# Neutral Slant Submanifolds of a Para-Kähler Manifold

Authors: Yılmaz Gündüzalp
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We define and study both neutral slant and semineutral slant submanifolds of an almost para-Hermitian manifold and, in particular, of a para-Kähler manifold. We give characterization theorems for neutral slant and semi-neutral slant submanifolds. We also investigate the integrability conditions for the distributions involved in the definition of a semi-neutral slant submanifold when the ambient manifold is a para-Kähler manifold.


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## Body

## 1. Introduction

The geometry of slant submanifolds was initiated by Chen, as a generalization of both holomorphic and totally real submanifolds in complex geometry [1, 2]. Since then, many mathematicians have studied these submanifolds. Slant submanifolds have been studied by many geometers in various manifolds [3–5]. In particular, Papaghiuc [6] introduced semislant submanifolds. Lotta [7, 8] defined and studied slant submanifolds in contact geometry. Cabrerizo et al. studied slant, semislant, and bislant submanifolds in contact geometry [9, 10]. Recently, Arslan et al. [11] studied these submanifolds in the setting of neutral Kähler manifolds.In this paper we define and study both neutral slant and semineutral slant submanifolds of an almost para-Hermitian manifold and, in particular, of a para-Kähler manifold. The paper is organized as follows. In Section2, we review some formulas and definitions for an almost para-Hermitian manifold and their submanifolds. In Section 3, we define neutral slant submanifolds for an almost para-Hermitian manifold and give theorem for a neutral slant submanifold. In the last section, we define and study semineutral slant submanifolds of an almost para-Hermitian manifold. We give theorems for a semineutral slant submanifold. In the last part of Section 4, we obtain that the distributions are integrable and their leaves are totally geodesic in semineutral slant submanifold under the condition ∇f=0. Finally, the paper contains some examples.


## 2. Preliminaries

An almost para-Hermitian manifold(M¯,g,J) is a smooth manifold endowed with an almost paracomplex structure J and a pseudo-Riemannian metric g compatible in the sense that
(1)J2=I,g(JX,Y)+g(X,JY)=0,X,Y∈Γ(TM¯),
where Γ(TM) is the module of differentiable vector fields on M. It follows that the metric g is neutral; that is, it has signature (m,m), and the eigenbundles TM¯± are totally isotropic with respect to g.An almost para-Hermitian manifoldM¯ is called a para-Kähler manifold if
(2)(∇¯XJ)Y=0,∀X,Y∈Γ(TM¯),
where ∇¯ is the Levi-Civita connection on M¯ [12, 13].LetM be an isometrically immersed submanifold of an almost para-Hermitian manifold M¯. We denote the Levi-Civita connections on M and M¯ by ∇ and ∇¯, respectively. Then, the Gauss and Weingarten formulas are given by
(3)∇¯XY=∇XY+h(X,Y),∇¯XN=-ANX+∇X⊥N,
for any X,Y∈Γ(TM) and N∈Γ(TM⊥), where ∇⊥ is the connection in the normal bundle TM⊥,h is the second fundamental form of M, and AN is the shape operator. The second fundamental form h and the shape operator AN are related by
(4)g(ANX,Y)=g(h(X,Y),N),
where the induced pseudo-Riemannian metric on M is denoted by the same symbol g.Let us consider thatM is an immersed submanifold of an almost para-Hermitian manifold M¯. For any X∈Γ(TM) and N∈Γ(TM⊥), we put
(5)JX=fX+ωX,(6)JN=BN+CN,
where fX (resp., ωX) is tangential (resp., normal) part of JX and BN (resp., CN) is tangential (resp., normal) part of JN. From (1) and (5), we have
(7)g(fX,Y)=-g(X,fY),
for any X,Y∈Γ(TM).The submanifoldM is said to be invariant if ω is identically zero; that is, JX=fX∈Γ(TM), for any X∈Γ(TM). On the other hand, M is said to be anti-invariant submanifold if f is identically zero; that is, JX=ωX∈Γ(TM⊥), for any X∈Γ(TM).For anyX∈Γ(TM), by a direct calculation, we have
(8)X=f2X+BωX,thatis,I=f2+Bω,(9)ωfX+CωX=0,thatis,ωf+Cω=0.Similarly, for anyN∈Γ(TM⊥), we have
(10)N=ωBN+C2N,thatis,I=ωB+C2,fBN+BCN=0,thatis,fB+BC=0.Now, letM be an immersed submanifold of an almost para-Kähler manifold M¯. For any X,Y∈Γ(TM), from ∇¯XJY=J(∇¯XY), taking into account (3), (5), and (6), then we have
(11)∇XfY+h(X,fY)-AωYX+∇X⊥ωY=f∇XY+ω∇XY+Bh(X,Y)+Ch(X,Y).Comparing the tangential and normal components of (11), respectively, we get
(12)(∇Xf)Y=AωYX+Bh(X,Y),(13)(∇Xω)Y=Ch(X,Y)-h(X,fY),
where the covariant derivations of f and ω are, respectively, defined by
(14)(∇Xf)Y=∇XfY-f∇XY,(∇Xω)Y=∇X⊥ωY-ω∇XY,
for any X,Y∈Γ(TM).LetM be a submanifold of a para-Hermitian manifold M¯. A tangent vector X∈TM is said to be spacelike (resp., timelike) if g(X,X)>0 (resp., g(X,X)<0). If X is a spacelike vector (resp., timelike), we have ∥X∥=g(X,X) (resp., ∥X∥=-g(X,X)) [11].


## 3. Neutral Slant Submanifolds of Almost Para-Hermitian Manifolds

In this section, we study neutral slant immersions of an almost para-Hermitian manifoldM¯. First, we present definition of a neutral slant submanifold of an almost para-Hermitian manifold following Chen’s [1] definition for a Hermitian manifold. Let M be a semi-Riemannian manifold isometrically immersed in an almost para-Hermitian manifold M¯. For each nonzero spacelike vector X tangent to M at x, the angle θ(X), 0≤θ(X)≤π/2 between JX and TxM is called the Wirtinger angle of X. Then, M is said to be neutral slant if the angle θ(X) is a constant, which is independent of the choice of x∈M and X∈Γ(TM). The angle θ of a neutral slant immersion is called the slant angle of the immersion. Thus, the invariant and anti-invariant immersions are neutral slant immersions with slant angle θ=0 and θ=π/2, respectively. A neutral slant immersion which is neither invariant nor anti-invariant is called a proper neutral slant immersion.We note that our definition is quite different from Chen’s definition for slant submanifold [1], and the slant submanifold is given by Arslan et al. [11].Next we give a useful characterization of neutral slant submanifolds in an almost para-Hermitian manifold.Theorem 1.
LetM be a submanifold of a para-Hermitian manifold M¯. Then,(i)
Mis neutral slant if and only if there exists a constant λ∈[0,1] such that f2=λI. Furthermore, in this case, if θ is the slant angle of M, it satisfies λ=cos2θ;(ii)
M is a neutral slant submanifold if and only if there exists a constant λ∈[0,1] such that B2ω=λI. Furthermore, in this case, if θ is the slant angle of M, it satisfies λ=sin2θ.Proof.
(i) Suppose thatM is a neutral slant submanifold. For any X∈Γ(TM), we can write
(15)cosθ(X)=∥fX∥∥JX∥,
where θ(X) is the slant angle. By using (7), (15), and (1), we get
(16)g(f2X,X)=-g(fX,fX)=-cos2θ(X)g(JX,JX)=cos2θ(X)g(X,X),
for all X∈Γ(TM). Since g is a neutral metric, from (16), we have
(17)f2X=cos2θ(X)X,X∈Γ(TM).
Letλ=cos2θ. Then it is obvious that λ∈[0,1].
Conversely, let us assume that there exists a constantλ∈[0,1] such that f2=λI is satisfied. From (7), (17), and (1), we get
(18)cosθ(X)=g(JX,fX)∥JX∥∥fX∥=-g(X,f2X)∥JX∥∥fX∥=-λg(X,J2X)∥JX∥∥fX∥=λg(JX,JX)∥JX∥∥fX∥,
for all X∈Γ(TM). Thus we have
(19)cosθ(X)=λ∥JX∥∥fX∥.
Since cosθ(X)=∥fX∥/∥JX∥, then by using the last equation we obtain cos2θ(X)=λ, which implies that θ(X) is a constant and so M is a neutral slant.
(ii) From (8) and (i), we have (ii).Corollary 2.
LetM be a neutral slant submanifold of an almost para-Hermitian manifold M¯ with slant angle θ. Then, for any X,Y∈Γ(TM), we have
(20)g(fX,fY)=-cos2θg(X,Y),(21)g(ωX,ωY)=-sin2θg(X,Y).Proof.
From Theorem1(i) and (7), we get
(22)g(fX,fY)=-g(f2X,Y),g(fX,fY)=-cos2θg(X,Y),
for any X,Y∈Γ(TM). On the other hand, from (1), (5), and (20), we obtain
(23)g(JX,JY)=g(fX+ωX,fY+ωY),-g(X,Y)=g(fX,fY)+g(ωX,ωY).
This completes the proof.Now, we give some examples of the neutral slant submanifolds in almost para-Hermitian manifolds inspirited by Chen [1].Note that given a semi-Euclidean spaceRn2n with coordinates (x1,…,x2n) on Rn2n, we can naturally choose an almost paracomplex structure J on Rn2n as follows:
(24)J(∂∂x2i)=∂∂x2i-1,J(∂∂x2i-1)=∂∂x2i,
where i=1,…,n. Let Rn2n be a semi-Euclidean space of signature (+,-,+,-,…) with respect to the canonical basis (∂/∂x1,…,∂/∂x2n).Example 3.
Consider a submanifoldM in R24 given by
(25)φ(u,v)=(ucosα,v,usinα,0).
It is easy to see thatM is a neutral slant submanifold with the slant angle α.Example 4.
Consider a submanifoldM in R24 given by
(26)x(u,v)=(usinα,vcosβ,ucosα,vsinβ),
where α and β are constant. Then M is a neutral slant submanifold with the slant angle cosθ=|sin(α+β)|.Remark 5.
ConsiderMp2p a neutral submanifold of an almost para-Hermitian manifold (M¯,g,J), in fact a neutral manifold M¯n2n, with
(27)|g(fX,JX)|≤∥fX∥∥JX∥.M is called a neutral slant submanifold if the Wirtinger angle between JX and TxM is constant, for all X∈TxM a spacelike vector field and all x∈M. It is well defined, because that angle can be measured as usual, it the same angle between JX and fX, and they both are timelike vector fields.
In fact, if that conditions hold, it would be the same angle betweenJY and TxM for Y∈TxM a timelike vector, both JY and fY would be spacelike vector fields. This condition is equivalent to
(28)|g(fX,fX)|≤|g(JX,JX)|,
or ∥fX∥≤∥JX∥, in fact it is equivalent to Theorem 1 condition f2X=cos2θI.


## 4. Semineutral Slant Submanifolds of Almost Para-Hermitian Manifolds

Definition 6.
Let(M¯,g) be an almost para-Hermitian manifold with an almost paracomplex structure J. A differentiable distribution on M¯ is called a neutral slant distribution if for each p∈M¯ and each nonzero spacelike vector X∈Γ(Dp), the angle θp between JX and Dp is a constant, that is, independent of the choice of p∈M¯ and X∈Γ(Dp). In this case, we call the constant angle θp the slant angle of the distribution Dp.
LetM be an immersed submanifold of an almost para-Hermitian manifold M¯ and D a differentiable distribution on M. We denote the orthogonal distribution to D on M by D⊥. Then, for all X∈Γ(TM), we write
(29)JX=P1fX+P2fX+ωX,
where P1 and P2 are orthogonal projections on D and D⊥, respectively.
Next, we will give a sufficient and necessary condition for a distribution to be slant.Theorem 7.
LetM be a submanifold of an almost para-Hermitian manifold M¯ and D a differentiable distribution on M. Then D is a neutral slant distribution if and only if there exists a constant λ∈[0,1] such that
(30)(P1f)2=λI.
Furthermore, in such case, ifθ is the slant angle of D then λ=cos2θ.Proof.
We suppose thatD is a neutral slant distribution on M. Then, from (29), we have
(31)cosθ(X)=g(JX,P1fX)∥JX∥∥P1fX∥=-g(X,(P1f)2X)∥JX∥∥P1fX∥=∥P1fX∥∥JX∥,
which implies that
(32)∥P1fX∥=cosθ(X)∥JX∥,
for any X∈Γ(D). By using (29), (32), and (1), we have
(33)g(X,(P1f)2X)=-g(P1fX,P1fX)=-cos2θ(X)g(JX,JX)=cos2θ(X)g(X,X),∀X∈Γ(D).
Sinceg is a neutral metric, we obtain
(34)(P1f)2X=cos2θ(X)X,∀X∈Γ(D).
If we putλ=cos2θ, then we have (30).
Conversely, letλ∈[0,1] be a constant such that (30) is satisfied. Then, from (1) we have
(35)cosθ(X)=g(JX,P1fX)∥JX∥∥P1fX∥=-g(X,(P1f)2X)∥JX∥∥P1fX∥=-λg(X,X)∥JX∥∥P1fX∥,
for any X∈Γ(D). Thus we get
(36)cosθ(X)=λ∥JX∥∥P1fX∥.
On the other hand, sincecosθ(X)=∥P1fX∥/∥JX∥, then we obtain cos2θ=λ, which implies that θ is a constant and D is a neutral slant distribution. This completes the proof.Definition 8.
M is called a bineutral slant submanifold of an almost para-Hermitian manifold M¯ if there exist two orthogonal distributions D1 and D2 on M such that (i)
TM admits the orthogonal direct decomposition TM=D1⊕D2;(ii)
Di is a neutral slant distribution with slant angle θi for i=1,2. Given a bineutral slant submanifold M, we can write, for any X∈Γ(TM),
(37)X=P1X+P2X,
where Pi denotes the component of X in Di for any i=1,2. In particular, if X∈Γ(Di), then we obtain Xi=PiX. If we define fi=Pi∘f, then we have
(38)JX=f1X+f2X+ωX,
for any X∈Γ(TM).
We note that semi-invariant submanifolds are particular cases of bineutral slant submanifolds with slant anglesθ1=0 and θ2=π/2.Theorem 9.
LetM be a bineutral slant submanifold with angles θ1=θ2=θ. If g(JX,Y)=0, for any X∈Γ(D1) and Y∈Γ(D2), then M is slant with angle θ.Proof.
Sinceg(JX,Y)=0, for any X∈Γ(D1) and Y∈Γ(D2), we have g(fX,Y)=0; that is, fX∈Γ(D1). Similarly, for Y∈Γ(D2), we find. Then for any X∈Γ(TM),X can be written as follows: X=X1+X2 such that X1∈Γ(D1) and X2∈Γ(D2) and cos2θ1=∥fX1∥2/∥JX1∥2,  cos2θ2=∥fX2∥2/∥JX2∥2. Since θ1=θ2=θ, we get
(39)g(fX,fX)g(JX,JX)=g(fX1,fX1)+g(fX2,fX2)g(JX1,JX1)+g(JX2,JX2)=cos2θ,
which gives assertion of the theorem.
Now, as a generalization of semi-invariant submanifolds, we can define semineutral slant submanifolds of an almost para-Hermitian manifold.Definition 10.
M is called a semineutral slant submanifold of an almost para-Hermitian manifold M¯ if there exist two orthogonal distributions D1 and D2 on M such that (i)
TM admits the orthogonal direct sum TM=D1⊕D2,(ii)
the distributionD1 is invariant; that is, J(D1)=D1,(iii)
the distributionD2 is neutral slant with slant angle θ≠0. In this case, we call θ the slant angle of submanifold M.
It is obvious that the invariant and anti-invariant distributions of a semineutral slant submanifold are neutral slant distributions with the slant anglesθ=0 and θ=π/2, respectively.
Now, letM be a semineutral slant submanifold of an almost para-Hermitian manifold M¯. Let M be a semislant submanifold with d1dim(D1) and d2dim(D2). Then we have the following particular cases.(i)
Ifd2=0, then M is an invariant submanifold.(ii)
Ifd1=0 and θ=π/2, then M is an anti-invariant submanifold.(iii)
Ifd1=0 and θ≠π/2, then M is a proper neutral slant submanifold with slant angle θ.(iv)
Ifd1·d2≠0 and θ≠π/2, then M is a proper semineutral slant submanifold.
We now give an example of bineutral slant submanifolds.Example 11.
Letx(u,v,t,s)=(usinα,v,ucosα,0, s,tsinβ, 0,tcosβ), where α and β are constant. Then, M is a 4-dimensional submanifold of M¯=R48.
By defining(40)D1=〈sinα∂∂x1+cosα∂∂x3,∂∂x2〉,D2=〈∂∂x5,sinβ∂∂x6+cosβ∂∂x8〉,
we have that TM=D1⊕D2 and D1,D2 are neutral slant with slant angles cos-1(|sinα|) and cos-1(|sinβ|), respectively. Thus M is a bineutral slant submanifold of M¯.
Now, letM be a semineutral slant submanifold of an almost para-Hermitian manifold M¯ and Pi, (i=1,2), denoting the orthogonal projections on Di, (i=1,2). Then, for any X∈Γ(TM), applying J to (37), we have
(41)JX=fP1X+fP2X+ωP2X,
where
(42)JP1X=fP1X,ωP1X=0.
From (41) and (42), we have
(43)fX=JP1X+fP2X.
By puttingY=P1Y in (20) and Y=P2Y in (21), we get
(44)g(fX,fP1Y)=-cos2θg(X,P1Y),X,Y∈Γ(TM),g(ωX,ωP2Y)=-sin2θg(X,P2Y),X,Y∈Γ(TM),
respectively.
We give a characterization for the semineutral slant submanifolds of an almost para-Hermitian manifold.Theorem 12.
LetM be an immersed submanifold of an almost para-Hermitian manifold M¯. Then M is a semineutral slant submanifold if and only if there exists a constant λ∈[0,1) such that D={X∈TM∣f2X=λX} is a distribution. Furthermore, in this case, λ=cos2θ, where θ denotes slant angle of M.Proof.
LetM be a semineutral slant submanifold and TM=D1⊕D2, where D1 is invariant and D2 is neutral slant. We put λ=cos2θ, where θ denotes slant angle of M. For any X∈Γ(D), if X∈Γ(D1), then we have
(45)X=J2X=f2X=λX,
which implies that λ=1. But this is a contradiction that λ∈[0,1). Therefore we obtain D⊆D2. On the other hand, since D2 is a neutral slant distribution, it follows from Theorem 7 that f2X=(fP2)2X=λX, which means that D2⊆D. Thus D=D2 is a distribution.
Conversely, we can consider the orthogonal direct decompositionTM=D⊕D⊥. It is obvious that fD⊆D, from which we have g(JX,Y)=-g(X,JY)=-g(X,fY)=0 for any X∈Γ(D⊥) and Y∈Γ(D). Hence D⊥ is an invariant distribution. Finally, Theorem 7 imply that D is a neutral slant distribution, with slant angle θ satisfying λ=cos2θ.We can easily present some examples of the above situation.Example 13.
x
(
u
,
v
,
t
,
r
)
=
(
u
,
0
,
u
,
v
sin
θ
,
0, vcosθ, t,s), θ≠π/2 defines a four-dimensional proper semineutral slant submanifold M, with slant angle cos-1(|sinθ/2|), in R48.
Moreover, it is easy to see that(46)X1=∂∂x7,X3=∂∂x1+∂∂x3,X2=∂∂x8,X4=sinθ∂∂x4+cosθ∂∂x6
from a local orthogonal frame of TM. Then, we can define D1=Span{X1,X2} and D2=Span{X3,X4}.Example 14.
x
(
u
,
v
,
t
,
s
)
=
(
u
,
v
,
t
sin
α
,
s
cos
β, tcosα, ssinβ,0,0) defines a four-dimensional proper semineutral slant submanifold M, with slant angle cosθ=|sin(α+β)|, in R48, where α and β are constant.
Moreover it is easy to see that(47)X1=∂∂x1,X3=sinα∂∂x3+cosα∂∂x5X2=∂∂x2,X4=cosβ∂∂x4+sinβ∂∂x6
from a local orthogonal frame of TM. Then we can define D1=Span{X1,X2} and D2=Span{X3,X4}.
Then, it is easy to show that all conditions of Theorem12 are satisfied.Next, we will give useful characterizations for integrable conditions of distributions.Theorem 15.
LetM be a semineutral slant submanifold of a para-Kähler manifold M¯. Then we have the following: (a)
the distributionD1 is integrable if and only if
(48)h(X,fY)=h(fX,Y),
for any X,Y∈Γ(D1),(b)
the distributionD2 is integrable if and only if
(49)P1(∇XfP2Y-∇YfP2X)=P1(AωP2YX-AωP2XY),
for any X,Y∈Γ(D2).Proof.
From (2), we get
(50)∇¯XJY=J∇¯XY,
for all X,Y∈Γ(TM¯).(a)
By using Gauss-Weingarten formulas, (5), and (6) in (50), we have
(51)∇XfY+h(X,fY)=f∇XY+ω∇XY+Bh(X,Y)+Ch(X,Y),
for anyX,Y∈Γ(D1). From (41) and (51), we obtain
(52)∇XfY+h(X,fY)=fP1∇XY+fP2∇XY+ωP2∇XY+Bh(X,Y)+Ch(X,Y).
By equating the normal part of the last equation, we have(53)h(X,fY)=ωP2∇XY+Ch(X,Y).
If we change the role ofX and Y in (53), we write
(54)h(fX,Y)=ωP2∇YX+Ch(Y,X).
Sinceh is symmetric, from (53) and (54), we get
(55)h(X,fY)-h(fX,Y)=ωP2[X,Y],∀X,Y∈Γ(D1).
Assume that the distributionD1 is integrable. Then, for any X,Y∈Γ(D1), we have [X,Y]∈Γ(D1) which implies that ωP2[X,Y]=0. Thus from (55) we obtain (48).
Conversely, if (48) is satisfied, then from (55), we have ωP2[X,Y]=0, for any X,Y∈Γ(D1), which implies that P2[X,Y]=0. Then we conclude that [X,Y]∈Γ(D1).(b)
From (41) and Gauss-Weingarten formulae, we have
(56)∇¯XJY=∇XJP1Y+h(X,JP1Y)+∇XfP2Y+h(X,fP2Y)-AωP2YX+∇X⊥ωP2Y,for all X,Y∈Γ(TM). On the other hand, by using (5) and (6), we write
(57)J∇¯XY=f∇XY+ω∇XY+Bh(X,Y)+Ch(X,Y).
By using (56) and (57) in (50), we get
(58)∇XfP2Y+h(X,fP2Y)-AωP2YX+∇X⊥ωP2Y=f∇XY+ω∇XY+Bh(X,Y)+Ch(X,Y),
for any X,Y∈Γ(D2). Since h is symmetric we obtain
(59)f[X,Y]=∇XfP2Y-∇YfP2X+AωP2XY-AωP2YX
which gives
(60)P1f[X,Y]=P1{∇XfP2Y-∇YfP2X}-P1{AωP2YX-AωP2XY}.
Let the distributionD2 be integrable. Then P1f[X,Y]=0, for all X,Y∈Γ(D2), and hence from (60), the equation (49) is obvious.
Conversely, if (49) is satisfied then P1f[X,Y]=0; that is, [X,Y]∈Γ(D2) for any X,Y∈Γ(D2). This completes the proof.Definition 16.
LetM be a semi-invariant submanifold of an almost para-Hermitian manifold M¯. Then we say that(i)
M is D1-geodesic if
(61)h(X,Y)=0,∀X,Y∈Γ(D1),(ii)
M is D2-geodesic if
(62)h(X,Y)=0,∀X,Y∈Γ(D2),(iii)
M is mixed geodesic if
(63)h(X,Y)=0,∀X∈Γ(D1),Y∈Γ(D2).Lemma 17.
LetM be a mixed-geodesic semineutral slant submanifold of a para-Kähler manifold M¯. Then the distribution D1 is integrable if and only if
(64)JANX=-ANJX,
for any X∈Γ(D1) and N∈Γ(T⊥M).Proof.
SinceM is a mixed-geodesic submanifold, from (4) we find that ANX has no component on D2. By using (4) and (1), we obtain
(65)g(JANX,Y)=-g(ANX,JY)=-g(h(X,JY),N),g(ANJX,Y)=g(h(JX,Y),N).
Thus, we can write(66)g(JANX+ANJX,Y)=g(h(JX,Y)-h(X,JY),N),
for all X,Y∈Γ(D1). Taking into account Theorem 15(a) and the last equation, the proof is completed.Theorem 18.
LetM be a semineutral slant submanifold of a para-Kähler manifold M¯. If ∇ω=0, then M is a mixed-geodesic submanifold. Furthermore,(a)
ifX,Y∈Γ(D1), then either M is a D1-geodesic submanifold or h(X,Y) is an eigenvector of C2 with the eigenvalue 1,(b)
ifX,Y∈Γ(D2), then either M is a D2-geodesic submanifold or h(X,Y) is an eigenvector of C2 with the eigenvalue cos2θ.Proof.
If∇ω=0, then from (13) we get Ch(X,Y)=h(X,fY), for all X,Y∈Γ(TM). Since D1 is an invariant and D2 is a neutral slant distribution with the slant angle θ, we obtain
(67)C2h(X,Y)=Ch(X,fY)=h(X,f2Y)=h(X,cos2θY)=cos2θh(X,Y),C2h(X,Y)=C2h(Y,X)=Ch(Y,fX)=h(Y,f2X)=h(Y,X)=h(X,Y),
for any X∈Γ(D1),Y∈Γ(D2). By using (67) we get
(68)sin2θh(X,Y)=0,
which implies that h(X,Y)=0, for any X∈Γ(D1), Y∈Γ(D2), that is, M is mixed-geodesic. Similarly, we obtain
(69)C2h(X,Y)=h(X,Y),
for all X,Y∈Γ(D1), and
(70)C2h(X,Y)=cos2θh(X,Y),
for all X,Y∈Γ(D2). This completes the proof.Proposition 19.
LetM be a semineutral slant submanifold of a para-Kähler manifold M¯. Then ∇ω=0 if and only if
(71)ACNZ=-ANfZ,
for all Z∈Γ(TM),N∈Γ(T⊥M).Proof.
From (13) and (1),we get
(72)g((∇Xω)Z,N)=g(Ch(X,Z)-h(X,fZ),N)=-g(h(X,Z),CN)-g(h(X,fZ),N),
for any X,Z∈Γ(TM),N∈Γ(T⊥M). Taking into account (4), we get
(73)g((∇Xω)Z,N)=-g(ACNZ+ANfZ,X),
which completes the proof.Proposition 20.
LetM be a semineutral slant submanifold of a para-Kähler manifold M¯. Then ∇f=0 if and only if
(74)AωP2YZ=AωP2ZY,
for all Y,Z∈Γ(TM).Proof.
From (12) and (1) we have
(75)g((∇Xf)Y,Z)=g(AωYX+Bh(X,Y),Z)=g(AωP2YX,Z)-g(h(X,Y),ωP2Z)=g(AωP2YX,Z)-g(AωP2ZX,Y),
for any X,Y,Z∈Γ(TM). Since the A is symmetric then we obtain from the last equation
(76)g((∇Xf)Y,Z)=g(AωP2YZ-AωP2ZY,X).
This completes the proof.Proposition 21.
LetM be a semineutral slant submanifold of a para-Kähler manifold M¯. If  ∇f=0 then the distributions are integrable and their leaves are totally geodesic in M.Proof.
Since∇f=0, then from (12) we obtain Bh(X,Y)=0 for any X∈Γ(TM) and Y∈Γ(D1). By using (1) and (5), we have
(77)0=(Bh(X,Y),Z)=g(Jh(X,Y),Z)=-g(h(X,Y),JZ),
where X,Z∈Γ(TM) and Y∈Γ(D1). Thus one can easily see that
(78)g(h(X,Y),ωP2Z)=0,(79)g(Jh(X,Y),ωP2Z)=0.
SinceM¯ is a para-Kähler manifold, taking into account (78), we get
(80)0=g(Jh(X,Y),ωP2∇XY)0=-g(ωP2∇XY,ωP2∇XY)0=sin2θg(P2∇XY,P2∇XY),
which gives P2∇XY=0; that is, ∇XY∈Γ(D1). Now, let Y∈Γ(D1) and V∈Γ(D2). Since D1 is orthogonal to D2, the induced metric on M is the neutral metric, and it is easy to see that ∇ZV∈Γ(D2). Hence the proof is complete.



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*Source: 752650-2013-09-11.xml*