#### Abstract

We introduce anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds onto semi-Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a semi-Riemannian submersion, and check the harmonicity of such submersions. We also obtain curvature relations between the base manifold and the total manifold.

#### 1. Introduction

The theory of Riemannian submersion was introduced by O’Neill and Gray in [1, 2], respectively. Later, Riemannian submersions were considered between almost complex manifolds by Watson in [3] under the name of almost Hermitian submersion. He showed that if the total manifold is a Kähler manifold, then the base manifold is also a Kähler manifold. Since then, Riemannian submersions have been used as an effective tool to describe the structure of a Riemannian manifold equipped with a differentiable structure. Presently, there is an extensive literature on the Riemannian submersions with different conditions imposed on the total space and on the fibres. For instance, Riemannian submersions between almost contact manifolds were studied by Chinea in [4] under the name of almost contact submersions. Riemannian submersions have been also considered for quaternionic Kähler manifolds [5] and para-quaternionic Kähler manifolds [6, 7]. This kind of submersions have been studied with different names by many authors (see [8–14], and more).

On the other hand, para-complex manifolds, almost para-Hermitian manifolds, and para-Kähler manifolds were defined by Libermann [15] in 1952. In fact, such manifolds arose in [16]. Indeed, Rashevskij introduced the properties of para-Kähler manifolds, when he considered a metric of signature defined from a potential function the so-called scalar field on a -dimensional locally product manifold called by him stratified space.

Semi-Riemannian submersions were introduced by O’Neill in his book [17]. It is known that such submersions have their applications in Kaluza-Klein theories, Yang-Mills equations, strings, and supergravity. For applications of semi-Riemannian submersions, see [18]. Since almost para-Hermitian manifolds are semi-Riemannian manifolds, one should consider semi-Riemannian submersions between such manifolds.

The paper is organized as follows. In Section 2 we recall some notions needed for this paper. In Section 3 we give the definition of anti-invariant semi-Riemannian submersions, provide an example and investigate the geometry of leaves of the distributions. We give necessary and sufficient conditions for such submersions to be totally geodesic or harmonic. We also find necessary and sufficient conditions for a Lagrangian semi-Riemannian submersion, a special anti-invariant semi-Riemannian submersion, to be totally geodesic. Moreover, we obtain decomposition theorems for the total manifold of such submersions. Finally, we obtain curvature relations between the base manifold and the total manifold.

#### 2. Preliminaries

In this section, we define almost para-Hermitian manifolds, recall the notion of semi-Riemannian submersions between semi-Riemannian manifolds, and give a brief review of basic facts of semi-Riemannian submersions.

An almost para-Hermitian manifold is a manifold endowed with an almost para-complex structure and a semi-Riemannian metric such that for , tangent to , where is the identity map. The dimension of is even and the signature of is (), where . Consider an almost para-Hermitian manifold and denote by the Levi-Civita connection on with respect to . Then is called a para-Kähler manifold if is parallel with respect to ; that is, for , tangent to [19].

Let and be two connected semi-Riemannian manifolds of index () and () respectively, with . A semi-Riemannian submersion is a smooth map which is onto and satisfies the following conditions:(i) is onto for all ;(ii)the fibres , are semi-Riemannian submanifolds of ;(iii) preserves scalar products of vectors normal to fibres.

The vectors tangent to the fibres are called vertical and those normal to the fibres are called horizontal. We denote by the vertical distribution, by the horizontal distribution and by and the vertical and horizontal projection. A horizontal vector field on is said to be *basic* if is -related to a vector field on . It is clear that every vector field on has a unique horizontal lift to and is *basic*.

We recall that the sections of , respectively , are called the vertical vector fields, respectively, horizontal vector fields. A semi-Riemannian submersion determines two tensor fields and on , by the formulas for any , , where and are the vertical and horizontal projections (see [20, 21]). From (3), one can obtain for any , , , . Moreover, if is basic then .

We note that for , , coincides with the second fundamental form of the immersion of the fibre submanifolds and for , , reflecting the complete integrability of the horizontal distribution . It is known that is alternating on the horizontal distribution: , for , and is symmetric on the vertical distribution: , for , .

We now recall the following result which will be useful for later.

Lemma 1 (see [17, 21]). * If is a semi-Riemannian submersion and , basic vector fields on , -related to and on , then one has the following properties: *(1)* is a basic vector field and ;*(2)* is a basic vector field -related to , where and are the Levi-Civita connection on and ;*(3)*, for any and for any basic vector field .*

Let and be (semi-)Riemannian manifolds and is a smooth map. Then the second fundamental form of is given by
for , , where we denote conveniently by the Levi-Civita connections of the metrics and . Recall that is said to be *harmonic* if *trace* and is called a *totally geodesic* map if for , [22]. It is known that the second fundamental form is symmetric.

#### 3. Anti-Invariant Semi-Riemannian Submersions

In this section, we define anti-invariant semi-Riemannian submersions from a para-Kähler manifold onto a semi-Riemannian manifold, investigate the integrability of distributions, and obtain a necessary and sufficient condition for such submersions to be totally geodesic map.

*Definition 2. *Let be an almost para-Hermitian manifold and a semi-Riemannian manifold. Suppose that there exists a semi-Riemannian submersion such that is anti-invariant with respect to ; that is, . Then we say is an anti-invariant semi-Riemannian submersion.

Let be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . First of all, from Definition 2, we have . We denote the complementary orthogonal distribution to in by . Then we have

It is easy to see that is an invariant distribution of , under the endomorphism . Thus, for , we have where and . On the other hand, since and is a semi-Riemannian submersion, using (10) we derive , for every and , which implies that

Note that given a semi-Euclidean space with coordinates , we can canonically choose an almost para-complex structure on as follows: where .

Also the neutral metric compatible with is

We now give an example of an anti-invariant semi-Riemannian submersion.

*Example 3. *Let be a map defined , . Then, by direct calculations
Then it is easy to see that is a semi-Riemannian submersion. Moreover and imply that . As a result, is an anti-invariant semi-Riemannian submersion.

Lemma 4. * Let be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then one has
**
for , and .*

*Proof. *For and , using (1) we have
due to and . Hence which is (15). Since is a para-Kähler manifold, using (15) we get
for , and . Then using (6) we have
Since , we obtain (16).

We now study the integrability of the distribution and then we investigate the geometry of leaves of and . We note that it is known that the distribution is integrable.

Theorem 5. *Let be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then the following assertions are equivalent to each other: *(i)* is integrable. *(ii)*. *(iii)*,
** for , and .*

*Proof. *For and , we see from Definition 2, and . Thus using (1) and (2) we obtain
Then from (10) we get
Since is a semi-Riemannian submersion, we have
Thus, from (8) and (16) we obtain
which proves (i)(ii). On the other hand, using (8) we have
Then (6) implies that
Since , this shows that (ii)(iii).

Theorem 6. * Let be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then the following assertions are equivalent to each other: *(i)* defines a totally geodesic foliation on .*(ii)*. *(iii)*,
** for , and .*

*Proof. *From (1), (2) and (6) we obtain
for , and . Then by (16) we have
which shows (i)(ii). On the other hand from (6) and (8) we get
This shows (ii)(iii).

Theorem 7. * Let be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then the following assertions are equivalent to each other: *(i)* defines a totally geodesic foliation on .*(ii)*. *(iii)*,
** for and , .*

*Proof. *Using (1) and (2) we have . Hence we get . Then a semi-Riemannian submersion and (8) imply that
which is (i)(ii). By direct calculation, we derive
Using (10) we obtain
Hence we have
Since , using (4) and (6), we get
This shows (ii)(iii).

We say that an anti-invariant semi-Riemannian submersion is a Lagrangian semi-Riemannian submersion if . If , then is called a proper anti-invariant semi-Riemannian submersion.

We note that the anti-invariant semi-Riemannian submersion given in Example 3 is a Lagrangian semi-Riemannian submersion.

If is a Lagrangian submersion, then (11) implies that .

From Theorem 5 we have the following.

Corollary 8. *Let be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then the following assertions are equivalent to each other: *(i)* is integrable. *(ii)*. *(iii)*,
** for , .*

Theorem 9. *Let be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then the following assertions are equivalent to each other: *(i)* defines a totally geodesic foliation on .*(ii)*. *(iii)*,
** for and .*

*Proof. *(i)(ii) is clear from Theorem 7. We only prove (ii)(iii). From (8), we get
for and , . Then using (5) we have
Since , we get (ii)(iii).

We note that a differentiable map between two semi-Riemannian manifolds is called totally geodesic if .

Theorem 10. * Let be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then is a totally geodesic map if and only if
*

*Proof. *First of all, we recall that the second fundamental form of a semi-Riemannian submersion satisfies
For , , by using (2), (5), and (8), we get
On the other hand, from (1), (2), and (8), we have
for . Then using (7), we get
Since is nonsingular, proof comes from (37)–(40).

We give a necessary and sufficient condition for a Lagrangian semi-Riemannian submersion to be harmonic.

Theorem 11. *Let be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then is harmonic if and only if Trace for .*

*Proof. * From [23] we know that is harmonic if and only if has minimal fibres. Thus is harmonic if and only if . On the other hand, from (2), (4), and (5), we obtain
for any , . Using (41), we get
for any . Thus using the properties of the O’Neill tensor we have
Since is symmetric, we obtain

Denote by the canonical foliations on a product manifold with natural projections onto , . Let be the projection of onto and also set , .

Let , be semi-Riemannian manifolds and be positive differentiable functions. A double-twisted product is the differentiable manifold with a semi-Riemannian metric defined by for all vectors and tangent to . In particular, if then is called the twisted product of with twisting function . Moreover, if only depends on the points of then is called the warped product of and with warping function .

In [24], the relations between the twisted and warped product structures in semi-Riemannian geometry are studied. There, essentially the following is proven (cf. [24], Proposition 3).

Let be a semi-Riemannian metric tensor on the manifold and assume that the canonical foliations and intersect perpendicularly everywhere. Then is the metric tensor of(1)a twisted product if is a totally geodesic foliation and is a totally umbilic foliation,(2)a warped product if is a totally geodesic foliation and is a spheric foliation; that is, it is umbilical and its mean curvature vector field is parallel,(3)a usual product of semi-Riemannian manifolds if and only if f and are totally geodesic foliations.

Now, we obtain decomposition theorems by using the existence of anti-invariant semi-Riemannian submersions.

Theorem 12. * Let be a para-Kähler manifold and be a semi-Riemannian manifold. If there exists a Lagrangian semi-Riemannian submersion from onto such that is locally twisted product manifold of the form , then is a usual (locally product) manifold, where and are integral manifolds of the distribution and .*

*Proof. *Suppose that is a Lagrangian semi-Riemannian submersion and is a locally twisted product of the form . Then is a totally geodesic foliation and is a totally umbilical foliation. We denote the second fundamental form of by . Then we get for and . Since is totally umbilical we have
where is the mean curvature vector field of . On the other hand, from (2), we derive . Then using (7), we obtain
Thus, from (46) and (47), we have
Hence, we arrive at
Then using (7) we get
Thus (2) implies that
Hence, we obtain
Then, since is alternating on the horizontal distribution, we have which implies that . Since is a semi-Riemannian metric and , we conclude that . This shows that is totally geodesic, so is usual product of Riemannian manifolds. Thus the proof is complete.

Theorem 13. * Let be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then is a locally twisted product manifold of the form if and only if
**
for , and , where and are integral manifolds of the distribution and .*

*Proof. *From (2) and (5), we obtain
for and , . Since is skew-symmetric, we get
This implies that is totally umbilical if and only if
where is a function on . Then by direct computations, it is easy to see that this is equivalent to
Thus the proof is complete.

#### 4. Curvature Relations for Anti-Invariant Semi-Riemannian Submersions

In this section, we are going to obtain curvature relations of anti-invariant semi-Riemannian submersions.

Let be a semi-Riemannian manifold. The sectional curvature of a 2-plane in , , spanned by , is defined by It is clear that the above definition makes sense only for nondegenerate planes, that is, those satisfying .

Lemma 14. *Let be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Then for , and , , we have the following relations:
*

*Proof. *From (2) and (5) we have
Using (4) we get
Then (10) implies that
Taking the vertical and horizontal parts of this equation, we obtain (59) and (60). The other assertions can be obtained in a similar way.

Using both Lemma 14 and pages 13-14 of [21], we obtain the following.

Theorem 15. * Let be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Let and be nonzero nonlightlike orthogonal unit vertical vectors. Then, we have
**
where , and is the para-holomorphic sectional curvature of .*

From Theorem 15 we have the following result.

Corollary 16. * Let be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold to a semi-Riemannian manifold . Let and be nonzero nonlightlike orthogonal unit vertical vectors. Then, if and only if , where is the sectional curvature of .*

#### Conflict of Interests

The author declares that he has no conflict of interests.

#### Acknowledgment

The author is grateful to the referee(s) for their valuable comments and suggestions.