Bi-slant submanifolds of para Hermitian manifolds
aa r X i v : . [ m a t h . G M ] J a n BI-SLANT SUBMANIFOLDS OF PARA HERMITIAN MANIFOLDS
PABLO ALEGRE AND ALFONSO CARRIAZO
Abstract.
In this paper we introduce the notion of bi-slant submanifolds of a para Hermitianmanifold. They naturally englobe CR, semi-slant and hemi-slant submanifolds. We study theirfirst properties and present a whole gallery of examples.2000
Mathematics Subject Classification : 53C40, 53C50.1.
Introduction
In [12], B.-Y. Chen introduced slant submanifolds of an almost Hermitian manifold, as those submanifolds forwhich the angle θ between JX and the tangent space is constant, for any tangent vector field X . They plays anintermediate role between complex submanifolds ( θ = 0) and totally real ones ( θ = π/ Preliminaries
Let f M be a 2 n -dimensional semi-Riemannian manifold. If it is endowed with a structure ( J, g ), where J is a(1 ,
1) tensor, and g is a semi-defined metric, satisfying(2.1) J X = X, g ( JX, Y ) + g ( X, JY ) = 0 , for any vector fields X, Y on f 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 submanifold of ( f M, J, g ). The Gauss and Weingarten formulas are given by(2.2) e ∇ X Y = ∇ X Y + h ( X, Y ) , (2.3) e ∇ X V = − A V X + ∇ ⊥ X N, Mathematics Subject Classification.
Key words and phrases. semi-Riemannian manifold, para Hermitian manifold, para Kaehler manifold, para-complex, totally real, CR, slant, bi-slant, semi-slant and hemi-slant or anti-slant submanifolds.Both authors are partially supported by the MINECO-FEDER grant MTM2014-52197-P. They are membersof the PAIDI group FQM-327 (Junta de Andaluc´ıa, Spain). The second author is also a member of the Institutode Matem´aticas de la Universidad de Sevilla (IMUS). for any tangent vector fields
X, Y and any normal vector field V , where h is the second fundamental form of M , A V is the Weingarten endomorphism associated with V and ∇ ⊥ is the normal connection.And the Gauss and Codazzi equations are given by(2.4) 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 )) , (2.5) ( e R ( X, Y ) Z ) ⊥ = ( e ∇ X h )( Y, Z ) − ( e ∇ Y h )( X, Z ) , for any vectors fields X, Y, Z, W tangent to M .For every tangent vector field X , we write(2.6) JX = P X + F X, where
P X is the tangential component of JX and F X is the normal one. And for every normal vector field V , JV = tV + fV, where tV and fV 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 Gaussand Weingarten formulas (2.2) and (2.3)(2.7) ( ∇ X P ) Y = ∇ X P Y − P ∇ X Y = A F Y X + th ( X, Y ) , (2.8) ( ∇ X F ) Y = ∇ ⊥ X F Y − F ∇ X Y = − h ( X, P Y ) + fh ( X, Y ) , for all tangent vector fields X, Y .In [2], we introduced the notion of slant submanifolds of para Hermitian manifolds, taking into account thatwe can not measure the angle for light-like vector fields:
Definition 2.1. [2] A submanifold M of a para Hermitian manifold ( f M, J, g ) is called slant submanifold if forevery space-like or time-like tangent vector field X , the quotient g ( P X, P X ) /g ( JX, JX ) is constant.
Remark . It is clear that, if M is a para-complex submanifold, then P ≡ J , and so the above quotient is equalto 1. On the other hand, if M is totally real, then P ≡ proper slant .Three cases can be distinguished, corresponding to three different types of proper slant submanifolds: Definition 2.3. [2] Let M be a proper slant submanifold of a para Hermitian manifold ( f M, J, g ). We say thatit is oftype 1 if for any space-like (time-like) vector field X , P X is time-like (space-like), and | P X || JX | > X , P X is time-like (space-like), and | P X || JX | < X , P X is space-like (time-like).These three types can be characterized as follows:
Theorem 2.4. [2]
Let M be a submanifold of a para Hermitian manifold ( f M, J, g ) . Then, M is slant of type 1 if and only if for any space-like (time-like) vector field X , P X is time-like (space-like),and there exists a constant λ ∈ (1 , + ∞ ) such that (2.9) P = λId. We write λ = cosh θ , with θ > . M is slant of type 2 if and only if for any space-like (time-like) vector field X , P X is time-like (space-like),and there exists a constant λ ∈ (0 , such that (2.10) P = λId. We write λ = cos θ , with < θ < π . I-SLANT SUBMANIFOLDS OF PARA HERMITIAN MANIFOLDS 3 M is slant of type 3 if and only if for any space-like (time-like) vector field X , P X is space-like (time-like),and there exists a constant λ ∈ ( −∞ , such that (2.11) P = λId. We write λ = − sinh θ , with θ > .In every case, we call θ the slant angle .Remark . It was proved in [2] that conditions (2.9), (2.10) and (2.11) 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. Also,that 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 P . In [2] we did not consider thatcase, but it will be useful in the present study. Theorem 2.6.
Let M be a submanifold of a para Hermitian manifold ( f M , J, g ) . Then, M is a para-complex submanifold if and only if P = Id. M is a totally real submanifold if and only if P = 0 . Proof. If M is para-complex, P = J = Id directly. Conversely, if P = Id , from g ( JX, JX ) = g ( P X, P X ) + g ( F X, F X ) , we have − g ( X, J X ) = − g ( X, P X ) + g ( F X, F X ) , then − g ( X, X ) = − g ( X, X ) + g ( F X, F X ) , and hence g ( F X, F X ) = 0, which implies F = 0.The second statement can be proved in a similar way. (cid:3) Slant distributions
In [19], 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 non zero vector X ∈ D x , x ∈ e N , the angle between JX and the vector space D x is constant, that is, is independent of the point x . If P D X is the projection of JX over D , they can be characterized as P D = λI . This, together with the definition of slantsubmanifolds of a para Hermitian manifold, aims us to give the following: Definition 3.1.
A differentiable distribution D on a para Hermitian manifold ( f M, J, g ) is called a slant distri-bution if for every non light-like X ∈ D , the quotient g ( P D X, P D X ) /g ( JX, JX ) is constant .A distribution is called invariant if it is slant with slant angle 0, that is if g ( P D X, P D X ) /g ( JX, JX ) = 1 forall non light-like X ∈ D . And it is called anti-invariant if P D X = 0 for all X ∈ D . In other case it is called proper slant distribution .With this definition every one dimensional distribution defines an anti-invariant distribution in f M , so we arejust going to take under study non trivial slant distributions, that is with dimensions greater than 1. Just like forslant submanifolds, we can consider three cases depending on the casual character of the implied vector fields.Obviously, a submanifold M is a slant submanifold if and only if T M is a slant distribution.
Definition 3.2.
Let D be a proper slant distribution of a para Hermitian manifold ( ˜ M , J, g ). We say that it isoftype 1 if for every space-like (time-like) vector field X , P D X is time-like (space-like), and | P D X || JX | > X , if P D X is time-like (space-like), and | P D X || JX | < X , P D X is space-like (time-like). P. ALEGRE AND A. CARRIAZO
Theorem 3.3.
Let D be a distribution of a para Hermitian metric manifold f M . Then, D is a slant distribution of type 1 if and only for any space-like (time-like) vector field X , P D X istime-like (space-like), and there exits a constant λ ∈ (1 , + ∞ ) such that (3.1) P D = λI Moreover, in such a case, λ = cosh θ . D is a slant distribution of type 2 if and only for any space-like (time-like) vector field X , P D X istime-like (space-like), and there exits a constant λ ∈ (0 , such that (3.2) P D = λI Moreover, in such a case, λ = cos θ . D is a slant distribution of type 3 if and only for any space-like (time-like) vector field X , P D X isspace-like (time-like), and there exits a constant λ ∈ (0 , + ∞ ) such that (3.3) P D = λI Moreover, in such a case, λ = sinh θ .In each case, we call θ the slant angle .Proof. In the first case, if D is a slant distribution of type 1, for any space-like tangent vector field X ∈ D , P D X is time-like, and, by virtue of (2.1), JX also is. Moreover, they satisfy | P D X | / | JX | >
1. So, there exists θ > θ = | P D X || JX | = p − g ( P D X, P D X ) p − g ( JX, JX ) . If we now consider P D X , then, in a similar way, we obtain:(3.5) cosh θ = | P D X || JP D X | = | P D X || P D X | . On the one hand,(3.6) g ( P D X, X ) = g ( JP D X, X ) = − g ( P D X, JX ) = − g ( P D X, P D X ) = | P D X | . Therefore, using (3.4), (3.5) and (3.6) g ( P D X, X ) = | P D X | = | P D X || JX | = | P D X || X | . On the other hand, since both X and P D X are space-like, it follows that they are collinear, that is P D X = λX .Finally, from (3.4) we deduce that λ = cosh θ .Everything works in a similar way for any time-like tangent vector field Y ∈ D , but now, P D Y and JY arespace-like and so, instead of (3.4) we should write:cosh θ = | P D Y || JY | = p g ( P D Y, P D Y ) p g ( JY, JY ) . Since P D X = λX , for any space-like or time-like X ∈ D , it also holds for light-like vector fields and so we havethat P D = λId D .The converse is just a simple computation.In the second case, if D is a slant distribution of type 2, for any space-like or time-like vector field X ∈ D , | P D X | / | JX | <
1, and so there exists θ > θ = | P D X || JX | = p − g ( P D X, P D X ) p − g ( JX, JX ) . Proceeding as before, we can prove that g ( P D X, X ) = | P D X || X | and, as both X and P D X are space-likevector fields, it follows that they are collinear, that is P D X = λX . Again, the converse is a direct computation.Finally, if D is a slant distribution of type 3, for any space-like vector field X ∈ D , P D X is also space-like, andthere exists θ > θ = | P D X || JX | = p g ( P D X, P D X ) p − g ( JX, JX ) . I-SLANT SUBMANIFOLDS OF PARA HERMITIAN MANIFOLDS 5
Once more, we can prove that g ( P D X, X ) = | P D X || X | and P D X = λX . And again, the converse is a directcomputation. (cid:3) Remember that an holomorphic distribution satisfies JD = D , so every holomorphic distribution is a slantdistribution with angle 0, but the converse is not true. And it is called totally real distribution if JD ⊆ T ⊥ M ,therefore every totally real distribution is anti-invariant but the converse does not always hold. For holomorphicand totally real distributions the following necessary conditions are easy to prove: Theorem 3.4.
Let D be a distribution of a submanifold of a para Hermitian metric manifold f M . If D is a holomorphic distribution then | P D X | = | JX | , for all X ∈ D . If D is a totally real distribution then | P D X | = 0 , for all X ∈ D . However the converse results do not hold if D is not T M ; in such a case
T M = D ⊕ ν , and for a unit vectorfield X JX = P D X + P ν X + F X.
Therefore from g ( JX, JX ) = g ( P D X, P D X ) + g ( P ν X, P ν X ) + g ( F X, F X ) , and | P D X | = | JX | , in the case that P D X is also space-like, it is only deduced that g ( P ν X, P ν X ) + g ( F X, F X ) = − , or, in the case it is time-like, g ( P ν X, P ν X ) + g ( F X, F X ) = 0 . So in general
F X = 0, and D is not invariant.Similarly it can be shown that the converse of the second statement does not always hold. Theorem 3.5.
Let M be a submanifold of a para Hermitian metric manifold f M . The maximal holomorphic distribution is characterized as D = { X/F X = 0 } . The maximal totally real distribution is characterized as D ⊥ = { X/P X = 0 } .Proof. For the first statement, if a distribution D is holomorphic, obviusly F ⌉ D = 0. For the converse, consider D = { X/F X = 0 } . We should prove that it is a holomorphic distribution. Let X ∈ D be, JX = T X is tangentto M , and g ( F JX, V ) = g ( J X, V ) = g ( X, V ) = 0 , for all V ∈ T ⊥ M . Therefore F JX = 0. That implies JX ∈ D for all X ∈ D , so D is holomorphic.The second statement is trivial. (cid:3) Bi-slant, semi-slant and hemi-slant submanifolds. Definition and examples.
In [19], semi-slant submanifolds of an almost Hermitian manifold were introduced as those submanifolds whosetangent space could be decomposed as a direct sum of two distributions, one totally real and the other a slantdistribution. In [10], anti-slant submanifolds were introduced as those whose tangent space is decomposed as adirect sum of an anti-invariant and a slant distribution; they were called hemi-slant submanifolds in [23]. Finally,in [9], the authors defined bi-slant submanifolds with both distributions slant ones.
Definition 4.1.
A submanifold M of a para Hermitian manifold ( f M, J, g ) is called a bi-slant submanifold if thetangent space admits a decomposition
T M = D ⊕ D with both D and D slant distributions.It is called semi-slant submanifold if T M = D ⊕ D with D a holomorphic distribution and D a properslant distribution. In such a case, we will write D = D T .And it is called hemi-slant submanifold if T M = D ⊕ D with D a totally real distribution and D a properslant distribution. In such a case, we will write D = D ⊥ . Remark . As we have said before, being holomorphic (totally real) is a stronger condition than being slantwith slant angle 0 ( π/ P. ALEGRE AND A. CARRIAZO
We write π i the projections over D i and P i = π i ◦ P , i = 1 , R : J = , g = − − , and J = , g = − − . Using the examples of slant submanifolds of R given in [2] and making products, we can obtain examples ofbi-slant submanifolds in R . To present different examples with all the combinations of slant distributions, weconsider the following para Kaehler structures over R : J = (cid:18) J ΘΘ J (cid:19) , g = (cid:18) g ΘΘ g (cid:19) ,J = (cid:18) J ΘΘ J (cid:19) , g = (cid:18) g ΘΘ g (cid:19) ,J = (cid:18) J ΘΘ J (cid:19) , g = (cid:18) g ΘΘ g (cid:19) , Example . For any a, b, c, d ∈ R with a + b = 1, and c + d = 1, x ( u , v , u , v ) = ( au , v , bu , u , cu , v , du , u )defines a bi-slant submanifold in ( R , J , g ), with slant distributions D = Span (cid:26) ∂∂u , ∂∂v (cid:27) and D = Span (cid:26) ∂∂u , ∂∂v (cid:27) .We can see the different types in the following table: D D type 1 a + b > , b < c + d > , c < a + b > , b > c + d > , c > a + b < c + d < R , J , g ) P = a − a + b Id P = c − c + d Id Remark . The decomposition of
T M in two slant distributions it is not unique, for example, if we choose˜ D = Span (cid:26) ∂∂u , ∂∂v (cid:27) and ˜ D = Span (cid:26) ∂∂u , ∂∂v (cid:27) in the previous example, both distributions are anti-invariant, that is P ( ˜ D ) = ˜ D and P ( ˜ D ) = ˜ D ; therefore P = 0 and P = 0. However they are not totally realdistributions. Example . Taking a = 0 in the previous example we obtain a semi-slant submanifold, and taking b = 1 weobtain a hemi-slant submanifold. Example . For any a, b, c, d with a − b = 1, c − d = 1 x ( u , v , u , v ) = ( u , av , bv , v , u , cv , dv , v ) , defines a bi-slant submanifold, with slant distributions D = Span (cid:26) ∂∂u , ∂∂v (cid:27) and D = Span (cid:26) ∂∂u , ∂∂v (cid:27) . Wecan see the different types in the following table: I-SLANT SUBMANIFOLDS OF PARA HERMITIAN MANIFOLDS 7 D D type 1 b − a < , b > d − c < , d > b − a < , b < d − c < , d < b − a > d − c > R , J , g ) P = a a − b Id P = c c − d Id type 1 b − a > , a > d − c < , d > b − a > , a < d − c < , d < b − a < d − c > R , J , g ) P = a a − b Id P = c c − d Id type 1 b − a > , a > d − c > , c > b − a > , a < d − c > , c < b − a < d − c < R , J , g ) P = a a − b Id P = c c − d Id 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 and2 are neutral distributions. Taking this into account the only possible cases are the following:i) M s +11 with T M = D ⊕ D , where D is a one dimensional, time-like, anti-invariant distribution and D is a space-like, type 3 slant distribution.ii) M s +21 with T M = D ⊕ D , where D is a two dimensional, neutral, slant distribution of type 1 or 2,and D is a space-like, type 3 slant distribution.With examples 4.3 and 4.6 we can obtain examples for the ii) case. It only remains to construct a case i)example. Example . Consider in R the almost para Hermitian structure given by J = J ΘΘ 0 11 0 , g = g ΘΘ 1 00 − . For any k > x ( u, v, w ) = ( u, k cosh v, v, k sinh v, w, R , J , g ) with D = Span (cid:26) ∂∂w (cid:27) a totally real distribution and D =Span (cid:26) ∂∂u , ∂∂v (cid:27) a type 3 slant distribution with P = 1 k − Id ⌉ D .We can present a bi-slant submanifold, with the same angle for both slant distributions, that is not a slantsubmanifold. Example . The submanifold of ( R , J , g ) defined by x ( u , v , u , v ) = ( u , v + u , u , u , u , v , √ u , u − v ) , is a bi-slant submanifold. The slant distributions are D = Span (cid:26) ∂∂u , ∂∂v (cid:27) and D = Span (cid:26) ∂∂u , ∂∂v (cid:27) , with P = 12 Id and P = 12 Id . 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.
P. ALEGRE AND A. CARRIAZO
Proposition 5.1.
Let M be a semi-slant submanifolds of a para Hermitian manifold. Both the holomorphic andthe slant distributions are P invariant.Proof. Let be
T M = D T ⊕ D the decomposition with D holomorphic and D the slant distribution. Of course D T is invariant as JD T = D T implies P D T = D T . Now, consider X ∈ D , JX = P X + P X + F X.
Given Y ∈ D T , g ( P X, Y ) = g ( JX, Y ) = − g ( X, JY ) = 0, as D T is invariant. Moreover, for all Z ∈ D , g ( P X, Z ) = 0. Therefore P X = 0, and P X = P X , so P D ⊆ D . (cid:3) Theorem 5.2.
Let M be a semi-slant submanifold of a para Kaehler manifold. The holomorphic distribution isintegrable if and only if h ( X, JY ) = h ( JX, Y ) for all X, Y ∈ D T .Proof. For
X, Y ∈ D T , P X = JX , F X = 0,
P Y = JY and F Y = 0. From (2.8) it follows F [ X, Y ] = h ( X, P Y ) − h ( Y, P X ). Then, [
X, Y ] ∈ D T , that is D T is integrable, if and only if h ( X, JY ) = h ( JX, Y ). (cid:3) Theorem 5.3.
Let M be a semi-slant submanifold of a para Kaehler manifold. The slant distribution is integrableif and only if (5.1) π ( ∇ X P Y − ∇ Y P X ) = π ( A F Y X − A F X Y ) , for all X, Y ∈ D , where π is the projection over the invariant distribution D T .Proof. From (2.7), P ∇ X Y = π ( ∇ X P Y − th ( X, Y ) − A F Y X ). Then P [ X, Y ] = π ( ∇ X P Y − ∇ Y P X + A F X Y − A F Y X ) . Then (5.1) is equivalent to P [ X, Y ] = 0. As P [ X, Y ] = π P [ X, Y ] = 0, it holds if and only if P [ X, Y ] ∈ D .Finally, from Theorem 5.1 D is P invariant, so we obtain [ X, Y ] ∈ D . (cid:3) Now we study conditions for the involved distributions being totally geodesic.
Proposition 5.4.
Let M be a semi-slant submanifold of a para Kaehler manifold f M . If the holomorphic distri-bution D T is totally geodesic then ( ∇ X P ) Y = 0 , and ∇ X Y ∈ D T for any X, Y ∈ D T .Proof. For a para Kaehler manifold taking
X, Y ∈ D T , (2.7)-(2.8) leads to(5.2) ∇ X P Y − P ∇ X Y − th ( X, Y ) = 0 , (5.3) − F ∇ X Y + h ( X, P Y ) − fh ( X, Y ) = 0 . If D T is totally geodesic, ( ∇ X P ) Y = 0 and F ∇ X Y = 0, which imply the result. (cid:3) Note that for semi-slant submanifolds of para Kaehler manifolds, on the opposite that for Kaehler manifolds[19].
Proposition 5.5.
Let M be a semi-slant submanifold of a para Kaehler manifold f M . The slant distribution D is totally geodesic if and only if ( ∇ X F ) Y = 0 , and ( ∇ X P ) Y = A F Y X for any X, Y ∈ D .Proof. If D is a totally geodesic distribution, from (2.7) and (2.8), taking X, Y ∈ D (5.4) ∇ X P Y − A F Y X − P ∇ X Y = 0 , (5.5) ∇ ⊥ X F Y − F ∇ X Y = 0 . which implies the given conditions. On the converse, if ( ∇ X P ) Y = A F Y X , then th ( X, Y ) = 0, which implies Jh ( X, Y ) = fh ( X, Y ). From (2.8) and ∇ F = 0, it holds h ( X, P Y ) = nh ( X, Y ). Then for
P Y ∈ D λh ( X, Y ) = h ( X, P Y ) = f h ( X, Y ) = J h ( X, Y ) = h ( X, Y ) , and as D is a proper slant distribution, λ = 1, it must be h ( X, Y ) = 0 for all
X, Y ∈ D . (cid:3) I-SLANT SUBMANIFOLDS OF PARA HERMITIAN MANIFOLDS 9
Given two orthogonal distributions D and D over a submanifold, it is called D − D - mixed totally geodesic if h ( X, Y ) = 0 for all X ∈ D , Y ∈ D . Proposition 5.6.
Let M be a semi-slant submanifold of a para Hermitian manifold f M . M is mixed totallygeodesic if and only if A N X ∈ D i for any X ∈ D i , N ∈ T ⊥ M , i = 1 , .Proof. If M is D T − D mixed totally geodesic, for any X ∈ D T , Y ∈ D , g ( A N X, Y ) = g ( h ( X, Y ) , N ) = 0 , which implies A N X ∈ D T . The same proof is valid for X ∈ D and for the converse. (cid:3) Proposition 5.7.
Let M be a semi-slant submanifold of a para Kaehler manifold f M . If ∇ F = 0 , then either M is D T − D -mixed totally geodesic or h ( X, Y ) is a eigenvector of f associated with the eigenvalue 1, for all X ∈ D T , Y ∈ D .Proof. Let be X ∈ D T , Y ∈ D , if ∇ F = 0, from (2.8), fh ( X, Y ) = h ( X, P Y ).As D T is holomorphic, that is J -invariant, D is P -invariant. Therefore, f h ( X, Y ) = fh ( X, P Y ) = h ( X, P Y ) = h ( X, P Y ) = λh ( X, Y ) , with λ = cosh θ (cos θ, sinh θ respectively). But also f h ( Y, X ) = fh ( Y, P X ) = h ( Y, P X ) = h ( Y, X ) . From both equations, either h ( X, Y ) = 0 or it is a eigenvalue of f associated with λ = 1. (cid:3) Proposition 5.8.
Let M be a mixed totally geodesic semi-slant submanifold of a para Kaehler manifold f M . If D T is integrable, then P A N X = A N P X , for all X ∈ D T and N ∈ T ⊥ M .Proof. From Theorem 5.2, h ( X, JY ) = h ( Y, JX ) for all
X, Y ∈ D T , g ( JA N X, Y ) = − g ( A N X, P Y ) = − g ( N, h ( X, P Y )) = − g ( N, h ( Y, P X )) = − g ( A N P Y, Y ) . And given Z ∈ D , g ( JA N X, Z ) = − g ( A N X, P Z ) = − g ( N, h ( X, P Z )) = 0 , because M is mixed totally geodesic. From both equations P A N X = A N P X what finishes the proof. (cid:3)
Finally the mixed-totally geodesic characterization can be summarized with
Theorem 5.9.
Let M be a proper semi-slant submanifold of a para Kaehler manifold f M . M is D T − D -mixedtotally geodesic if and only if ( ∇ X P ) Y = A F Y X and ( ∇ X F ) Y = 0 , for all X, Y in different distributions.Proof.
On the one hand, if M is D T − D -mixed totally geodesic, let be X, Y belonging to different distributions.From (2.7) and (2.8), both conditions are deduced.On the other hand, from (2.7) and ( ∇ X P ) Y = A F Y X , it is deduced th ( X, Y ) = 0. And from (2.8) and( ∇ X F ) Y = 0 it is deduced(5.6) h ( X, P Y ) = fh ( X, Y ) , for all X, Y in different distributions.Therefore, for X ∈ D T and Y ∈ D f h ( X, Y ) = h ( X, P Y ) = λh ( X, Y )and also f h ( Y, X ) = h ( Y, P X ) = h ( Y, X ) . As M is a proper semi-slant submanifold, λ = 1, and h ( X, Y ) = 0 so M is mixed totally geodesic. (cid:3) Hemi-slant submanifolds of a para Kaehler manifold.
We will also study the integrability of the involved distributions for a hemi-slant submanifold.
Proposition 6.1.
Let M be a hemi-slant submanifolds of a para Hermitian manifold. The slant distribution is P invariant.Proof. Let be
T M = D ⊥ ⊕ D the decomposition with D ⊥ totally real and D the slant distribution. Consider X ∈ D , JX = P X + P X + F X.
Given Y ∈ D ⊥ , g ( P X, Y ) = g ( JX, Y ) = − g ( X, JY ) = 0, as D ⊥ is totally real, therefore P D ⊆ D . As P = λId , given X ∈ D , X = P (cid:0) λ X (cid:1) , then X ∈ P D and it is proved that P D = D . (cid:3) Lemma 6.2.
Let M be a hemi-slant submanifold of a para Kaehler manifold. The totally real distribution isintegrable if and only if A F X Y = A F Y X for all X, Y ∈ D ⊥ .Proof. For
X, Y ∈ D ⊥ , P X = 0, JX = F X , P Y = 0 and JY = F Y . From (2.7) it follows P [ X, Y ] = A F X Y − A F Y X . Then [ X, Y ] ∈ D ⊥ , that is D ⊥ is integrable, if and only if A F X Y = A F Y X . (cid:3) The following result was known for hemi-slant submanifolds of Kaehler manifolds, [23]. We obtain the equiv-alent one for hemi-slant submanifolds of para Kaehler manifolds:
Theorem 6.3.
Let M be a hemi-slant submanifold of a para Kaehler manifold. The totally real distribution isalways integrable.Proof. From the previous lemma it is enough to prove g ( A F X
Y, Z ) = g ( A F Y
X, Z ), for
X, Y ∈ D ⊥ and Z tangent.Then, g ( A F X
Y, Z ) = g ( h ( Y, Z ) , F X ) = g ( − th ( Y, Z ) , X ) =using (2.7) = g ( P ∇ Z Y + A F Y
Z, X ) = g ( A F Y
Z, X ) = g ( A F Y
X, Z ) , which finishes the proof. (cid:3) Now we study the integrability of the slant distribution.
Theorem 6.4.
Let M be a hemi-slant submanifold of a para Kaehler manifold. The slant distribution is integrableif and only if (6.1) π ( ∇ X P Y − ∇ Y P X ) = π ( A F Y X − A F X Y ) , for all X, Y ∈ D , where π is the projection over the totally real distribution D ⊥ . The proof is analogous to the one of Theorem 5.3.
Lemma 6.5.
Let M be a hemi-slant submanifold of a para Kaehler manifold f M . The totally real distribution D ⊥ is totally geodesic if and only if ( ∇ X F ) Y = 0 , and P ∇ X Y = − A F Y X for any X, Y ∈ D ⊥ .Proof. From (2.7) and (2.8) for
X, Y ∈ D ⊥ (6.2) − P ∇ X Y − A F Y X − th ( X, Y ) = 0 , (6.3) ∇ ⊥ X F Y − F ∇ X Y − fh ( X, Y ) = 0 , which imply the given conditions. (cid:3) The same proof of Lemma 5.5 is valid for the slant distribution of a hemi-slant distribution.
Lemma 6.6.
Let M be a hemi-slant submanifold of a para Kaehler manifold f M . The slant distribution D istotally geodesic if and only if ( ∇ X F ) Y = 0 , and P ∇ X Y = − A F Y X for any X, Y ∈ D . I-SLANT SUBMANIFOLDS OF PARA HERMITIAN MANIFOLDS 11
Remember that the classical De RhamWu Theorem, [25] [20], says that two orthogonally, complementaryand geodesic foliations (called a direct product structure) in a complete and simply connected semi-Riemannianmanifold give rise to a global decomposition as a direct product of two leaves. Therefore, from the previouslemmas it is directly deduced:
Theorem 6.7.
Let M be a complete and simply connected hemi-slant submanifold of a para Kaehler manifold f M .Then, M is locally the product of the integral submanifolds of the slant distributions if and only if ( ∇ X F ) Y = 0 ,and P ∇ X Y = − A F Y X for both any X, Y ∈ D ⊥ or X, Y ∈ D . Finally, we can also study when a hemi-slant submanifold is mixed totally geodesic. We get a result similar toProposition 5.9, but now the proof is much more easier.
Proposition 6.8.
Let M be a hemi-slant submanifold of a para Kaehler manifold f M . M is D ⊥ − D -mixedtotally geodesic if and only if ( ∇ X P ) Y = A F Y X and ( ∇ X F ) Y = 0 , for all X, Y in different distributions.Proof.
Again, if M is D ⊥ − D -mixed totally geodesic, and X, Y belong to different distributions, from (2.7) and(2.8), both conditions are deduced.Now, if we suppose both conditions, from (2.7) and (2.8), it is deduced th ( x, Y ) = 0 and h ( X, P Y ) = fh ( X, Y ).So, taking X ∈ D and Y ∈ D ⊥ , we get th ( X, Y ) = 0 and fh ( X, Y ) = 0. Therefore h ( X, Y ) = 0 and M is mixedtotally geodesic. (cid:3) CR-submanifolds of a para Kaehler manifold.
CR-submanifolds have been intensively studied in many environments. Moreover, there are also some worksabout CR submanifolds of para Kaehler manifolds, [17]. A submanifold M of an almost para Hermitian manifoldis called a CR-submanifold if the tangent bundle admits a decomposition
T M = D ⊕ D ⊥ with D an holomorphicdistribution, that is JD = D , and D ⊥ a totally real one, that is JD ⊆ T ⊥ M .Now we make a study similar to the one made for generalized complex space forms in [4].Examples of CR-submanifolds can be obtained from Example 4.3. Taking a = 1 , d = 0, D = Span (cid:26) ∂∂u , ∂∂v (cid:27) is a totally real distribution and D = Span (cid:26) ∂∂u , ∂∂v (cid:27) is an holomorphic distribution. Moreover:1) D is type 1 if b < D is type 2 if b > D is type 2 if c > D is type 3 if c < a = 0 , d = 1 we can obtain 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(7.1) e R ( X, Y ) Z = c { g ( X, Z ) Y − g ( Y, Z ) X + g ( X, JZ ) JY − g ( Y, JZ ) JX + 2 g ( X, JY ) JZ } ;such a manifold is called a para complex space form . Theorem 7.1.
Let M be a slant submanifold of a para Kaehler space form f M ( c ) . Then, M is a proper CRsubmanifold if and only if the maximal holomorphic subspace D p = T p M T JT p M , p ∈ M , defines a non trivialdifferentiable distribution on M such as e R ( D, D, D ⊥ , D ⊥ ) = 0 , where D ⊥ denotes the orthogonal complementary of D on T M . Proof. If M is a CR submanifold, from (7.1) e R ( X, Y ) Z = 2 g ( X, JY ) JZ, for all
X, Y ∈ D and Z ∈ D ⊥ , and this is normal to M ; therefore the equality holds.On the other hand, let D p = T p M T JT p M be and suppose e R ( D, D, D ⊥ , D ⊥ ) = 0 . Again from (7.1), e R ( X, JX, Z, W ) = c g ( X, X ) g ( JZ, W ) , for every X ∈ D , Z, W ∈ D ⊥ . Taking X = 0 a non-light-like vector, it follows that g ( JZ, W ) = 0. Then JZ isorthonormal to D ⊥ and it is normal. Therefore D ⊥ is anti-invariant and M is a CR submanifold. (cid:3) There is a well known result for CR submanifolds of a complex space form f M ( c ) [4] establishing that if theinvariant 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 XH ( X ) = R ( X, JX, JX, X ) ≤ c. The situation in the semi Riemannian case, for a para complex space form is completely different. From (7.1)and (2.4), for every non-light-like tangent unit vector field X it holds R ( X, JX, JX, X ) = c + g ( h ( X, X ) , h ( JX, JX )) − g ( h ( X, JX ) , h ( X, JX )) . Now, if D is integrable, from Theorem 5.2, h ( JX, JX ) = h ( X, J X ) = h ( X, X ), and then H ( X ) = c + k h ( X, X ) k − k h ( X, JX ) k . A submanifold is called totally umbilical if there exists a normal vector field L such as h ( X, Y ) = g ( X, Y ) L forall tangent vector fields X, Y . Totally geodesic submanifolds are particular cases with L = 0. Theorem 7.2.
There not exits proper CR totally umbilical submanifolds of a para complex space form f M ( c ) with c = 0 .Proof. From (7.1) it follows( e R ( X, Y ) Z ) ⊥ = c { g ( X, JZ ) F Y − g ( Y, JZ ) F X + 2 g ( X, JY ) F Z } , for all X, Y, Z tangent vectors fields. Supposing M is a proper CR submanifold we can choose two non-light-likevector fields X ∈ D and Z ∈ D ⊥ ; for them( e R ( X, JX ) Z ) ⊥ = c g ( X, X ) F Z.
But for a totally umbilical submanifold, Codazzi’s equation (2.5) gives( e R ( X, Y ) Z ) ⊥ = ∇ ⊥ X g ( Y, Z ) L − g ( ∇ X Y, Z ) L − g ( Y, ∇ X Z ) L − ∇ ⊥ Y g ( X, Z ) L + g ( ∇ Y X, Z ) L + g ( X, ∇ Y Z ) L = 0 . Comparing both equations, if c = 0, it follows F Z = 0 which is a contradiction. (cid:3)
Moreover the same proof is valid for asserting:
Corollary 7.3.
There not exits proper semi slant totally umbilical submanifolds of a para complex space form f M ( c ) with c = 0 . I-SLANT SUBMANIFOLDS OF PARA HERMITIAN MANIFOLDS 13
References [1]
P. Alegre . Slant submanifolds of Lorentzian Sasakian and para Sasakian manifolds.
Taiwanese J. Math. (2013), 897-910.[2] P. Alegre and A. Carriazo.
Slant submanifolds of para Hermitian manifolds.
Mediterr. J. Math. (2017)14: 214. https://doi.org/10.1007/s00009-017-1018-3[3]
K. Arslan, A. Carriazo, B. Y. Chen and C. Murathan.
On slant submanifolds of neutral Kaehlermanifolds.
Taiwanesse J. Math. No. 2 (2010), 561-584.[4]
M. Barros and F. Urbano.
CR-submanifolds of generalized complex space forms.
An. Stiint. Al. I. Cuza.Univ. Iasi. (1979), 855-863.[5] A. Bejancu.
CR submanifolds of a Kaehler manifold.
Trans. Amer. Math. Soc. (1979), 333-345.[6]
D. E. Blair.
Contact Manifolds in Riemannian Geometry . Lecture Notes in Math. 509, Springer-Verlag,New York, 1976.[7]
D. E. Blair.
Riemannian Geometry of Contact and Symplectic Manifolds . Progress in Mathematics, 203,Birkh¨auser Boston, Inc. Boston, MA, 2002.[8]
J.L. Cabrerizo, A. Carriazo, L.M. Fern´andez and M. Fern´andez.
Slant submanifolds in Sasakianmanifolds.
Glasgow Math. J. (2000), 125-138.[9] J.L. Cabrerizo, A. Carriazo, L.M. Fern´andez and M. Fern´andez.
Semi-slant submanifolds of aSasakian manifolds.
Geometriae Dedicata (1999), 183-199.[10] A. Carriazo.
Bi-slant immersions.
Proc. ICRAMS , Kharagpur, India (2000), 88-97.[11]
A. Carriazo and M. J. P´erez-Garc´ıa.
Slant submanifolds in neutral almost contact pseudo-metric man-ifolds.
Differential Geom. Appl. Part A (2017), 71-80.[12]
B. Y. Chen.
Slant inmersions.
Bull. Austral. Math. Soc. (1990), 135-147.[13] B.-Y. Chen and O. Garay.
Classification of quasi-minimal surfaces with parallel mean curvature vector inpseudo-Euclidean 4-space E . Results Math. (2009), 23-38.[14] B.Y. Chen and I. Mihai.
Classification of quasi-minimal slant surfaces in Lorentzian complex space forms.
Acta Math. Hungar.
No. 4 (2009), 307-328.[15]
M. A. Khan, K. Singh and V. A. Khan.
Slant submanifolds of LP-contact manifolds.
Diff. Geo. - Dy-namical Systems (2010), 102-108.[16] H. Li andX. Liu.
Semi-slant submanifolds of a locally product manifold.
Georgian Math. J. (2005), no.2, 273-282.[17] A. Mihai and R. Rosca.
Skew-symmetric vector fields on a CR-submanifold of a para-Kaehlerian manifold.
I J M M S . (2004), 535-540.[18] B. O’Neill.
Semi-Riemannian Geometry with Aplications to Relativity.
Pure and Applied Mathematics 103.Academic Press, New York, 1983.[19]
N. Papaghiuc.
Semi-slant submanifolds of a Kaehlerian manifold.
An. Stiint. Al. I. Cuza. Univ. Iasi. (1994), 55-61.[20] R. Ponge and H. Reckziegel.
Twisted products in pseudo-Riemannian geometry.
Geom. Dedicata. (1993), 15-25.[21] G. S.Ronsse.
Generic and skew CR-submanifolds of a Kaehler manifold.
Bull. Inst. Math. Acad. Sinica (1990), 127-141.[22] B. Sahin.
Slant submanifolds of an almost product Riemannian manifold.
J. Korean Math. Soc. (2006),no. 4, 717-732.[23] B. Sahin.
Warped product submanifolds of Kaehler manifolds with a slant factor.
Annales Polonici Mathe-matici (2009), no. 3, 207226.[24] H.M. Tas¸tan and S. Gerdan.
Hemi-slant submanifolds of a locally conformal K¨ahler manifold.
Interna-tional Electronic J. Geo. (2015) no. 2, 46-56.[25] H. Wu.
On the de Rham decomposition theorem.
Illinois J. Math. (1964) 291-311. Departamento de Econom´ıa, M´etodos Cuantitativos e Historia Econ´omica, ´Area de Estad´ısticae Investigaci´on Operativa. Universidad Pablo de Olavide. Ctra. de Utrera km. 1, 41013 Sevilla,Spain
E-mail address , Corresponding author: [email protected]
Departamento de Geometr´ıa y Topolog´ıa. Universidad de Sevilla. c/ Tarfia s/n, 41012 Sevilla,Spain
E-mail address ::