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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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