Bi-slant submanifolds of para Hermitian manifolds

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.


Introduction
In [12], 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 plays 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 [8], and B. Sahin did in almost product manifolds in [22]. 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 [13] and [14]. K. Arslan, A. Carriazo, B.-Y. Chen and C. Murathan defined slant submanifolds of a neutral Kaehler manifold in [3], while A. Carriazo and M. J. Pérez-García did in neutral almost contact pseudo-metric manifolds in [11]. Moreover, M. A. Khan, K. Singh and V. A. Khan introduced slant submanifolds in LP-contact manifolds in [15], and P. Alegre studied slant submanifolds of Lorentzian Sasakian and para Sasakian manifolds in [1]. Finally, slant submanifolds of para Hermitian manifolds were defined in [2].
On the other hand, some generalizations of both slant and CR submanifolds have also been defined in different ambient spaces, such as semi-slant [19] and [9], hemi-slant [23], bi-slant [10] or generic submanifolds [21].
In this paper, we continue on this line, introducing semi-slant, hemi-slant and bi-slant submanifolds of para Hermitian manifolds.
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.
And the Gauss and Codazzi equations are given by (2.4) R(X, Y, Z, W ) = R(X, Y, Z, W ) + g(h(X, Z), h(Y, W )) − g(h(Y, Z), h(X, W )), for any vectors fields X, Y, Z, W tangent to M . For every tangent vector field X, we write where P X is the tangential component of JX and F X is the normal one. And for every normal vector field 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 (2.2) and (2.3) In [2], we introduced the notion of slant submanifolds of para Hermitian manifolds, taking into account that we can not measure the angle for light-like vector fields: A submanifold M of a para Hermitian manifold ( M , J, g) is called slant submanifold if for every space-like or time-like tangent vector field X, the quotient g(P X, P X)/g(JX, JX) is constant.
Remark 2.2. It is clear that, if M is a para-complex submanifold, then P ≡ J, and so the above quotient is equal to 1. On the other hand, if M is totally real, then P ≡ 0 and the quotient equals 0. 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 proper slant.
Three cases can be distinguished, corresponding to three different types of proper slant submanifolds: Let M be a proper slant submanifold of a para Hermitian manifold ( M , J, g). We say that it is of type 1 if for any space-like (time-like) vector field X, P X is time-like (space-like), and |P X| |JX| > 1, type 2 if for any space-like (time-like) vector field X, P X is time-like (space-like), and |P X| |JX| < 1, type 3 if for any space-like (time-like) vector field X, P X is space-like (time-like).
These three types can be characterized as follows: Let M be a submanifold of a para Hermitian manifold ( M , J, g). Then, 1) 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 2 = λId.
We write λ = cosh 2 θ, with θ > 0. 2) 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, 1) such that (2.10) P 2 = λId.
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 ∈ (−∞, 0) such that (2.11) P 2 = λId.
Remark 2.5. 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 2 . In [2] we did not consider that case, but it will be useful in the present study. Proof. If M is para-complex, P 2 = J 2 = Id directly. Conversely, if P 2 = Id, from g(JX, JX) = g(P X, P X) + g(F X, F X), we have −g(X, J 2 X) = −g(X, P 2 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.

Slant distributions
In [19], N. Papaghiuc introduced slant distributions in a Kaehler manifold. Given an almost Hermitian manifold, ( N , J, g), and a differentiable distribution D, it is called a slant distribution if for any non zero vector X ∈ Dx, x ∈ N , the angle between JX and the vector space Dx is constant, that is, is independent of the point x. If PDX is the projection of JX over D, they can be characterized as P 2 D = λI. This, together with the definition of slant submanifolds of a para Hermitian manifold, aims us to give the following: With this definition every one dimensional distribution defines an anti-invariant distribution in M , so we are just going to take under study non trivial slant distributions, that is with dimensions greater than 1. Just like for slant 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.
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 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 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, PDX is time-like, and, by virtue of (2.1), JX also is. Moreover, they satisfy |PDX|/|JX| > 1. So, there exists θ > 0 such that If we now consider PDX, then, in a similar way, we obtain: On the one hand, On the other hand, since both X and P 2 D X are space-like, it follows that they are collinear, that is P 2 D X = λX. Finally, from (3.4) we deduce that λ = cosh 2 θ.
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 (3.4) we should write: Since P 2 D X = λX, for any space-like or time-like X ∈ D, it also holds for light-like vector fields and so we have that P 2 D = λIdD. 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, |PDX|/|JX| < 1, and so there exists θ > 0 such that Proceeding as before, we can prove that g(P 2 D X, X) = |P 2 D X||X| and, as both X and P 2 D X are space-like vector fields, it follows that they are collinear, that is P 2 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, PDX is also space-like, and there exists θ > 0 such that Once more, we can prove that g(P 2 D X, X) = |P 2 D X||X| and P 2 D X = λX. And 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 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 holomorphic and totally real distributions the following necessary conditions are easy to prove: 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 vector field X JX = PDX + PνX + F X. Therefore from g(JX, JX) = g(PDX, PDX) + g(PνX, PνX) + g(F X, F X), and |PDX| = |JX|, in the case that PDX is also space-like, it is only deduced that Similarly it can be shown that the converse of the second statement does not always hold. In [19], 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 [10], 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 [23]. Finally, in [9], the authors defined bi-slant submanifolds with both distributions slant ones. It is called semi-slant submanifold if T M = D1 ⊕ D2 with D1 a holomorphic distribution and D2 a proper slant distribution. In such a case, we will write D1 = DT .
And it is called hemi-slant submanifold if T M = D1 ⊕ D2 with D1 a totally real distribution and D2 a proper slant distribution. In such a case, we will write D1 = D ⊥ . Remark 4.2. As we have said before, being holomorphic (totally real) is a stronger condition than being slant with slant angle 0 (π/2).
We write πi the projections over Di and Pi = πi • P , i = 1, 2. Let us consider two different para Kaehler structures over R 4 : Using the examples of slant submanifolds of R 4 given in [2] and making products, we can obtain examples of bi-slant submanifolds in R 8 . To present different examples with all the combinations of slant distributions, we consider the following para Kaehler structures over R 8 : We can see the different types in the following table: 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:  We can present a bi-slant submanifold, with the same angle for both slant distributions, that is not a slant submanifold.
Example 4.8. The submanifold of (R 8 , J2, g2) defined by It is always interesting to study the integrability of the involved distributions.
for all X, Y ∈ D2, where π1 is the projection over the invariant distribution DT . .
Now we study conditions for the involved distributions being totally geodesic. Proof. For a para Kaehler manifold taking X, Y ∈ DT , (2.7)-(2.8) leads to If DT is totally geodesic, (∇X P )Y = 0 and F ∇X Y = 0, which imply the result.
Note that for semi-slant submanifolds of para Kaehler manifolds, on the opposite that for Kaehler manifolds [19]. Proof. If D2 is a totally geodesic distribution, from (2.7) and (2.8), taking X, Y ∈ D2 which implies the given conditions. On the converse, if (∇XP )Y = AF Y X, then th(X, Y ) = 0, which implies Jh(X, Y ) = f h(X, Y ). From (2.8) and ∇F = 0, it holds h(X, P Y ) = nh(X, Y ). Then for P Y ∈ D2 and as D2 is a proper slant distribution, λ = 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 D1 − D2-mixed totally geodesic if h(X, Y ) = 0 for all X ∈ D1 , Y ∈ D2.
Proposition 5.6. Let M be a semi-slant submanifold of a para Hermitian manifold M . M is mixed totally geodesic if and only if AN X ∈ Di for any X ∈ Di, N ∈ T ⊥ M , i = 1, 2.
Proof. If M is DT − D2 mixed totally geodesic, for any X ∈ DT , Y ∈ D2, g(AN X, Y ) = g(h(X, Y ), N ) = 0, which implies AN X ∈ DT . The same proof is valid for X ∈ D2 and for the converse.
As DT is holomorphic, that is J-invariant, D2 is P -invariant. Therefore, From both equations, either h(X, Y ) = 0 or it is a eigenvalue of f 2 associated with λ = 1.
for all X, Y in different distributions. Therefore, for X ∈ DT and Y ∈ D2 and also f 2 h(Y, X) = h(Y, P 2 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.
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.
The following result was known for hemi-slant submanifolds of Kaehler manifolds, [23]. We obtain the equivalent 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 is always integrable.
Proof. From the previous lemma it is enough to prove g(AF X Y, Z) = g(AF Y X, Z), for X, Y ∈ D ⊥ and Z tangent. Then, g(AF X Y, Z) = g(h(Y, Z), F X) = g(−th(Y, Z), X) = using (2.7) = g(P ∇ZY + AF Y Z, X) = g(AF Y Z, X) = g(AF Y X, Z), which finishes the proof. Now we study the integrability of the slant distribution.
for all X, Y ∈ D2, where π1 is the projection over the totally real distribution D ⊥ .
The proof is analogous to the one of Theorem 5.3. Remember that the classical De RhamWu Theorem, [25] [20], says that two orthogonally, 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 6.7. Let M be a complete and simply connected hemi-slant submanifold of a para Kaehler manifold 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 = −AF Y 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 5.9, but now the proof is much more easier. CR-submanifolds have been intensively studied in many environments. Moreover, there are also some works about CR submanifolds of para Kaehler manifolds, [17]. A submanifold M of an almost para Hermitian manifold is called a CR-submanifold if the tangent bundle admits a decomposition T M = D ⊕ D ⊥ with D an holomorphic distribution, 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, D1 = Span ∂ ∂u1 , ∂ ∂v1 is a totally real distribution and D2 = Span ∂ ∂u2 , ∂ ∂v2 is an holomorphic distribution. Moreover: For a para Kaehler manifold with constant holomorphic curvature for every non-light-like vector field, that is R(X, JX, JX, X) = c, the curvature tensor is given by