Anti-Invariant Semi-Riemannian Submersions from Almost Para-Hermitian Manifolds

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 (𝑚,𝑚) defined from a potential function the so-called scalar field on a 2𝑚-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 𝑀 endowed with an almost para-complex structure 𝑃≠±𝐼 and a semi-Riemannian metric 𝑔 such that 𝑃²=𝐼,𝑔(𝑃𝑋,𝑃𝑌)=−𝑔(𝑋,𝑌) (1) for 𝑋, 𝑌 tangent to 𝑀, where 𝐼 is the identity map. The dimension of 𝑀 is even and the signature of 𝑔 is (𝑚,𝑚), where dim𝑀=2𝑚. Consider an almost para-Hermitian manifold (𝑀,𝑃,𝑔) and denote by ∇ the Levi-Civita connection on 𝑀 with respect to 𝑔. Then 𝑀 is called a para-Kähler manifold if 𝑃 is parallel with respect to ∇; that is, ∇_𝑋𝑃𝑌=0 (2) for 𝑋, 𝑌 tangent to 𝑀 [19].

Let (𝑀,𝑔) and (𝑁,𝑔^′) be two connected semi-Riemannian manifolds of index 𝑠 (0≤𝑠≤dim𝑀) and 𝑠^′ (0≤𝑠^′≤dim𝑁) respectively, with 𝑠>𝑠^′. A semi-Riemannian submersion is a smooth map 𝜋∶𝑀→𝑁 which is onto and satisfies the following conditions:(i)𝜋_∗𝑝∶𝑇_𝑝𝑀→𝑇_𝜋(𝑝)𝑁 is onto for all 𝑝∈𝑀;(ii)the fibres 𝜋⁻¹(𝑝^′), 𝑝^′∈𝑁 are semi-Riemannian submanifolds of 𝑀;(iii)𝜋_∗ 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 𝒱 the vertical distribution, by ℋ the horizontal distribution and by 𝑣 and ℎ the vertical and horizontal projection. A horizontal vector field 𝑋 on 𝑀 is said to be basic if 𝑋 is 𝜋-related to a vector field 𝑋^′ on 𝑁. It is clear that every vector field 𝑋^′ on 𝑁 has a unique horizontal lift 𝑋 to 𝑀 and 𝑋 is basic.

We recall that the sections of 𝒱, respectively ℋ, are called the vertical vector fields, respectively, horizontal vector fields. A semi-Riemannian submersion 𝜋∶𝑀→𝑁 determines two (1,2) tensor fields 𝑇 and 𝐴 on 𝑀, by the formulas 𝑇(𝐸,𝐹)=𝑇_𝐸𝐹=ℎ∇_𝑣𝐸𝑣𝐹+𝑣∇_𝑣𝐸𝐴()ℎ𝐹,𝐸,𝐹=𝐴_𝐸𝐹=𝑣∇_ℎ𝐸ℎ𝐹+ℎ∇_ℎ𝐸𝑣𝐹 (3) for any 𝐸, 𝐹∈Γ(𝑇𝑀), where 𝑣 and ℎ are the vertical and horizontal projections (see [20, 21]). From (3), one can obtain ∇_𝑈𝑊=𝑇_𝑈𝑊+∇_𝑈∇𝑊;_𝑈𝑋=𝑇_𝑈𝑋+ℎ∇_𝑈𝑋;∇_𝑋𝑈=𝑣∇_𝑋𝑈+𝐴_𝑋∇𝑈;_𝑋𝑌=𝐴_𝑋𝑌+ℎ∇_𝑋𝑌, (4)(5)(6)(7) for any 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂), 𝑈, 𝑊∈Γ(ker𝜋_∗). Moreover, if 𝑋 is basic then ℎ(∇_𝑈𝑋)=ℎ(∇_𝑋𝑈)=𝐴_𝑋𝑈.

We note that for 𝑈, 𝑉∈Γ(ker𝜋_∗), 𝑇_𝑈𝑉 coincides with the second fundamental form of the immersion of the fibre submanifolds and for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂), 𝐴_𝑋𝑌=(1/2)𝑣[𝑋,𝑌] reflecting the complete integrability of the horizontal distribution ℋ. It is known that 𝐴 is alternating on the horizontal distribution: 𝐴_𝑋𝑌=−𝐴_𝑌𝑋, for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑇 is symmetric on the vertical distribution: 𝑇_𝑈𝑉=𝑇_𝑉𝑈, for 𝑈, 𝑉∈Γ(ker𝜋_∗).

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

Lemma 1 (see [17, 21]). If 𝜋∶𝑀→𝑁 is a semi-Riemannian submersion and 𝑋, 𝑌 basic vector fields on 𝑀, 𝜋-related to 𝑋^′ and 𝑌^′ on 𝑁, then one has the following properties: (1)ℎ[𝑋,𝑌] is a basic vector field and 𝜋_∗ℎ[𝑋,𝑌]=[𝑋^′,𝑌^′]∘𝜋;(2)ℎ(∇_𝑋𝑌) is a basic vector field 𝜋-related to (∇^′_𝑋′𝑌^′), where ∇ and ∇^′ are the Levi-Civita connection on 𝑀 and 𝑁;(3)[𝐸,𝑈]∈Γ(ker𝜋_∗), for any 𝑈∈Γ(ker𝜋_∗) and for any basic vector field 𝐸.

Let (𝑀,𝑔) and (𝑁,𝑔^′) be (semi-)Riemannian manifolds and 𝜋∶𝑀→𝑁 is a smooth map. Then the second fundamental form of 𝜋 is given by ∇𝜋_∗(𝑋,𝑌)=∇^𝜋_𝑋𝜋_∗𝑌−𝜋_∗∇_𝑋𝑌 (8) for 𝑋, 𝑌∈Γ(𝑇𝑀), where we denote conveniently by ∇ the Levi-Civita connections of the metrics 𝑔 and 𝑔^′. Recall that 𝜋 is said to be harmonic if trace  (∇𝜋_∗)=0 and 𝜋 is called a totally geodesic map if (∇𝜋_∗)(𝑋,𝑌)=0 for 𝑋, 𝑌∈Γ(𝑇𝑀) [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 (𝑀,𝑔,𝑃) be an almost para-Hermitian manifold and (𝑁,𝑔^′) a semi-Riemannian manifold. Suppose that there exists a semi-Riemannian submersion 𝜋∶𝑀→𝑁 such that ker𝜋_∗ is anti-invariant with respect to 𝑃; that is, 𝑃(ker𝜋_∗)⊆(ker𝜋_∗)^⟂. Then we say 𝜋 is an anti-invariant semi-Riemannian submersion.

Let 𝜋∶(𝑀,𝑔,𝑃)→(𝑁,𝑔^′) be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). First of all, from Definition 2, we have 𝑃(ker𝜋_∗)^⟂∩(ker𝜋_∗)≠0. We denote the complementary orthogonal distribution to 𝑃(ker𝜋_∗) in (ker𝜋_∗)^⟂ by 𝜇. Then we have ker𝜋_∗^⟂=𝑃ker𝜋_∗⊕𝜇. (9)

It is easy to see that 𝜇 is an invariant distribution of (ker𝜋_∗)^⟂, under the endomorphism 𝑃. Thus, for 𝑋∈Γ((ker𝜋_∗)^⟂), we have 𝑃𝑋=𝐵𝑋+𝐶𝑋, (10) where 𝐵𝑋∈Γ(ker𝜋_∗) and 𝐶𝑋∈Γ(𝜇). On the other hand, since 𝜋_∗((ker𝜋_∗)^⟂)=𝑇𝑁 and 𝜋 is a semi-Riemannian submersion, using (10) we derive 𝑔^′(𝜋_∗𝑃𝑉,𝜋_∗𝐶𝑋)=0, for every 𝑋∈Γ((ker𝜋_∗)^⟂) and 𝑉∈Γ(ker𝜋_∗), which implies that 𝑇𝑁=𝜋_∗𝑃ker𝜋_∗⊕𝜋_∗𝜇. (11)

Note that given a semi-Euclidean space 𝑅⁴₂ with coordinates (𝑥₁,…,𝑥₄), we can canonically choose an almost para-complex structure 𝑃 on 𝑅⁴₂ as follows: 𝑃𝑎₁𝜕𝜕𝑥₁+𝑎₂𝜕𝜕𝑥₂+𝑎₃𝜕𝜕𝑥₃+𝑎₄𝜕𝜕𝑥₄=−𝑎₂𝜕𝜕𝑥₁−𝑎₁𝜕𝜕𝑥₂+𝑎₄𝜕𝜕𝑥₃+𝑎₃𝜕𝜕𝑥₄, (12) where 𝑎₁,…,𝑎₄∈𝑅.

Also the neutral metric compatible with 𝑃 is 𝑔=𝑑𝑥₁²−𝑑𝑥₂²+𝑑𝑥₃²−𝑑𝑥₄². (13)

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

Example 3. Let 𝜋∶𝑅⁴₂→𝑅²₁ be a map defined 𝜋(𝑥₁,𝑥₂,𝑥₃,𝑥₄)=((𝑥₁+𝑥₃)/√2, (𝑥₂+𝑥₄)/√2). Then, by direct calculations ker𝜋_∗=Span𝑍₁𝜕=−𝜕𝑥₁+𝜕𝜕𝑥₃,𝑍₂𝜕=−𝜕𝑥₂+𝜕𝜕𝑥₄,ker𝜋_∗^⟂=Span𝑋₁=𝜕𝜕𝑥₁+𝜕𝜕𝑥₃,𝑋₂=𝜕𝜕𝑥₂+𝜕𝜕𝑥₄. (14) Then it is easy to see that 𝜋 is a semi-Riemannian submersion. Moreover 𝑃𝑍₁=𝑋₂ and 𝑃𝑍₂=𝑋₁ imply that 𝑃(ker𝜋_∗)=(ker𝜋_∗)^⟂. As a result, 𝜋 is an anti-invariant semi-Riemannian submersion.

Lemma 4. Let 𝜋 be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then one has 𝑔()𝑔∇𝐶𝑌,𝑃𝑉=0,_𝑋𝐶𝑌,𝑃𝑉=−𝑔𝐶𝑌,𝑃𝐴_𝑋𝑉 (15)(16) for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑉∈Γ(ker𝜋_∗).

Proof. For 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑉∈Γ(ker𝜋_∗), using (1) we have 𝑔(𝐶𝑌,𝑃𝑉)=𝑔(𝑃𝑌−𝐵𝑌,𝑃𝑉)=𝑔(𝑃𝑌,𝑃𝑉) (17) due to 𝐵𝑌∈Γ(ker𝜋_∗) and 𝑃𝑉∈Γ((ker𝜋_∗)^⟂). Hence 𝑔(𝑃𝑌,𝑃𝑉)=−𝑔(𝑌,𝑉)=0 which is (15). Since 𝑀 is a para-Kähler manifold, using (15) we get 𝑔∇_𝑋𝐶𝑌,𝑃𝑉=−𝑔𝐶𝑌,𝑃∇_𝑋𝑉 (18) for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑉∈Γ(ker𝜋_∗). Then using (6) we have 𝑔∇_𝑋𝐶𝑌,𝑃𝑉=−𝑔𝐶𝑌,𝑃𝐴_𝑋𝑉−𝑔𝐶𝑌,𝑃𝑣∇_𝑋𝑉. (19) Since 𝑃𝑣∇_𝑋𝑉∈Γ(𝑃ker𝜋_∗), we obtain (16).

We now study the integrability of the distribution (ker𝜋_∗)^⟂ and then we investigate the geometry of leaves of ker𝜋_∗ and (ker𝜋_∗)^⟂. We note that it is known that the distribution ker𝜋_∗ is integrable.

Theorem 5. Let 𝜋 be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then the following assertions are equivalent to each other: (i)(ker𝜋_∗)^⟂ is integrable. (ii)𝑔^′((∇𝜋_∗)(𝑋,𝐵𝑌),𝜋_∗𝑃𝑉)=𝑔^′((∇𝜋_∗)(𝑌,𝐵𝑋),𝜋_∗𝑃𝑉)−𝑔(𝐶𝑌,𝑃𝐴_𝑋𝑉)+𝑔(𝐶𝑋,𝑃𝐴_𝑌𝑉). (iii)𝑔(𝑃𝑉,𝐴_𝑌𝐵𝑋−𝐴_𝑋𝐵𝑌)=−𝑔(𝐶𝑌,𝑃𝐴_𝑋𝑉)+𝑔(𝐶𝑋,𝑃𝐴_Y𝑉), for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑉∈Γ(ker𝜋_∗).

Proof. For 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑉∈Γ(ker𝜋_∗), we see from Definition 2, 𝑃𝑉∈Γ((ker𝜋_∗)^⟂) and 𝑃𝑌∈Γ(ker𝜋_∗⊕𝜇). Thus using (1) and (2) we obtain 𝑔([𝑋,𝑌],𝑉)=−𝑔∇_𝑋𝑃𝑌,𝑃𝑉+𝑔∇_𝑌𝑃𝑋,𝑃𝑉. (20) Then from (10) we get 𝑔([𝑋,𝑌],𝑉)=−𝑔∇_𝑋𝐵𝑌,𝑃𝑉−𝑔∇_𝑋∇𝐶𝑌,𝑃𝑉+𝑔_𝑌𝐵𝑋,𝑃𝑉+𝑔∇_𝑌𝐶𝑋,𝑃𝑉. (21) Since 𝜋 is a semi-Riemannian submersion, we have 𝑔([𝑋,𝑌],𝑉)=−𝑔^′𝜋_∗∇_𝑋𝐵𝑌,𝜋_∗𝑃𝑉−𝑔∇_𝑋𝐶𝑌,𝑃𝑉+𝑔^′𝜋_∗∇_𝑌𝐵𝑋,𝜋_∗𝑃𝑉+𝑔∇_𝑌𝐶𝑋,𝑃𝑉. (22) Thus, from (8) and (16) we obtain 𝑔([𝑋,𝑌],𝑉)=𝑔^′∇𝜋_∗(𝑋,𝐵𝑌)−∇𝜋_∗(𝑌,𝐵𝑋),𝜋_∗𝑃𝑉+𝑔𝐶𝑌,𝑃𝐴_𝑋𝑉−𝑔𝐶𝑋,𝑃𝐴_𝑌𝑉 (23) which proves (i)⇔(ii). On the other hand, using (8) we have ∇𝜋_∗(𝑋,𝐵𝑌)−∇𝜋_∗(𝑌,𝐵𝑋)=−𝜋_∗∇_𝑋𝐵𝑌−∇_𝑌𝐵𝑋. (24) Then (6) implies that ∇𝜋_∗(𝑋,𝐵𝑌)−∇𝜋_∗(𝑌,𝐵𝑋)=−𝜋_∗𝐴_𝑋𝐵𝑌−𝐴_𝑌𝐵𝑋. (25) Since 𝐴_𝑌𝐵𝑋−𝐴_𝑋𝐵𝑌∈Γ((ker𝜋_∗)^⟂), this shows that (ii)⇔(iii).

Theorem 6. Let 𝜋 be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then the following assertions are equivalent to each other: (i)(ker𝜋_∗)^⟂ defines a totally geodesic foliation on 𝑀.(ii)𝑔(𝐴_𝑋𝐵𝑌,𝑃𝑉)=𝑔(𝐶𝑌,𝑃𝐴_𝑋𝑉). (iii)𝑔^′((∇𝜋_∗)(𝑋,𝐵𝑌),𝜋_∗𝑃𝑉)=−𝑔(𝐶𝑌,𝑃𝐴_𝑋𝑉), for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑉∈Γ(ker𝜋_∗).

Proof. From (1), (2) and (6) we obtain 𝑔∇_𝑋𝑌,𝑉=−𝑔𝐴_𝑋𝐵𝑌,𝑃𝑉−𝑔∇_𝑋𝐶𝑌,𝑃𝑉 (26) for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑉∈Γ(ker𝜋_∗). Then by (16) we have 𝑔∇_𝑋𝑌,𝑉=−𝑔𝐴_𝑋𝐵𝑌,𝑃𝑉+𝑔𝑃𝐴_𝑋𝑉,𝐶𝑌 (27) which shows (i)⇔(ii). On the other hand from (6) and (8) we get 𝑔𝐴_𝑋𝐵𝑌,𝑃𝑉=𝑔^′−∇𝜋_∗(𝑋,𝐵𝑌),𝜋_∗𝑃𝑉. (28) This shows (ii)⇔(iii).

Theorem 7. Let 𝜋 be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then the following assertions are equivalent to each other: (i)(ker𝜋_∗) defines a totally geodesic foliation on 𝑀.(ii)𝑔^′((∇𝜋_∗)(𝑉,𝑃𝑋),𝜋_∗𝑃𝑊)=0. (iii)𝑇_𝑉𝐵𝑋+𝐴_𝐶𝑋𝑉∈Γ(𝜇), for 𝑋∈Γ((ker𝜋_∗)^⟂) and 𝑉, 𝑊∈Γ(ker𝜋_∗).

Proof. Using (1) and (2) we have 𝑔(∇_𝑉𝑊,𝑋)=−𝑔(∇_𝑉𝑃𝑊,𝑃𝑋). Hence we get 𝑔(∇_𝑉𝑊,𝑋)=𝑔(ℎ∇_𝑉𝑃𝑋,𝑃𝑊). Then a semi-Riemannian submersion 𝜋 and (8) imply that 𝑔∇_𝑉𝑊,𝑋=−𝑔^′∇𝜋_∗(𝑉,𝑃𝑋),𝜋_∗𝑃𝑊 (29) which is (i)⇔(ii). By direct calculation, we derive 𝑔∇_𝑉𝑃𝑋,𝑃𝑊=−𝑔^′∇𝜋_∗(𝑉,𝑃𝑋),𝜋_∗𝑃𝑊. (30) Using (10) we obtain 𝑔∇_𝑉𝐵𝑋+∇_𝑉𝐶𝑋,𝑃𝑊=−𝑔^′∇𝜋_∗(𝑉,𝑃𝑋),𝜋_∗𝑃𝑊. (31) Hence we have 𝑔∇_𝑉𝐵𝑋+[𝑉,𝐶𝑋]+∇_𝐶𝑋𝑉,𝑃𝑊=−𝑔^′∇𝜋_∗(𝑉,𝑃𝑋),𝜋_∗𝑃𝑊. (32) Since [𝑉,𝐶𝑋]∈Γ(ker𝜋_∗), using (4) and (6), we get 𝑔𝑇_𝑉𝐵𝑋+𝐴_𝐶𝑋𝑉,𝑃𝑊=−𝑔^′∇𝜋_∗(𝑉,𝑃𝑋),𝜋_∗𝑃𝑊. (33) This shows (ii)⇔(iii).

We say that an anti-invariant semi-Riemannian submersion is a Lagrangian semi-Riemannian submersion if 𝑃(ker𝜋_∗)=(ker𝜋_∗)^⟂. If 𝜇≠{0}, then 𝜋 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 𝜋 is a Lagrangian submersion, then (11) implies that 𝑇𝑁=𝜋_∗(𝑃(ker𝜋_∗)).

From Theorem 5 we have the following.

Corollary 8. Let 𝜋 be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then the following assertions are equivalent to each other: (i)(ker𝜋_∗)^⟂ is integrable. (ii)(∇𝜋_∗)(𝑋,𝑃𝑌)=(∇𝜋_∗)(𝑃𝑋,𝑌). (iii)𝐴_𝑋𝑃𝑌=𝐴_𝑌𝑃𝑋, for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂).

Theorem 9. Let 𝜋 be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then the following assertions are equivalent to each other: (i)(ker𝜋_∗) defines a totally geodesic foliation on 𝑀.(ii)(∇𝜋_∗)(𝑉,𝑃𝑋)=0. (iii)𝑇_𝑉𝑃𝑊=0, for 𝑋∈Γ((ker𝜋_∗)^⟂) and 𝑉,𝑊∈Γ(ker𝜋_∗).

Proof. (i)⇔(ii) is clear from Theorem 7. We only prove (ii)⇔(iii). From (8), we get 𝑔∇_𝑉𝑃𝑊,𝑃𝑋=𝑔^′∇𝜋_∗(𝑉,𝑃𝑋),𝜋_∗𝑃𝑊 (34) for 𝑋∈Γ((ker𝜋_∗)^⟂) and 𝑉, 𝑊∈Γ(ker𝜋_∗). Then using (5) we have 𝑔^′∇𝜋_∗(𝑉,𝑃𝑋),𝜋_∗𝑃𝑊=𝑔𝑇_𝑉𝑃𝑊,𝑃𝑋. (35) Since 𝑇_𝑉𝑃𝑊∈Γ(ker𝜋_∗), we get (ii)⇔(iii).

We note that a differentiable map 𝜋 between two semi-Riemannian manifolds is called totally geodesic if ∇𝜋_∗=0.

Theorem 10. Let 𝜋 be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then 𝜋 is a totally geodesic map if and only if 𝑇_𝑊𝑃𝑉=0,∀𝑊,𝑉∈Γker𝜋_∗,𝐴_𝑋𝑃𝑊=0,∀𝑋∈Γker𝜋_∗^⟂. (36)

Proof. First of all, we recall that the second fundamental form of a semi-Riemannian submersion satisfies ∇𝜋_∗(𝑋,𝑌)=0,𝑋,𝑌∈Γker𝜋_∗^⟂. (37) For 𝑊, 𝑉∈Γ(ker𝜋_∗), by using (2), (5), and (8), we get ∇𝜋_∗(𝑊,𝑉)=−𝜋_∗𝑃𝑇_𝑊𝑃𝑉. (38) On the other hand, from (1), (2), and (8), we have ∇𝜋_∗(𝑋,𝑊)=−𝜋_∗𝑃∇_𝑋𝑃𝑊 (39) for 𝑋∈Γ((ker𝜋_∗)^⟂). Then using (7), we get ∇𝜋_∗(𝑋,𝑊)=−𝜋_∗𝑃𝐴_𝑋𝑃𝑊. (40) Since 𝑃 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 𝜋 be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then 𝜋 is harmonic if and only if Trace  𝑃𝑇_𝑉=0 for 𝑉∈Γ(ker𝜋_∗).

Proof. From [23] we know that 𝜋 is harmonic if and only if 𝜋 has minimal fibres. Thus 𝜋 is harmonic if and only if ∑^𝑟_𝑖=1𝑇_𝑒𝑖𝑒_𝑖=0. On the other hand, from (2), (4), and (5), we obtain 𝑇_𝑉𝑃𝑊=𝑃𝑇_𝑉𝑊 (41) for any 𝑉, 𝑊∈Γ(ker𝜋_∗). Using (41), we get ^𝑟_𝑖=1𝑔𝑇_𝑒𝑖𝑃𝑒_𝑖,𝑉=−^𝑟_𝑖=1𝑔𝑇_𝑒𝑖𝑒_𝑖,𝑃𝑉 (42) for any 𝑉∈Γ(ker𝜋_∗). Thus using the properties of the O’Neill tensor 𝑇 we have ^𝑟_𝑖=1𝑔𝑇_𝑒𝑖𝑉,𝑃𝑒_𝑖=^𝑟_𝑖=1𝑔𝑇_𝑒𝑖𝑒_𝑖,𝑃𝑉. (43) Since 𝑇 is symmetric, we obtain ^𝑟_𝑖=1𝑔𝑇_𝑉𝑒_𝑖,𝑃𝑒_𝑖=^𝑟_𝑖=1𝑔𝑇_𝑒𝑖𝑒_𝑖,𝑃𝑉. (44)

Denote by {𝐿_𝑖} the canonical foliations on a product manifold 𝑀₁×𝑀₂ with natural projections 𝑝_𝑖 onto 𝑀_𝑖, (𝑖=1,2). Let 𝑃_𝑖 be the projection of 𝑇(𝑀₁×𝑀₂) onto 𝑇𝐿_𝑖 and also set 𝑃^⟂_𝑖=𝐼−𝑃_𝑖, (𝑖=1,2).

Let (𝑀_𝑖,𝑔_𝑖), (𝑖=1,2) be semi-Riemannian manifolds and 𝜆_𝑖∶𝑀₁×𝑀₂→𝑅 be positive differentiable functions. A double-twisted product 𝑀₁×_{(𝜆1,𝜆2)}𝑀₂ is the differentiable manifold 𝑀₁×𝑀₂ with a semi-Riemannian metric defined by 𝑔(𝑋,𝑌)=𝜆²₁𝑔₁𝑃₁𝑋,𝑃₁𝑌+𝜆²₂𝑔₂𝑃₂𝑋,𝑃₂𝑌 (45) for all vectors 𝑋 and 𝑌 tangent to 𝑀₁×𝑀₂. In particular, if 𝜆₁=1 then ₁𝑀₁×_𝜆2𝑀₂=𝑀₁×_𝜆2𝑀₂ is called the twisted product of (𝑀₂,𝑔₂) with twisting function 𝜆₂. Moreover, if 𝜆₂ only depends on the points of 𝑀₁ then 𝑀₁×_𝜆2𝑀₂ is called the warped product of (𝑀₁,𝑔₁) and (𝑀₂,𝑔₂) with warping function 𝜆₂.

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 𝑔 be a semi-Riemannian metric tensor on the manifold 𝑀=𝑀₁×𝑀₂ and assume that the canonical foliations 𝐿₁ and 𝐿₂ intersect perpendicularly everywhere. Then 𝑔 is the metric tensor of(1)a twisted product 𝑀₁×_𝜆𝑀₂ if 𝐿₁ is a totally geodesic foliation and 𝐿₂ is a totally umbilic foliation,(2)a warped product 𝑀₁×_𝜆𝑀₂ if 𝐿₁ is a totally geodesic foliation and 𝐿₂ 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 𝐿₁ and 𝐿₂ are totally geodesic foliations.

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

Theorem 12. Let (𝑀,𝑔,𝑃) be a para-Kähler manifold and (𝑁,𝑔^′) be a semi-Riemannian manifold. If there exists a Lagrangian semi-Riemannian submersion from 𝑀 onto 𝑁 such that 𝑀 is locally twisted product manifold of the form 𝑀_ker𝜋∗×_𝑓𝑀_(ker𝜋∗)^⟂, then 𝑀 is a usual (locally product) manifold, where 𝑀_(ker𝜋∗)^⟂ and 𝑀_ker𝜋∗ are integral manifolds of the distribution (ker𝜋_∗)^⟂ and ker𝜋_∗.

Proof. Suppose that 𝜋∶(𝑀,𝑔,𝑃)→(𝑁,𝑔^′) is a Lagrangian semi-Riemannian submersion and 𝑀 is a locally twisted product of the form 𝑀_ker𝜋∗×_𝑓𝑀_(ker𝜋∗)^⟂. Then 𝑀_ker𝜋∗ is a totally geodesic foliation and 𝑀_(ker𝜋∗)^⟂ is a totally umbilical foliation. We denote the second fundamental form of 𝑀_(ker𝜋∗)^⟂ by ℎ. Then we get 𝑔(∇_𝑋𝑌,𝑉)=𝑔(ℎ(𝑋,𝑌),𝑉) for 𝑋,𝑌∈Γ((ker𝜋_∗)^⟂ and 𝑉∈Γ(ker𝜋_∗). Since 𝑀_(ker𝜋∗)^⟂ is totally umbilical we have 𝑔∇_𝑋𝑌,𝑉=𝑔(𝐻,𝑉)𝑔(𝑋,𝑌), (46) where 𝐻 is the mean curvature vector field of 𝑀_(ker𝜋∗)^⟂. On the other hand, from (2), we derive 𝑔(∇_𝑋𝑌,𝑉)=𝑔(∇_𝑋𝑃𝑉,𝑃𝑌). Then using (7), we obtain 𝑔∇_𝑋𝑌,𝑉=𝑔𝐴_𝑋𝑃𝑉,𝑃𝑌. (47) Thus, from (46) and (47), we have 𝐴_𝑋𝑃𝑉=−𝑔(𝐻,𝑉)𝑃𝑋. (48) Hence, we arrive at 𝑔𝐴_𝑋𝑃𝑉,𝑃𝑋=𝑔(𝐻,𝑉)𝑔(𝑋,𝑋). (49) Then using (7) we get 𝑔∇_𝑋𝑃𝑉,𝑃𝑋=𝑔(𝐻,𝑉)𝑔(𝑋,𝑋). (50) Thus (2) implies that −𝑔∇_𝑋𝑉,𝑋=𝑔(𝐻,𝑉)𝑔(𝑋,𝑋). (51) Hence, we obtain 𝑔∇_𝑋𝑋,𝑉=𝑔(𝐻,𝑉)𝑔(𝑋,𝑋). (52) Then, since 𝐴 is alternating on the horizontal distribution, we have 𝐴_𝑋𝑋=0 which implies that 𝑔(𝐻,𝑉)𝑔(𝑋,𝑋)=0. Since 𝑔 is a semi-Riemannian metric and 𝐻∈Γ(ker𝜋_∗), we conclude that 𝐻=0. This shows that (ker𝜋_∗)^⟂ is totally geodesic, so 𝑀 is usual product of Riemannian manifolds. Thus the proof is complete.

Theorem 13. Let 𝜋 be a Lagrangian semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then 𝑀 is a locally twisted product manifold of the form 𝑀_(ker𝜋∗)^⟂×_𝑓𝑀_ker𝜋∗ if and only if 𝑇_𝑉𝑃𝑋=−𝑔𝑋,𝑇_𝑉𝑉𝑔(𝑉,𝑉)⁻¹𝑃𝑉 (53) for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑉∈Γ(ker𝜋_∗), where 𝑀_(ker𝜋∗)^⟂ and 𝑀_ker𝜋∗ are integral manifolds of the distribution (ker𝜋_∗)^⟂ and ker𝜋_∗.

Proof. From (2) and (5), we obtain 𝑔∇_𝑉𝑊,𝑋=−𝑔𝑇_𝑉𝑃𝑊,𝑃𝑋 (54) for 𝑋∈Γ((ker𝜋_∗)^⟂) and 𝑉, 𝑊∈Γ(ker𝜋_∗). Since 𝑇_𝑉 is skew-symmetric, we get 𝑔∇_𝑉𝑊,𝑋=𝑔𝑇_𝑉𝑃𝑋,𝑃𝑊. (55) This implies that ker𝜋_∗ is totally umbilical if and only if 𝑋(𝜆)𝑃𝑉=𝑇_𝑉𝑃𝑋, (56) where 𝜆 is a function on 𝑀. Then by direct computations, it is easy to see that this is equivalent to 𝑇_𝑉𝑃𝑋=−𝑔𝑋,𝑇_𝑉𝑉𝑔(𝑉,𝑉)⁻¹𝑃𝑉. (57) 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 (𝑀,𝑔) be a semi-Riemannian manifold. The sectional curvature 𝐾 of a 2-plane in 𝑇_𝑝𝑀, 𝑝∈𝑀, spanned by {𝑋,𝑌}, is defined by 𝐾()=𝑅()𝑋,𝑌𝑋,𝑌,𝑋,𝑌𝑔(𝑋,𝑋)𝑔(𝑌,𝑌)−𝑔(𝑋,𝑌)². (58) It is clear that the above definition makes sense only for nondegenerate planes, that is, those satisfying 𝑄(𝑋,𝑌)=𝑔(𝑋,𝑋)𝑔(𝑌,𝑌)−𝑔(𝑋,𝑌)²≠0.

Lemma 14. Let 𝜋 be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Then for 𝑋, 𝑌∈Γ((ker𝜋_∗)^⟂) and 𝑈, 𝑉∈Γ(ker𝜋_∗), we have the following relations: 𝐵𝑇_𝑈𝑉=𝑇_𝑈𝑃𝑉;ℎ∇_𝑈𝑃𝑉=𝐶𝑇_𝑈𝑉+𝑃∇_𝑈𝑉;𝑃𝐴_𝑋𝑌+𝐶ℎ∇_𝑋𝑌=𝐴_𝑋𝐵𝑌+ℎ∇_𝑋𝐶𝑌;𝐵ℎ∇_𝑋𝑌=𝑣∇_𝑋𝐵𝑌+𝐴_𝑋𝐶𝑌;𝐵𝐴_𝑋𝑈=𝐴_𝑋𝑃𝑈. (59)(60)(61)(62)(63)

Proof. From (2) and (5) we have 𝑃∇_𝑈𝑉=ℎ∇_𝑈𝑃𝑉+𝑇_𝑈𝑃𝑉. (64) Using (4) we get 𝑃𝑇_𝑈𝑉+𝑃∇_𝑈𝑉=ℎ∇_𝑈𝑃𝑉+𝑇_𝑈𝑃𝑉. (65) Then (10) implies that 𝐵𝑇_𝑈𝑉+𝐶𝑇_𝑈𝑉+𝑃∇_𝑈𝑉=ℎ∇_𝑈𝑃𝑉+𝑇_𝑈𝑃𝑉. (66) 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 𝜋 be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Let 𝑈 and 𝑉 be nonzero nonlightlike orthogonal unit vertical vectors. Then, we have 𝐾(𝑈∧𝑉)=𝐾^∗(𝑃𝑈∧𝑃𝑉)+𝜖_𝑈𝜖_𝑉3𝑔𝐴_𝑃𝑈𝑉,𝐴_𝑃𝑈𝑉+3𝑔𝐶𝐴_𝑃𝑈𝑉,𝐶𝐴_𝑃𝑈𝑉𝐻(𝑈)=−𝑔∇_𝑃𝑈𝑇_𝑈𝑈,𝑃𝑈+𝑔𝑇_𝑈𝑈,𝑇_𝑈𝑈+𝑔𝐶𝑇_𝑈𝑈,𝐶𝑇_𝑈𝑈+𝑔𝐴_𝑃𝑈𝑈,𝐴_𝑃𝑈𝑈, (67) where 𝜖_𝑈=𝑔(𝑈,𝑈)∈{±1}, 𝜖_𝑉=𝑔(𝑉,𝑉)∈{±1} and 𝐻(𝑈)=𝐾(𝑈∧𝑃𝑈) is the para-holomorphic sectional curvature of 𝑀.

From Theorem 15 we have the following result.

Corollary 16. Let 𝜋 be an anti-invariant semi-Riemannian submersion from a para-Kähler manifold (𝑀,𝑔,𝑃) to a semi-Riemannian manifold (𝑁,𝑔^′). Let 𝑈 and 𝑉 be nonzero nonlightlike orthogonal unit vertical vectors. Then, 𝐾(𝑈∧𝑉)=𝐾^∗(𝑃𝑈∧𝑃𝑉) if and only if 𝑔(𝐴_𝑃𝑈𝑉,𝐴_𝑃𝑈𝑉)=−𝑔(𝐶𝐴_𝑃𝑈𝑉,𝐶𝐴_𝑃𝑈𝑉), where 𝐾^∗ is the sectional curvature of 𝑁.

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.