £(<N4, e3) = ^(4>e4, e5) = 0,

e3 and e4 are orthogonal to M, DL = span(e3, e4) is an anti-invariant distribution. Thus M is a 5-dimensional proper pseudo-slant submanifold of R7 with its usual almost contact metric structure.

REFERENCES

1. Bejancu A. CR-submanifold of a Kahler manifold. I. Proc. Amer. Math. Soc, 1978, vol. 69, pp. 135-142.

2. Bejancu A. Geometry of CR-submatiifolds. Dordrecht, D. Reidel Pub. Co, 1986. 172 p.

3. Bejancu A., Papaghiuc N. Semi-invariant submanifolds of a Sasakian manifold. An St Univ At I Cuza last supl, 1981, vol. XVII, no. 1, I-a, pp. 163-170.

4. Blair D.E. The theory of quasi Sasakian structures. /. Differential Geom., 1967, vol. 1, no. 3-4, pp. 331-345.

5. Carriazo A. New developments in slant submanifolds theory. Applicable Mathematics in the Golden Age. New Delhi, Narosa Publishing House, 2002, vol. 394, pp. 339-356.

6. Chen B.Y. Geometry of Slant Submanifolds. Leuven, Katholieke Universiteit Leuven, 1990. 126 p.

7. Cabrerizo J.L., Carriazo A., Fernandez L.M., Fernandez M. Slant Submanifolds in Sasakian manifolds. Glasgow Mathematical Journal, 2000, vol. 42, pp. 125-138. DOI: 10.1017/S0017089500010156.

8. Cabrerizo J.L., Carriazo A., Fernandez L.M., Fernandez M. Semi-slant submanifolds of a Sasakian manifold. Geom. Dedicata, 1999, vol. 78, iss. 2, pp. 183-19. DOI: 10.1023/A: 1005241320631.

9. Cabrerizo J.L., Carriazo A., Fernandez L.M., Fernandez M. Structure on a slant submanifold of a contact manifold. Indian Journal of Pure and Applied Mathematics, 2000, vol. 31, pp. 857-864.

10. Lotta A. Slant submanifolds in contact geometry. Bull. Math. Soc. Sci. Math. Roum., Nouv. Ser., 1996, vol. 39 (87), no. 1/4, pp. 183-198.

11. Khan V.A., Khan M.A. Pseudo-slant submanifolds of a Sasakian manifold. Indian Journal of Pure and Applied Mathematics, 2007, vol. 38, pp. 31-42.

12. Papaghiuc N. Semi-slant submanifolds of a Kaehlerian manifold. An. Stiint. Univ. Al. I. Cuza-Iasi, Ser. Noua, Mat., 1994, vol. 40, pp. 55-61.

13. Rahman S., Khan M.S., Horaira A. Contact CR-Warped Product Submanifolds of Nearly Quasi-Sasakian Manifold. Konuralp Journal of Mathematics, 2019, vol. 7, iss. 1, pp. 140-145.

О ПСЕВДО-НАКЛОННЫХ ПОДМНОГООБРАЗИЯХ БЛИЗКО КВАЗИ-САСАКИЕВЫХ МНОГООБРАЗИЙ

Шамсур Раман

Кандидат наук, Доцент Кафедры математики, Национальный Университет Урду им. Мауланы Азада shamsur@rediffmail.com

Политехнический Спутник Кампус Дарбханга, 846002 Бихар, Индия

Мод Садик Хан

Кандидат наук, Кафедра Математики, Нициональный колледж им. Н. ШИбли msadiqkhan.snc@gmail.com Азамгарх, Уттар-Прадеш 276001, Индия

Абу Хорайра

Кафедра Математики,

Нициональный колледж им. Н. ШИбли

aboohoraria@gmail.com

Азамгарх, Уттар-Прадеш 276001, Индия

Аннотация. В работе изучается геометрия псевдо-наклонных подмногообразий близко квази-сасакиевых многообразий. Доказано, что вполне омбилическое правильно наклонное многообразие близко квази-сасакиева многообразия является вполне геодезическим, если вектор средней кривизны Н е ц. Также получены условия интегрируемости распределения псевдо-наклонных подмногообразий близко квази-сасакиевых многообразий

Ключевые слова: близко квази-сасакиевы многообразия, наклонные многообразия, правильные наклонные подмногообразия, псевдо-наклонные подмногообразия.

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