On pseudo-slant submanifolds of nearly quasi-sasakian manifolds

DOI:

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 known spaces. The properties of slant submanifolds of an almost contact metric manifolds were studied by A. Lotta [10].

L. Cabrerizo et. al [8] 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 pseudoslant 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 (/, d,n,g), where / is a tensor field of type (1,1), d vector field, n is a 1-form and g is Riemannian metric on M such that

• (a)   /2 = -I + П ® d,   (b)  n(d) = 1,   (c)  n ° / =0

• (d)   /(d) = 0,   (e) n(X) = g(X, d),(1)

(/)    g( /X, /Y ) = g(X, Y ) - n(X)n(Y)

for any vector field X, Y tangent to M , where I is the identity on the tangent bundle ГМ of M . An almost contact metric structure (/, d, n, g) on M is called quasi-Sasakian manifold if

(Vx/) Y = n(Y)AX - g(AX, Y)d,(2)

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

(Vx/)Y + (VY/)X = n(Y)AX + n(X)AY - 2g(AX, Y)d,(3)

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

V x d =/AX.(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 P ^L M 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

V xY = VxY + h(X,Y),(5)

VxV = -Av X + VXV(6)

for any X, Y G PM and V G PMd , where V ^± is the connection on the normal bundle PM, , h is the second fundamental form and A _v is the Weingarten map associated with V as

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

For any X E PM and V E P ^L M , we write

/X = PX + VX (PX E PM and VX E PLM),(8)

/V = tV + nV (tV E PM and nV EP^M).(9)

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

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

for any X,Y E PM . If we put Q = P ² we have

(VxQ^Y = XxQY -Q^XY,(11)

(XxP)Y = XxPY - PXxY,(12)

(XxV)Y = XXVY -VXxY(13)

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

^x k = PAX,(14)

K(X, k) = VAX.

The mean curvature vector H of M is given by

H = — trace (K) = n

n

- DM, n “

2 = 1

where n is the dimension of M and e 1 ,e ₂ , ...,e _n is a local orthonormal frame of M . A submanifold M of an contact metric manifold M is said to be totally umbilical if

K(X,Y ) = g(X,Y)H,

where H is the mean curvature vector. A submanifold M is said to be totally geodesic if K(X, Y ) = 0, for each X, Y E r(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 0(x) E [0,n/2], between /X and PX is called the slant angle or the Wirtinger angle of M . If the slant angle is constant for each X E r(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, n/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 D ¹ and D ₀ on M such that

• 1)    PM admits the orthogonal direct decomposition PM = D ¹ ф D ₀ , E = r(D).

• 2)    The distribution D ¹ is anti-invariant i.e., /(D ¹ ) C P ¹ M.

• 3)    The distribution D ₀ is a slant with slant angle 0 = 0 , that is, the angle between /(D ₀ ) and D ₀ 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 antiinvariant.

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 D ¹ and D ₀ by d 1 and d ₂ , respectively, then we have the following cases:

• 1) If d ₂ = 0 then M is an anti-invariant submanifold.

• 2) If d 1 = 0 and 0 = 0, then M is an invariant submanifold.

• 3) If d 1 = 0 and 0 = 0, then M is a proper slant submanifold with slant angle 0.

• 4) If d 1 .d ₂ = 0 and 0 G [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 G PM. Then M is slant if and only if there exists a constant Л G [0,1] such that

P2 = -Л{/ - n 0 E}

Furthermore, in such a case if 0 is the slant angle of M then Л = cos 2 0.

Corollary 1. Let M be a slant submanifold of nearly quasi-Sasakian manifold M with slant angle 0 . Then for any X,Y G r(PM ) we have

g(PX, PY) = cos20(g(X, Y) - n(X)n(Y)),(19)

g(VX, VY) = srn20(g(X, Y) - n(X)n(Y)).(20)

Let M a proper pseudo slant submanifold of a contact metric manifold M and the projections on D1 and D0 by P1 and P2, respectively, then for any vector field X G r(PM), we can write

X = P1X + P2X + n(X)E.(21)

Now applying / on both sides of equation (3.4), we obtain

• /X = /P i X + /P 2 X

that is,

PX + VX = VP1X + PP2X + VP2X.(22)

We can easily to see

PX = PP2X, VX = VP1X + VP2 X and

/P1X = VP1X, TP1X = 0, /P2X = TP2X + VP2X,(24)

TP2X G r(D0).(25)

If we denote the orthogonal complementary of f PM in D ^L M by ^, then the normal bundle P ^L M can be decomposed as follows

P ^ M = V (D ^ ® V(D _e ) ф ^,

where ^ is an invariant sub bundle of PMd as N (D ^± ) and N (D _e ) are orthogonal distribution on M . Indeed, g(Z,X ) = 0 for each Z E F(D ^± ) and X E F(D _e ). Thus, by equation (1) and (25), we can write

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

that is, the distributions V (D ^± ) and V(D _e ) 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 E D ^± we have

A _$y X - A _fX Y = X x (PY ) + h(X, PY ) - A _vy X + X ^ ( VY ) - - P( X x Y ) - V( X x Y ) - V(h(X,Y)).

Proof. In view of (7), we get

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

From (5) and (29), we get

g(A _f Y X, Z ) = - g(f X z X, Y ) + g(f X _Z X, Y )

= -g(fXzX,Y) since fX zX EPM(30)

= g(( X z f)X,Y) - g( X z fX,Y).

Now, for X E D 1 , fX E P ^± M . Hence, from (6) we have

XzfX = -AfxZ + X^fX.(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(A jx Z,Y )= g(A jx Y,Z ).

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

g(A _f Y X, Z ) - g(A _f x Y, Z ) = n(Z)g(AX, Y ) + n(X)g(AZ, Y ) -

- 2g(AZ,X)n(Y)+ g(( X x f )Y,Z) =

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

+ g( X x (PY) + X x (PY) - f ( X x Y) - f (h(X,Y)),Z) =          (33)

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

+ g( X x (PY ) + h(X,PY ) - A vy (X) + X x (VY) -

- P( X x Y ) - V( X x Y ) - P(h(X, Y ) - V(h(X, Y ))), Z ).

Since X,Y,Z E D ^L an orthonormal distribution to the distribution ( ^ ) it follows that n(X) = n(Y) = 0. Therefore, the above equation reduces to

A ,y X - A _fX Y = V x (PY ) + h(X, PY ) - A vy X + V x (VY ) -- P ( V x Y ) - V( V x Y ) - V (h(X, Y )).

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

( V x P ) Y = A vy X + A vx Y + th(X, Y ) + n(Y )AX + n(X )AY -- 2g(AX, Y )^ + V y (PX ) + P (V y X ) + P(h(Y, X)).

Proof. Let X, Y E PM , we have

V x /Y = ( V x / )Y + / ( V x Y )

From (8) and (9), we obtain

V x PY + V x VY = ( V x / ) Y + /( V x Y + h(X, Y ))

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

V x PY + V x VY = ( V x / ) Y + P( V x Y ) + V( V x Y ) + th(X, Y ) + nh(X, Y )

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

V x PY + h(X, PY ) - A vy X + V x VY = n(Y)AX + n(Y)AX -- 2g(AX, Y )^ + P ( V x Y ) + V( V x Y ) + th(X, Y ) + nh(X, Y ) - -V y PX - h(Y, PX ) + A vx Y - V Y VX + P V y X + +V V y X + P(h(Y, X )) + V(h(Y, X )).

Comparing tangential and normal parts, we obtain

V x PY - A vy X = n(Y)AX + n(X )AY - 2g(AX, Y )^ + P ( V x Y) +

+th(X, Y ) + V y (PX) + A vx Y + P (V y X ) + P(h(Y, X )).          (36)

That is,

( V x P)Y = A vy X + A vx Y + th(X, Y ) + n(Y)AX + n(X )AY -- 2g(AX, Y )^ + V y (PX) + P (V y X ) + P(h(Y, X )).

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

A vw Z + A vz W + 2T V _Z W + 2th(W,Z ) = - n(W)AZ - n(Z)AW + 2g(AZ,W)^. (38)

Proof. Let Z,W G r(D ^± ) and using (3), we obtain

(7z f ) W + (7wf )Z = n(W)AZ + n(Z)AW - 2g(AZ, W)^, which is equivalent to

7 z fW - f 7 z W + 7 w fW - f 7 w Z = n(W)AZ + n(Z)AW - 2g(AZ, W )^.

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

n(W)AZ + n(Z)AW - 2g(AZ, W )^ = 7 z ^W - T 7 z W - V V z W - th(W, Z ) -- nh(W, Z ) + 77 w VZ - T V w Z - V 7 w Z - th(W, Z ) - nh(W, Z ).

So we have

n(W )AZ + n(Z )AW - 2g(AZ, W )^ = - A vw Z + 7 ^ VW - T V z W - V V z W -- 2th(W, Z ) - A vz W + 7 w VZ - T 7 w Z - V 7 w Z - 2nh(W, Z ).

Corresponding the tangent components of the last equation, we conclude

- n(W )AZ - n(Z )AW + 2g(AZ, W)^ = A vw Z + T 7 z W + 2th(W, Z ) + A vz W + T 7 w Z.

From the above equation, we can infer

- n(W )AZ - n(Z )AW + 2g(AZ, W)^ = A vw Z + A vz W + 2T 7 z W -

- T ( 7 z W - 7 w Z ) + 2th(W, Z )

T [Z, W ] = A vw Z + A vz W + 2T 7 _Z W + 2th(W, Z )+ +n(W)A + n(Z )AW - 2g(AZ, W )^

Thus [Z,W ] G r(D ^L ) 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 D _e is integrable if and only if for any X,Y G r(D ₆ )

P i {V _x PY - P 7 _y X + (7 _y P )X - A vx Y - A v _Y X - 2th(X, Y ) -- n(Y )AX - n(X )AY + 2g(AX,Y )^ } = 0.

Proof. For any X, Y G r(D ₆ ) and we denote the projections on D ^± and D ₀ by P 1 and P ₂ , respectively, then for any vector fields X, Y G r(D ₆ ), by using equation (4), we obtain

( 7 x f ) Y + ( 7 Y f )X = n(Y)AX + n(X )AY - 2g(AX, Y )^

7 x f Y - f 7 x Y + 7 Y fX - f 7 y X = n(Y)AX + n(X )AY - 2g(AX, Y)^.

By using equations (5), (6), (8), and (9), we can write

7 x PY + 7 x VY - f ( 7 x Y + h(X, Y )) + 7 y PX + 7 y VX -- f ( 7 y X + h(X, Y )) = n(Y )AX + n(X )AY - 2g(AX, Y )^+ + 7 x PY + h(X, PY ) - A vy X + 7 x VY - P 7 x Y -

- V 7 x Y - th(X, Y ) - nh(X, Y ) + 7 y PX + h(Y, PX ) -

- A _N x Y + 7 Y VX - P 7 y X - V 7 y X - th(X, Y ) -- nh(X, Y ) = n(Y)AX + n(X )AY - 2g(AX, Y )^

From tangential components of (40) reach

V x PY - P V x Y + ( V y P)X - A v XY - A v YX - 2th(X, Y ) = = n(Y ) AX + n(X ) AY - 2g(AX, Y )L,

P [X, Y ] = V x PY - P V x Y + ( V y P)X - A v XY - - A _v YX - 2th(X, Y ) - n(Y)AX - n(X ) AY + 2g(AX, Y )L.

Applying P 1 to (42), we get (39).

• Theorem 6.    Let M be a pseudo-slant submanifold of a nearly quasi Sasakian manifold M. Then the distribution D ^ ф L is integrable if and only if for any Z,W E r(D ^± ф L)

A fz W - A fw Z = 1[n(AZ )W - n(Z)AW + n(W)AZ - n(AW)Z].

Proof. For any Z,W E r(D ^± ф L) and U E r(PM ) , by using (7), we can write

2g(A _JZ W, U ) = g(h(U, W ), /Z) + g(h(U, W ), /Z ).

By using (5), we have

2g(A _f z W, U ) = g( V w U, /Z ) + g( V u W, /Z ) = = - g(/ V w UZ ) - g(/ V u W,Z).

So we have

2g(A _f z W, U ) = g(( V w / )U + ( V u /)W, Z ) -- g( V w /U,Z) - g( V u /W,Z).

By using equation (3), we obtain

2g(A fz W, U ) = g(n(U)AW + n(W )AU - 2g(AW, U)L, Z ) - g( V w /U, Z ) - g( V u /W, Z ) = = g(n(U)AW + n(W )AU - 2g(AW, U)L, Z ) - g( V w Z, /U) - g( - A fw U, Z ) =

= g(n(AW)Z + n(W)AZ - 2n(AW)Z, U ) - g(/ V WZ, U ) + g(A fw U, Z ) =

= g(n(AW)Z + n(W )AZ - 2n(AW )Z - P V w Z - th(Z, W),U ) + g(A fw Z, U )

which is equivalent to

2A fz W = n(W)AZ - n(AW)Z + A fw Z - P V w Z - th(Z, W ).       (43)

Take Z = W in (43), we infer

2A fw Z = n(Z)AW - n(AZ)W + A fz W - P V z W - th(W, Z ).       (44)

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

3(A fz W - A fw Z ) = P [Z, W ] - n(Z)AW + n(AZ)W + n(W)AZ - n(AW)Z, (45) thus the distribution D ^± ф L 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)    H G Г(^) ,

• 3)    M is proper pseudo-slant submanifold.

Proof. Let Z G r(D ^± and using (3), we obtain

( V z / )Z = n(Z )AZ - g(AZ,Z )^

V z VZ - /( V _z Z + h(Z, Z )) = n(Z )AZ - g(AZ, Z )^.

From the last equation, we have

• -AvzZ + V^VZ - NVzZ - th(Z, Z) - nh(Z, Z) = n(Z)AZ - g(AZ, Z)^.(46)

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

AvzZ + th(Z, Z) = -n(Z)AZ + g(AZ, Z)Pt,.(47)

Taking the product by Ж G r(D ^± ), we obtain

g(A vz Z + th(Z, Z ) + n(Z)AZ - g(AZ, Z)P^, Ж) = 0.

It implies that

g(h(Z, Ж), NZ) + g(th(Z, Z), Ж) + n(Z)g(AZ, Ж) - g(AZ, Z)g(P^, Ж) = 0.(48)

Since M is totally umbilical submanifold, we obtain

g(Z, Ж)g(H, VZ ) + g(Z, Z)g(tH, Ж ) + n(Z )g(AZ, Ж ) - g(AZ, Z)g(P^, Ж ) = 0, (49)

that is

-g(tH, Z)Ж + g(tH, Ж)Z + g(AZ, Ж)^ - g(P^, Ж)A.Z = 0.(50)

Here tH is either zero or Z and Ж are linearly dependent vector fields. If tH = 0, then dimV(D ^v ) = 1.

Otherwise H G Г(^). Since D g = 0, M is pseudo-slant submanifold. Since 0 = 0 and d 1 ,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 antiinvariant if H, V x H G Г(^).

Proof. Since the ambient space is nearly quasi Sasakian manifold, for any X E Г(Р М ), by using 3, we have

( X x /)X = n(X)AX - g(AX,X)^ X x / X - / X x X = n(X)AX - g(AX,X )^.

Using (5), (7), (8) and (12) in (51) and we get

X x PX - g(X, PX)H - A _vx X + X X VX = = / X x X + g(X, X)/H + n(X)AX - g(AX, X )^.

Applying product /H to the above equation we get

g( X X VX, /H ) = g(V X _xX, /H ) + g(X, X) \\ H \ 2 - g(AX, X)g(V^, /H )

taking into account (6), we get

g(X x VX, /H ) = g(X, X ) \\ H \ ² - g(AX, X)g(V^, /H ).

Now, for any X E Г(РМ ) , we obtain

X x /H = (X x /)H + / X x H.

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

• - A _$H X + X x /H = (X x /)H - PA _H X - VA h X + I^H + ^H.

Applying product VX to the above equation we get

g( X x /H, VX ) = g((X x /)H, VX ) - g(VA H X, VX),

g(X x /H, VX ) = g(( X x n)H + h(tH, X ) + VA h X, VX ) - g(V A h X, VX ).

By using (7), (17) and (20), we have

g(X x /H, VX ) = - sin ² 0 { g(X, X ) \\ H \ ² - g(h(X, ^), H )n(X) } .

From (15), we obtain

g(X x /H,VX ) = - sin ² 0 { g(X,X ) \\ H \ ² } , g(X x VH,/X ) = sin ² 0 { g(X,X ) \\ H \ ² } .

Thus, (54) and (57) imply

g(X,X) \ H ||² = sin ² 0 { g(X,X) \ H \ ² } , cos ² 0g(X,X) \\ H ||² = 0.

From (58), we conclude that g(X,X ) \\ H \ ² = 0, for any X E Г(РМ ). Since М

(58) proper

pseudo slant submanifold of nearly quasi Sasakian manifold we obtain H = 0. This tells us that М is totally geodesic in ЛМ.

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

• 1)    Н Е ^ .

• 2)    д ( Х рх ^,Х ) = 0 .

• 3)    п(( Х х Р )Х) = 0 .

• 4)    М is a anti-invariant submanifold.

• 5)    If М proper slant submanifold then, dim(M ) >  3 , for any Х Е Г(РМ) .

Proof. From equation (3) and M is nearly quasi Sasakian manifold, we have

Х х /Х - / Х х Х = п(Х)АХ - д(АХ, Х )^.

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

Х х РХ + К(Х , РХ ) - A _v ХХ + Х Х ХХ - Р Х х Х -

• -V ХхХ - ^Д (Х, Х) - пК(Х, Х) = п(Х)АХ - д(АХ, Х )^

tangential components of (59), we obtain

ХхРХ - РХхХ - ^Д (Х, Х) - AvхX = п(Х)АХ.(60)

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

g(AvхX, Х) = д(ЦХ, Х), ХХ) = д(Н, ХХ)д(Х, Х) = д(д(Н, ХХ)Х, Х) = 0.(61)

If Н Е Г(^), then from (60), we obtain

Х х РХ - Р Х х Х = п(Х)АХ.

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

д(ХхРХ, ^) - п(РХхХ) = п(Х)п(АХ)д(ХхРХ, ^) = 0.(62)

Interchanging X by PX in (62), we derive

д( Х рх Р 2 Х, ^) = 0 ^   д( Х рх ^, Р 2 Х ) = 0

by using (18), we have

д( Х рх ^, - cos ² 0(Х - п(Х)^) =0   ^   cos ² 0д( Х рх ^, (Х - п(Х)^) = 0.

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

д( Х рх ^, (Х - п(Х)^) = 0.

From which

д( Х рх ^,Х ) = п(Х)д( Х рх ^, ^).                         (63)

Now, we have д(^, ^) = 1. Taking the covariant derivative of above equation with respect to РХ for any Х Е Г(РМ ), we obtain д( Х _рх ^, £,) + д(^, Х _рх £,) = 0 which implies д( Х _рх ^, ^) = 0 and then (63) gives

д( Х рх ^,Х )=0.                             (64)

This proves 2) of theorem.

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

g( V p 2 _X h,TX ) = g( V cos 2 0 ( - x + n ( x ) h ) h,PX ) = 0, cos ² 6g( V ( - x ^PX ) = 0,

- cos ² 0g( V x h, PX ) + - cos ² 0n(X)g( V ^ h, PX ) = 0.

Since Vhh = 0, we obtain cos2 0g(Vxh,PX) = 0.

From (65) if cos 0 = 0, 0 = n/2 then M is an anti-invariant submanifold. On the other hand, g( V _x h,PX ) = 0, that is V _x h = 0. This implies that h is a the Killing vector field on M . If the vector field h is not Killing, then we can take at least two linearly independent vectors X and PX to span D ₀ , that is, the dim(M ) >  3.

Example 1. Suppose M be a submanifold of R7 with coordinates (x1, x2, x3, y1, y2, y3, w), defined by x1 = VSu sinh a, x2 = -x cosh a, x3 = s sinh z, y1 = x cosh a, y2 = 2x cosh a, y3 = -s sinh z, t = w, where u, x, s and z denote the arbitrary parameters. The tangent bundal space of M is spanned by tangent vectors.

r- . d                , d         , d           , d e1 = Vs sinh a——,  e2 = cosh a—--cosh a—--+ 2 cosh a——,

• , d d , d , d e3 = sinh z—--sinh z——, e4 = s cosh z—--s cosh z——.

^dx 3         ^дУ з                ^дх з           ^дУ з

For the almost contact metric structure ф of R ⁷ , choosing

ф(H") = TP’ ф(F") = -TF1 ф(lit)=0, 1 6 ^ 6 3, dx,    dy,      oy^      oxj dt and h = ,П = dt. For any vector field Ж = цF + v7-F + ЛF G T(R7), then we have Ot* x                 J                           F      j o^j      OW V / ’ дd ф2 = ht^--vj, dyj

g ^(ф Z^{, ф} Z) = h 2 + ^v 2 ,

g(Z,Z ) = h 2 + v 2 + Л ² , n(Z )= g(Z, h) = Л

^z = -ц,X - Vj A - A 9 + A 9 = -Z + n(z)h dxt     dy,    dt dt for any i,i = 1,2,3. It follows that g(фZ, фZ) = g(Z, Z) - n2(Z). Thus (ф, h,n,g) is an almost contact metric structure on R7. Thus we have фе1 = V3 sinh a——,

дд  д фе2 = - cosh a—--cosh a—--2 cosh a——,

3         3                 3           д фе3 = sinh z—--+ sinh z——, фе4 = s cosh z—--+ s cosh z——.

^ду з         ^д^ з                  ^дУ з           ^д^ з

By direct calculations, we can infer D ₀ = span { e 1 , e ₂ } is a slant distribution with slant angle 0 = cos ^-1 (^). Since

у(фе з , e i ) = у(фе з , e 2 ) = у(фе з , e 4 ) = у(фе з , e s ) = 0,

у(фе4, ei) = y(фe4, e2) = у(фе4, ез) = у(фе4, es) = 0, e3 and e4 are orthogonal to M, D± = 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.