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mathematics

Article

Bi-Slant Submanifolds of Para Hermitian Manifolds

Pablo Alegre 1,* and Alfonso Carriazo 2

1Departamento de Economía, Métodos Cuantitativos e Historia Económica, Área de Estadística e

Investigación Operativa, Universidad Pablo de Olavide, Ctra. de Utrera km. 1, 41013 Sevilla, Spain

2Departamento de Geometría y Topología, Universidad de Sevilla, c/Tarﬁa s/n, 41012 Sevilla, Spain

*Correspondence: psalerue@upo.es

Received: 26 June 2019; Accepted: 9 July 2019; Published: 11 July 2019

Abstract:

In this paper, we introduce the notion of bi-slant submanifolds of a para Hermitian manifold.

They naturally englobe CR, semi-slant, and hemi-slant submanifolds. We study their ﬁrst properties

and present a whole gallery of examples.

Keywords:

semi-Riemannian manifold; para Hermitian manifold; para Kaehler manifold;

para-complex; totally real; CR; slant; bi-slant; semi-slant; hemi-slant; anti-slant submanifolds

MSC: 53C15; 53C25; 53C40; 53C50

1. Introduction

In [

1

], B.-Y.Chen introduced slant submanifolds of an almost Hermitian manifold, as those

submanifolds for which the angle

θ

between

JX

and the tangent space is constant, for any tangent

vector ﬁeld

X

. They play an intermediate role between complex submanifolds (

θ=

0) and totally real

ones (

θ=π/

2). Since then, the study of slant submanifolds has produced an incredible amount of

results and examples in two different ways: various ambient spaces and more general submanifolds.

On the one hand, J. L. Cabrerizo, A. Carriazo, L. M. Fernández, and M. Fernández analyzed slant

submanifolds of a Sasakian manifold in [

2

], and B. Sahin did so in almost product manifolds in [

3

].

The study of slant submanifolds in a semi-Riemannian manifold has been also initiated: B.-Y. Chen,

O. Garay, and I. Mihai classiﬁed slant surfaces in Lorentzian complex space forms in [

4

,

5

]. K. Arslan, A.

Carriazo, B.-Y. Chen, and C. Murathan deﬁned slant submanifolds of a neutral Kaehler manifold in [

6

],

while A. Carriazo and M. J. Pérez-García did so in neutral almost contact pseudo-metric manifolds

in [

7

]. Moreover, M. A. Khan, K. Singh, and V. A. Khan introduced slant submanifolds in LP-contact

manifolds in [

8

], and P. Alegre studied slant submanifolds of Lorentzian Sasakian and para Sasakian

manifolds in [9]. Finally, slant submanifolds of para Hermitian manifolds were deﬁned in [10].

On the other hand, some generalizations of both slant and CR submanifolds have also been

deﬁned in different ambient spaces, such as semi-slant [

11

–

13

], hemi-slant [

14

,

15

], bi-slant [

16

], or

generic submanifolds [17].

In this paper, we continue on this line, introducing semi-slant, hemi-slant, and bi-slant

submanifolds of para Hermitian manifolds.

2. Preliminaries

Let

e

M

be a 2

n

-dimensional manifold. If it is endowed with a structure

(J

,

g)

, where

J

is a

(

1, 1

)

tensor and gis a semi-deﬁned metric, satisfying:

J2X=X,g(JX,Y) + g(X,JY) = 0, (1)

Mathematics 2019,7, 618; doi:10.3390/math7070618 www.mdpi.com/journal/mathematics

Mathematics 2019,7, 618 2 of 15

for any vector ﬁelds

X

,

Y

on

e

M

, it is called a para Hermitian manifold. It is said to be para Kaehler if,

in addition, e

∇J=0, where e

∇is the Levi–Civita connection of g.

Let now

M

be a semi-Riemannian submanifold of

(e

M

,

J

,

g)

. The Gauss and Weingarten formulas

are given by:

e

∇XY=∇XY+h(X,Y), (2)

e

∇XV=−AVX+∇⊥

XN, (3)

for any tangent vector ﬁelds

X

,

Y

and any normal vector ﬁeld

V

, where

h

is the second fundamental

form of

M

,

AV

is the Weingarten endomorphism associated with

V

, and

∇⊥

is the normal connection.

The Gauss and Codazzi equations are given by:

e

R(X,Y,Z,W) = R(X,Y,Z,W) + g(h(X,Z),h(Y,W)) −g(h(Y,Z),h(X,W)), (4)

(e

R(X,Y)Z)⊥= ( e

∇Xh)(Y,Z)−(e

∇Yh)(X,Z), (5)

for any vectors ﬁelds X,Y,Z,Wtangent to M.

For every tangent vector ﬁeld X, we write:

JX =PX +FX, (6)

where

PX

is the tangential component of

JX

and

FX

is the normal one. For every normal vector

ﬁeld V,

JV =tV +f V ,

where tV and f V are the tangential and normal components of JV, respectively.

For such a submanifold of a para Kaehler manifold, taking the tangent and normal part and using

the Gauss and Weingarten formulas (2) and (3):

(∇XP)Y=∇XPY −P∇XY=AFY X+th(X,Y), (7)

(∇XF)Y=∇⊥

XFY −F∇XY=−h(X,PY) + f h(X,Y), (8)

for all tangent vector ﬁelds X,Y.

In [

10

], we introduced the notion of slant submanifolds of para Hermitian manifolds, taking into

account that we cannot measure the angle for light-like vector ﬁelds:

Deﬁnition 1

([

10

])

.

A semi-Riemannian submanifold

M

of a para Hermitian manifold

(e

M

,

J

,

g)

is called slant

submanifold if for every space-like or time-like tangent vector ﬁeld

X

, the quotient

g(PX

,

PX)/g(JX

,

JX)

is constant.

Remark 1.

It is clear that, if

M

is a para-complex submanifold, then

P≡J

, and so, the above quotient is

equal to one. On the other hand, if

M

is totally real, then

P≡

0and the quotient equals zero. Therefore, both

para-complex and totally real submanifolds are particular cases of slant submanifolds. A neither para-complex

nor totally real slant submanifold will be called a proper slant.

Three cases can be distinguished, corresponding to three different types of proper slant

submanifolds:

Deﬁnition 2

([

10

])

.

Let

M

be a proper slant semi-Riemannian submanifold of a para Hermitian manifold

(e

M,J,g). We say that it is of:

Type 1

if for any space-like (time-like) vector ﬁeld X, PX is time-like (space-like), and |PX|

|JX|>1,

Mathematics 2019,7, 618 3 of 15

Type 2

if for any space-like (time-like) vector ﬁeld X, PX is time-like (space-like), and |PX|

|JX|<1,

Type 3

if for any space-like (time-like) vector ﬁeld X, PX is space-like (time-like).

These three types can be characterized as follows:

Theorem 1 ([10]).Let M be a semi-Riemannian submanifold of a para Hermitian manifold (e

M,J,g). Then,

(1) M

is a slant of Type 1 if and only if for any space-like (time-like) vector ﬁeld

X

,

PX

is time-like (space-like),

and there exists a constant λ∈(1, +∞)such that:

P2=λId. (9)

We write λ=cosh2θ, with θ>0.

(2) M

is a slant of Type 2 if and only if for any space-like (time-like) vector ﬁeld

X

,

PX

is time-like (space-like),

and there exists a constant λ∈(0, 1)such that:

P2=λId. (10)

We write λ=cos2θ, with 0<θ<2π.

(3) M

is a slant of Type 3 if and only if for any space-like (time-like) vector ﬁeld

X

,

PX

is space-like (time-like),

and there exists a constant λ∈(−∞, 0)such that:

P2=λId. (11)

We write λ=−sinh2θ, with θ>0.

In every case, we call θthe slant angle.

Remark 2.

It was proven in [

10

] that Conditions (9), (10), and (11) also hold for every light-like vector ﬁeld,

as every light-like vector ﬁeld can be decomposed as a sum of one space-like and one time-like vector ﬁeld.

Furthermore, every slant submanifold of Type 1 or 2 must be a neutral semi-Riemannian manifold.

Para-complex and totally real submanifolds can also be characterized by

P2

. In [

10

], we did not

consider that case, but it will be useful in the present study.

Theorem 2. Let M be a semi-Riemannian submanifold of a para Hermitian manifold (e

M,J,g). Then,

1) M is a para-complex submanifold if and only if P2=I d.

2) M is a totally real submanifold if and only if P2=0.

Proof. If Mis para-complex, P2=J2=Id directly. Conversely, if P2=Id, from:

g(JX,JX) = g(PX,PX) + g(FX,FX),

we have:

−g(X,J2X) = −g(X,P2X) + g(FX,FX),

then

−g(X,X) = −g(X,X) + g(FX,FX),

and hence, g(FX,FX) = 0, which implies F=0.

The second statement can be proven in a similar way.

Mathematics 2019,7, 618 4 of 15

3. Slant Distributions

In [

11

], N. Papaghiuc introduced slant distributions in a Kaehler manifold. Given an almost

Hermitian manifold,

(e

N

,

J

,

g)

, and a differentiable distribution

D

, it is called a slant distribution if for

any nonzero vector

X∈Dx

,

x∈e

N

, the angle between

JX

and the vector space

Dx

is constant, that is

it is independent of the point

x

. If

PDX

is the projection of

JX

over

D

, they can be characterized as

P2

D=λI

. This, together with the deﬁnition of slant submanifolds of a para Hermitian manifold, aims

us to give the following:

Deﬁnition 3.

A differentiable distribution

D

on a para Hermitian manifold

(e

M

,

J

,

g)

is called a slant

distribution if for every non-light-like X ∈D, the quotient g(PDX,PDX)/g(JX,JX)is constant.

A distribution is called invariant if it is a slant with slant angle zero, that is if

g(PDX

,

PDX)/g(JX

,

JX) =

1 for all non-light-like

X∈D

. It is called anti-invariant if

PDX=

0

for all X∈D. In other cases, it is called a proper slant distribution.

With this deﬁnition, every one-dimensional distribution deﬁnes an anti-invariant distribution in

e

M

, so we are just going to take under study non-trivial slant distributions, that is with dimensions

greater than one. Just like for slant submanifolds, we can consider three cases depending on the casual

character of the implied vector ﬁelds.

Obviously, a submanifold Mis a slant submanifold if and only if T M is a slant distribution.

Deﬁnition 4.

Let

D

be a proper slant distribution of a para Hermitian manifold

(˜

M

,

J

,

g)

. We say that it is of:

Type 1

if for every space-like (time-like) vector ﬁeld X, PDX is time-like (space-like), and |PDX|

|JX|>1,

Type 2

if for every space-like (time-like) vector ﬁeld X, if PDX is time-like (space-like), and |PDX|

|JX|<1,

Type 3

if for every space-like (time-like) vector ﬁeld X, PDX is space-like (time-like).

These slant distributions can be characterized as slant submanifolds as in Theorem 1[10].

Theorem 3. Let D be a distribution of a para Hermitian metric manifold e

M. Then,

(1) D

is a slant distribution of Type 1 if and only for any space-like (time-like) vector ﬁeld

X

,

PDX

is time-like

(space-like), and there exits a constant λ∈(1, +∞)such that:

P2

D=λI(12)

Moreover, in such a case, λ=cosh2θ.

(2) D

is a slant distribution of Type 2 if and only for any space-like (time-like) vector ﬁeld

X

,

PDX

is time-like

(space-like), and there exits a constant λ∈(0, 1)such that:

P2

D=λI(13)

Moreover, in such a case, λ=cos2θ.

(3) D

is a slant distribution of Type 3 if and only for any space-like (time-like) vector ﬁeld

X

,

PDX

is space-like

(time-like), and there exits a constant λ∈(0, +∞)such that:

P2

D=λI(14)

Moreover, in such a case, λ=sinh2θ.

In each case, we call θthe slant angle.

Mathematics 2019,7, 618 5 of 15

Proof.

If

D

is a slant distribution of Type 1, for any space-like tangent vector ﬁeld

X∈D

,

PDX

and

JX

are also time-like, where we have used

(1)

. They satisfy

|PDX|/|JX|>

1. Therefore, there exists

θ>

0

such that:

cosh θ=|PDX|

|JX|=p−g(PDX,PDX)

p−g(JX,JX). (15)

Considering PDXinstead of X, we obtain:

cosh θ=|P2

DX|

|JPDX|=|P2

DX|

|PDX|. (16)

Now,

g(P2

DX,X) = g(JPDX,X) = −g(PDX,JX) = −g(PDX,PDX) = |PDX|2. (17)

Therefore, using (15), (16), and (17):

g(P2

DX,X) = |PDX|2=|P2

DX||JX|=|P2

DX||X|.

Since both

X

and

P2

DX

are space-like, it follows that they are collinear, that is

P2

DX=λX

.

Finally, from (15), we deduce that λ=cosh2θ.

Everything works in a similar way for any time-like tangent vector ﬁeld

Y∈D

, but now,

PDY

and JY are space-like, and so, instead of (15), we should write:

cosh θ=|PDY|

|JY|=pg(PDY,PDY)

pg(JY,JY).

Since

P2

DX=λX

, for any space-like or time-like

X∈D

, it also holds for light-like vector ﬁelds,

and so, we have that P2

D=λIdD.

The converse is just a simple computation.

For the second case, let

D

be a slant distribution of Type 2, for any space-like or time-like vector

ﬁeld X∈D,|PDX|/|JX|<1, and so, there exists θ>0 such that:

cos θ=|PDX|

|JX|=p−g(PDX,PDX)

p−g(JX,JX).

Proceeding as before, we prove that

g(P2

DX

,

X) = |P2

DX||X|

, and as both

X

and

P2

DX

are space-like

vector ﬁelds, it follows that they are collinear, that is

P2

DX=λX

. The converse is just a direct

computation.

Finally, if

D

is a slant distribution of Type 3, for any space-like vector ﬁeld

X∈D

,

PDX

is also

space-like, and there exists θ>0 such that:

sinh θ=|PDX|

|JX|=pg(PDX,PDX)

p−g(JX,JX).

Once more, we can prove that

g(P2

DX

,

X) = |P2

DX||X|

and

P2

DX=λX

. Again, the converse is a

direct computation.

Remember that an holomorphic distribution satisﬁes

JD =D

, so every holomorphic distribution is

a slant distribution with angle zero, but the converse is not true. It is called a totally real distribution

if

JD ⊆T⊥M

; therefore, every totally real distribution is anti-invariant but the converse does not

always hold. For holomorphic and totally real distributions, the following necessary conditions are

easy to prove:

Theorem 4. Let D be a distribution of a submanifold of a para Hermitian metric manifold e

M.

Mathematics 2019,7, 618 6 of 15

(1) If D is a holomorphic distribution, then |PDX|=|JX|, for all X ∈D.

(2) If D is a totally real distribution, then |PDX|=0, for all X ∈D.

However, the converse results do not hold if

D

is not

TM

; in such a case

TM =D⊕ν

, and for a

unit vector ﬁeld X:

JX =PDX+PνX+FX.

Therefore from:

g(JX,JX) = g(PDX,PDX) + g(PνX,PνX) + g(FX,FX),

and |PDX|=|JX|, in the case that PDXis also space-like, it is only deduced that:

g(PνX,PνX) + g(FX,FX) = −2,

or, in the case it is time-like,

g(PνX,PνX) + g(FX,FX) = 0.

Therefore, in general FX 6=0, and Dis not invariant.

Similarly, it can be shown that the converse of the second statement does not always hold.

Theorem 5. Let M be a semi-Riemannian submanifold of a para Hermitian metric manifold e

M.

1) The maximal holomorphic distribution is characterized as D ={X/FX =0}.

2) The maximal totally real distribution is characterized as D⊥={X/PX =0}.

Proof.

For the ﬁrst statement, if a distribution

D

is holomorphic, obviously

FeD=

0. For the converse,

consider

D={X/FX =

0

}

. We should prove that it is a holomorphic distribution. Let

X∈D

,

JX =T X be tangent to M, and:

g(FJX,V) = g(J2X,V) = g(X,V) = 0,

for all V∈T⊥M. Therefore, FJX =0. That implies JX ∈Dfor all X∈D, so Dis holomorphic.

The second statement is trivial.

4. Bi-Slant, Semi-Slant and Hemi-Slant Submanifolds

In [

11

], semi-slant submanifolds of an almost Hermitian manifold were introduced as those

submanifolds whose tangent space could be decomposed as a direct sum of two distributions,

one totally real and the other a slant distribution. In [

16

], anti-slant submanifolds were introduced as

those whose tangent space is decomposed as a direct sum of an anti-invariant and a slant distribution;

they were called hemi-slant submanifolds in [

14

]. Finally, in [

12

], the authors deﬁned bi-slant submanifolds

with both distributions slant ones.

Deﬁnition 5.

A semi-Riemannian submanifold

M

of a para Hermitian manifold

(e

M

,

J

,

g)

is called a bi-slant

submanifold if the tangent space admits a decomposition

TM =D1⊕D2

with both

D1

and

D2

slant

distributions.

It is called a semi-slant submanifold if

TM =D1⊕D2

with

D1

a holomorphic distribution and

D2

a

proper slant distribution. In such a case, we will write D1=DT.

It is called a hemi-slant submanifold if

TM =D1⊕D2

with

D1

a totally real distribution and

D2

a

proper slant distribution. In such a case, we will write D1=D⊥.

Mathematics 2019,7, 618 7 of 15

Remark 3.

As we have said before, being holomorphic (totally real) is a stronger condition than being a slant

with slant angle zero (π/2).

We write πi, the projections over Diand Pi=πi◦P,i=1, 2.

Let us consider two different para Kaehler structures over R4:

J=

0100

1000

0001

0010

,g=

1 0 0 0

0−1 0 0

0 0 1 0

0 0 0 −1

,

and:

J1=

0010

0001

1000

0100

,g1=

1 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 −1

.

Using the examples of slant submanifolds of

R4

given in [

10

] and making products, we can obtain

examples of bi-slant submanifolds in

R8

. To present different examples with all the combinations of

slant distributions, we consider the following para Kaehler structures over R8:

J2= JΘ

ΘJ!,g2= gΘ

Θg!,

J3= J1Θ

ΘJ!,g3= g1Θ

Θg!,

J4= J1Θ

ΘJ1!,g4= g1Θ

Θg1!,

where Θis the corresponding null matrix.

Example 1. For any a,b,c,d∈Rwith a2+b26=1, and c2+d26=1,

x(u1,v1,u2,v2) = (au1,v1,bu1,u1,cu2,v2,du2,u2)

deﬁnes a bi-slant submanifold in

(R8

,

J2

,

g2)

, with slant distributions

D1=Span ∂

∂u1,∂

∂v1

and

D2=

Span ∂

∂u2,∂

∂v2. We can see the different types in the Table 1:

Table 1. Types for Example 1.

D1D2

Type 1 a2+b2>1, b2<1c2+d2>1, c2<1

Type 2 a2+b2>1, b2>1c2+d2>1, c2>1

time-like Type 3 a2+b2<1c2+d2<1

(R8,J2,g2)

P2

1=a2

−1+a2+b2Id1

P2

2=c2

−1+c2+d2Id2

Remark 4.

The decomposition of

TM

into two slant distributions is not unique, for example, if we choose

˜

D1=

Span ∂

∂u1,∂

∂v2

and

˜

D2=Span ∂

∂u2,∂

∂v1

in the previous example, both distributions are anti-invariant,

Mathematics 2019,7, 618 8 of 15

that is

P(˜

D1) = ˜

D2

and

P(˜

D2) = ˜

D1

; therefore,

P1=

0and

P2=

0. However, they are not totally

real distributions.

Example 2.

Taking

a=

0in the previous example, we obtain a semi-slant submanifold, and taking

b=

1, we

obtain a hemi-slant submanifold.

Example 3. For any a,b,c,d with a2−b26=1, c2−d26=1:

x(u1,v1,u2,v2) = (u1,av1,bv1,v1,u2,cv2,dv2,v2),

deﬁnes a bi-slant submanifold, with slant distributions

D1=Span ∂

∂u1,∂

∂v1

and

D2=Span ∂

∂u2,∂

∂v2

.

We can see the different types in the Table 2:

Table 2. Types for Example 3.

D1D2

Type 1 b2−a2<1, b2>1d2−c2<1, d2>1

Type 2 b2−a2<1, b2<1d2−c2<1, d2<1

space-like Type 3 b2−a2>1d2−c2>1

(R8,J2,g2)

P2

1=a2

1+a2−b2Id1

P2

2=c2

1+c2−d2Id2

Type 1 b2−a2>1, a2>1d2−c2<1, d2>1

Type 2 b2−a2>1, a2<1d2−c2<1, d2<1

space-like Type 3 b2−a2<1d2−c2>1

(R8,J3,g3)

P2

1=a2

1+a2−b2Id1

P2

2=c2

1+c2−d2Id2

Type 1 b2−a2>1, a2>1d2−c2>1, c2>1

Type 2 b2−a2>1, a2<1d2−c2>1, c2<1

space-like Type 3 b2−a2<1d2−c2<1

(R8,J4,g4)

P2

1=a2

1+a2−b2Id1

P2

2=c2

1+c2−d2Id2

Now, we are interested in those bi-slant submanifolds of an almost para Hermitian manifold that

are Lorentzian. Let us remember that the only odd-dimensional slant distributions are the totally real

ones and that Type 1 and 2 are neutral distributions. Taking this into account, the only possible cases

are the following:

(i)

M2s+1

1

with

TM =D1⊕D2

, where

D1

is a one-dimensional, time-like, anti-invariant distribution

and D2is a space-like, Type 3 slant distribution.

(ii)

M2s+2

1

with

TM =D1⊕D2

, where

D1

is a two-dimensional, neutral, slant distribution of Type 1 or

2 and D2is a space-like, Type 3 slant distribution.

With Examples 1and 3, we can obtain examples for Case (ii). It only remains to construct a

Case (i) example.

Example 4. Consider in R6the almost para Hermitian structure given by:

J5=

JΘ

Θ0 1

1 0

,g5=

gΘ

Θ1 0

0−1

,

with Θthe corresponding null matrix.

For any k >1,

x(u,v,w) = (u,kcosh v,v,ksinh v,w, 0)

Mathematics 2019,7, 618 9 of 15

deﬁnes a bi-slant submanifold in

(R6

,

J5

,

g5)

with

D1=Span ∂

∂w

a totally real distribution and

D2=Span ∂

∂u,∂

∂va Type 3 slant distribution with P2

2=1

k2−1Id|D2.

We can present a bi-slant submanifold, with the same angle for both slant distributions, that is not

a slant submanifold.

Example 5. The submanifold of (R8,J2,g2)deﬁned by:

x(u1,v1,u2,v2) = (u1,v1+u2,u1,u1,u2,v2,√3u2,u2−v1),

is a bi-slant submanifold. The slant distributions are

D1=Span ∂

∂u1,∂

∂v1

and

D2=Span ∂

∂u2,∂

∂v2

,

with P2

1=1

2Id1and P2

2=1

2Id2. It is not a slant submanifold.

5. Semi-Slant Submanifolds of a Para Kaehler Manifold

It is always interesting to study the integrability of the involved distributions.

Proof.

Let

M

be a semi-slant submanifold of a para Hermitian manifold. Both the holomorphic and

the slant distributions are Pinvariant.

Proof.

Let be

TM =DT⊕D2

the decomposition with

DT

holomorphic and

D2

the slant distribution.

Of course DTis invariant as JDT=DTimplies PDT=DT. Now, consider X∈D2,

JX =P1X+P2X+FX.

Given

Y∈DT

,

g(P1X

,

Y) = g(JX

,

Y) = −g(X

,

JY) =

0, as

DT

is invariant. Moreover, for all

Z∈D2,g(P1X,Z) = 0. Therefore P1X=0 and PX =P2X, so PD2⊆D2.

Theorem 6.

Let

M

be a semi-slant submanifold of a para Kaehler manifold. The holomorphic distribution is

integrable if and only if h(X,JY) = h(JX,Y)for all X,Y∈DT.

Proof.

For

X

,

Y∈DT

,

PX =JX

,

FX =

0,

PY =JY

, and

FY =

0. From (8), it follows that

F[X

,

Y] =

h(X

,

PY)−h(Y

,

PX)

. Then,

[X

,

Y]∈DT

, that is

DT

is integrable, if and only if

h(X

,

JY) = h(JX

,

Y)

.

Theorem 7.

Let

M

be a semi-slant submanifold of a para Kaehler manifold. The slant distribution is integrable

if and only if:

π1(∇XPY − ∇YPX) = π1(AFY X−AFX Y), (18)

for all X,Y∈D2, where π1is the projection over the invariant distribution DT.

Proof. From (7), P1∇XY=π1(∇XPY −th(X,Y)−AFY X). Then:

P1[X,Y] = π1(∇XPY −∇YPX +AFXY−AFY X).

Then, (18) is equivalent to

P1[X

,

Y] =

0. As

P1[X

,

Y] = π1P[X

,

Y] =

0, it holds if and only if

P[X

,

Y]∈

D2. Finally, from Theorem 5,D2is Pinvariant, so we obtain [X,Y]∈D2.

Now, we study the conditions for the involved distributions being totally geodesic.

Proof.

Let

M

be a semi-slant submanifold of a para Kaehler manifold

e

M

. If the holomorphic

distribution DTis totally geodesic, then (∇XP)Y=0, and ∇XY∈DTfor any X,Y∈DT.

Mathematics 2019,7, 618 10 of 15

Proof. For a para Kaehler manifold taking X,Y∈DT, (7)–(8), leads to:

∇XPY −P∇XY−th(X,Y) = 0, (19)

−F∇XY+h(X,PY)−f h(X,Y) = 0. (20)

If DTis totally geodesic, (∇XP)Y=0 and F∇XY=0, which imply the result.

Proof.

Let

M

be a semi-slant submanifold of a para Kaehler manifold

e

M

. The slant distribution

D2

is

totally geodesic if and only if (∇XF)Y=0, and (∇XP)Y=AFY Xfor any X,Y∈D2.

Proof. If D2is a totally geodesic distribution, from (7) and (8), taking X,Y∈D2:

∇XPY −AFY X−P∇XY=0, (21)

∇⊥

XFY −F∇XY=0. (22)

which implies the given conditions. On the converse, if

(∇XP)Y=AFY X

, then

th(X

,

Y) =

0, which

implies

Jh(X

,

Y) = f h(X

,

Y)

. From (8) and

∇F=

0, it holds that

h(X

,

PY) = nh(X

,

Y)

. Then, for

PY ∈D2:

λh(X,Y) = h(X,P2Y)=f2h(X,Y) = J2h(X,Y) = h(X,Y),

and as D2is a proper slant distribution, λ6=1, it must be h(X,Y) = 0 for all X,Y∈D2.

Given two orthogonal distributions

D1

and

D2

over a submanifold, it is called a

D1−D2

-mixed

totally geodesic if h(X,Y) = 0 for all X∈D1,Y∈D2.

Proof.

Let

M

be a semi-slant submanifold of a para Hermitian manifold

e

M

.

M

is a mixed totally

geodesic if and only if ANX∈Difor any X∈Di,N∈T⊥M,i=1, 2.

Proof. If Mis a DT−D2mixed totally geodesic, for any X∈DT,Y∈D2,

g(ANX,Y) = g(h(X,Y),N) = 0,

which implies ANX∈DT. The same proof is valid for X∈D2and for the converse.

Proof. Let Mbe a semi-slant submanifold of a para Kaehler manifold e

M. If ∇F=0, then either Mis

DT−D2

-mixed totally geodesic or

h(X

,

Y)

is a eigenvector of

f2

associated with the eigenvalue of one,

for all X∈DT,Y∈D2.

Proof. Let be X∈DT,Y∈D2, if ∇F=0, from (8), f h(X,Y) = h(X,PY).

As DTis holomorphic, that is J-invariant, D2is P-invariant. Therefore,

f2h(X,Y) = f h(X,PY) = h(X,P2Y) = h(X,P2

2Y) = λh(X,Y),

with λ=cosh2θ(cos2θ, sinh2θ, respectively). However, also:

f2h(Y,X) = f h(Y,PX) = h(Y,P2X) = h(Y,X).

From both equations, either h(X,Y) = 0 or it is an eigenvalue of f2associated with λ=1.

Proof.

Let

M

be a mixed totally geodesic semi-slant submanifold of a para Kaehler manifold

e

M

. If

DT

is integrable, then PANX=ANPX, for all X∈DTand N∈T⊥M.

Proof. From Theorem 6,h(X,JY) = h(Y,J X)for all X,Y∈DT,

g(JANX,Y) = −g(ANX,PY) = −g(N,h(X,PY)) = −g(N,h(Y,PX)) = −g(ANPY,Y).

Mathematics 2019,7, 618 11 of 15

Given Z∈D2,

g(JANX,Z) = −g(ANX,PZ) = −g(N,h(X,PZ)) = 0,

because

M

is mixed totally geodesic. From both equations,

PANX=ANPX

, which ﬁnishes

the proof.

Finally, the mixed totally geodesic characterization can be summarized with:

Theorem 8.

Let

M

be a proper semi-slant submanifold of a para Kaehler manifold

e

M

.

M

is a

DT−D2

-mixed

totally geodesic if and only if (∇XP)Y=AFY X and (∇XF)Y=0, for all X,Y in different distributions.

Proof.

On the one hand, if

M

is a

DT−D2

-mixed totally geodesic, let

X

,

Y

belong to different

distributions. From (7) and (8), both conditions are deduced.

On the other hand, from (7) and

(∇XP)Y=AFY X

, it is deduced

th(X

,

Y) =

0. From (8) and

(∇XF)Y=0, it is deduced:

h(X,PY) = f h(X,Y), (23)

for all X,Yin different distributions.

Therefore, for X∈DTand Y∈D2:

f2h(X,Y) = h(X,P2Y) = λh(X,Y)

and also:

f2h(Y,X) = h(Y,P2X) = h(Y,X).

As

M

is a proper semi-slant submanifold,

λ6=

1, and

h(X

,

Y) =

0, so

M

is a mixed totally geodesic.

6. Hemi-Slant Submanifolds of a Para Kaehler Manifold

We will also study the integrability of the involved distributions for a hemi-slant submanifold.

Proof.

Let

M

be a hemi-slant submanifold of a para Hermitian manifold. The slant distribution is

Pinvariant.

Proof.

Let be

TM =D⊥⊕D2

, the decomposition with

D⊥

totally real, and

D2

the slant distribution.

Consider X∈D2,

JX =P1X+P2X+FX.

Given

Y∈D⊥

,

g(PX

,

Y) = g(JX

,

Y) = −g(X

,

JY) =

0, as

D⊥

is totally real, therefore

PD2⊆D2

.

As P2

2=λId, given X∈D2,X=P1

λX, then X∈PD2, and it is proven that PD2=D2.

Lemma 1.

Let

M

be a hemi-slant submanifold of a para Kaehler manifold. The totally real distribution is

integrable if and only if AFX Y=AFY X for all X,Y∈D⊥.

Proof.

For

X

,

Y∈D⊥

,

PX =

0,

JX =FX

,

PY =

0, and

JY =FY

. From (7), it follows that

P[X

,

Y] =

AFX Y−AFY X. Then, [X,Y]∈D⊥, that is D⊥is integrable, if and only if AFX Y=AFY X.

The following result is known for hemi-slant submanifolds of Kaehler manifolds [

14

]. We obtain

the equivalent one for hemi-slant submanifolds of para Kaehler manifolds:

Theorem 9.

Let

M

be a hemi-slant submanifold of a para Kaehler manifold. The totally real distribution is

always integrable.

Proof.

From the previous lemma, it is enough to prove

g(AFX Y

,

Z) = g(AFY X

,

Z)

, for

X

,

Y∈D⊥

and

Ztangent. Then,

g(AFX Y,Z) = g(h(Y,Z),FX) = g(−th(Y,Z),X) =

Mathematics 2019,7, 618 12 of 15

using (7):

=g(P∇ZY+AFY Z,X) = g(AFY Z,X) = g(AF Y X,Z),

which ﬁnishes the proof.

Now, we study the integrability of the slant distribution.

Theorem 10.

Let

M

be a hemi-slant submanifold of a para Kaehler manifold. The slant distribution is integrable

if and only if:

π1(∇XPY − ∇YPX) = π1(AFY X−AFX Y), (24)

for all X,Y∈D2, where π1is the projection over the totally real distribution D⊥.

The proof is analogous to the one of Theorem 7.

Lemma 2.

Let

M

be a hemi-slant submanifold of a para Kaehler manifold

e

M

. The totally real distribution

D⊥

is totally geodesic if and only if (∇XF)Y=0, and P∇XY=−AF Y X for any X,Y∈D⊥.

Proof. From (7) and (8) for X,Y∈D⊥:

−P∇XY−AFY X−th(X,Y) = 0, (25)

∇⊥

XFY −F∇XY−f h(X,Y) = 0, (26)

which imply the given conditions.

The same proof of Lemma 5is valid for the slant distribution of a hemi-slant distribution.

Lemma 3.

Let

M

be a hemi-slant submanifold of a para Kaehler manifold

e

M

. The slant distribution

D2

is

totally geodesic if and only if (∇XF)Y=0, and P∇XY=−AF Y X for any X,Y∈D2.

Remember that the classical De Rham–Wu Theorem [

18

,

19

], says that two orthogonal,

complementary, and geodesic foliations (called a direct product structure) in a complete and simply

connected semi-Riemannian manifold give rise to a global decomposition as a direct product of two

leaves. Therefore, from the previous lemmas, it is directly deduced:

Theorem 11.

Let

M

be a complete and simply-connected hemi-slant submanifold of a para Kaehler manifold

e

M

.

Then,

M

is locally the product of the integral submanifolds of the slant distributions if and only if

(∇XF)Y=

0,

and P∇XY=−AFY X for both any X,Y∈D⊥or X,Y∈D2.

Finally, we can also study when a hemi-slant submanifold is mixed totally geodesic. We get a

result similar to Proposition 8, but now the proof is much easier.

Proof.

Let

M

be a hemi-slant submanifold of a para Kaehler manifold

e

M

.

M

is a

D⊥−D2

-mixed totally

geodesic if and only if (∇XP)Y=AFY Xand (∇XF)Y=0, for all X,Yin different distributions.

Proof.

Again, if

M

is a

D⊥−D2

-mixed totally geodesic and

X

,

Y

belong to different distributions,

from (7) and (8), both conditions are deduced.

Now, if we suppose both conditions, from (7) and (8), it is deduced that

th(x

,

Y) =

0 and

h(X

,

PY) = f h(X

,

Y)

. Therefore, taking

X∈D2

and

Y∈D⊥

, we get

th(X

,

Y) =

0 and

f h(X

,

Y) =

0.

Therefore, h(X,Y) = 0 and Mis a mixed totally geodesic.

Mathematics 2019,7, 618 13 of 15

7. CR-Submanifolds of a Para Kaehler Manifold

CR-submanifolds have been intensively studied in many environments [

20

]. Moreover, there

are also some works about CR submanifolds of para Kaehler manifolds [

21

]. A semi-Riemannian

submanifold

M

of an almost para Hermitian manifold is called a CR-submanifold if the tangent bundle

admits a decomposition

TM =D⊕D⊥

with

D

a holomorphic distribution, that is

JD =D

, and

D⊥

a

totally real one, that is J D ⊆T⊥M.

Now, we make a study similar to the one made for generalized complex space forms in [22].

Examples of CR-submanifolds can be obtained from Example 1. Taking

a=

1,

d=

0,

D1=Span ∂

∂u1,∂

∂v1

is a totally real distribution, and

D2=Span ∂

∂u2,∂

∂v2

is a holomorphic

distribution. Moreover:

(1) D1is Type 1 if b2<1

(2) D1is Type 2 if b2>1,

(3) D2is Type 2 if c2>1,

(4) D2is Type 3 if c2<1.

Therefore, we obtain examples of CR-submanifolds of Types 1-2, 1-3, 2-2, and 2-3. Taking

a=0, d=1 we can obtain the Types 2-1, 2-2, 3-1, and again 3-2 examples.

For a para Kaehler manifold with constant holomorphic curvature for every non-light-like vector

ﬁeld, that is e

R(X,JX,JX,X) = c, the curvature tensor is given by:

e

R(X,Y)Z=c

4{g(X,Z)Y−g(Y,Z)X+g(X,JZ)JY −g(Y,JZ)JX +2g(X,JY)JZ}; (27)

such a manifold is called a para complex space form.

Theorem 12.

Let

M

be a slant submanifold of a para Kaehler space form

e

M(c)

. Then,

M

is a proper CR

submanifold if and only if the maximal holomorphic subspace

Dp=TpMTJTpM

,

p∈M

, deﬁnes a non-trivial

differentiable distribution on M such as:

e

R(D,D,D⊥,D⊥) = 0,

where D⊥denotes the orthogonal complementary of D on TM.

Proof. If Mis a CR submanifold, from (27):

e

R(X,Y)Z=2g(X,JY)JZ,

for all X,Y∈Dand Z∈D⊥, and this is normal to M; therefore, the equality holds.

On the other hand, let

Dp=TpMTJTpM

be, and suppose

e

R(D

,

D

,

D⊥

,

D⊥) =

0. Again,

from (27),

e

R(X,JX,Z,W) = c

2g(X,X)g(JZ,W),

for every

X∈D

,

Z

,

W∈D⊥

. Taking

X6=

0, a non-light-like vector, it follows that

g(JZ

,

W) =

0. Then,

JZ

is orthonormal to

D⊥

, and it is normal. Therefore,

D⊥

is anti-invariant, and

M

is a

CR submanifold.

There is a well-known result for CR submanifolds of a complex space form

e

M(c)

[

22

] establishing

that if the invariant distribution is integrable, then the holomorphic sectional curvature determined by

a unit vector ﬁeld,

X∈D

, is upper bounded by the global holomorphic sectional curvature. That is,

for every unit vector ﬁeld X:

H(X) = R(X,JX,JX,X)≤c.

Mathematics 2019,7, 618 14 of 15

The situation in the semi-Riemannian case, for a para complex space form, is completely different.

From (27) and (4), for every non-light-like tangent unit vector ﬁeld X, it holds that:

R(X,JX,JX,X) = c+g(h(X,X),h(JX,JX)) −g(h(X,JX),h(X,JX)).

Now, if Dis integrable, from Theorem 6,h(JX,JX) = h(X,J2X) = h(X,X), and then:

H(X) = c+kh(X,X)k2−kh(X,JX)k2.

A submanifold is called totally umbilical if there exists a normal vector ﬁeld

L

such as

h(X

,

Y) =

g(X

,

Y)L

for all tangent vector ﬁelds

X

,

Y

.Totally geodesic submanifolds are particular cases with

L=

0.

Theorem 13. There do not exist proper CR totally umbilical submanifolds of a para complex space form e

M(c)

with c 6=0.

Proof. From (27), it follows that:

(e

R(X,Y)Z)⊥=c

4{g(X,JZ)FY −g(Y,JZ)FX +2g(X,JY)FZ},

for all

X

,

Y

,

Z

tangent vectors ﬁelds. Supposing

M

is a proper CR submanifold, we can choose two

non-light-like vector ﬁelds X∈Dand Z∈D⊥; for them:

(e

R(X,JX)Z)⊥=c

2g(X,X)FZ.

However, for a totally umbilical submanifold, Codazzi’s equation (5) gives:

(e

R(X,Y)Z)⊥=∇⊥

Xg(Y,Z)L−g(∇XY,Z)L−g(Y,∇XZ)L−∇⊥

Yg(X,Z)L+g(∇YX,Z)L+g(X,∇YZ)L=0.

Comparing both equations, if c6=0, it follows that FZ =0, which is a contradiction.

Moreover, the same proof is valid for asserting:

Corollary 1.

There do not exist proper semi-slant totally umbilical submanifolds of a para complex space form

e

M(c)with c 6=0.

Author Contributions: The authors contributed equally to this work.

Funding:

The work was supported by the MINECO-FEDER grant MTM2014-52197-P., the PAIDI group FQM-327

(Junta de Andalucía, Spain). The second author is also supported by the Instituto de Matemáticas de la Universidad

de Sevilla (IMUS).

Acknowledgments:

The authors would like to thank Benjamin Olea for his useful comments on some aspects of

this paper and the referees for their suggestions, which have improved it.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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