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

Authors: Yılmaz Gündüzalp
Journal: Journal of Function Spaces and Applications
 (2013)

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## 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.


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## Body

## 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 Section2 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 manifoldM endowed with an almost para-complex structure P≠±I and a semi-Riemannian metric g such that
(1)P2=I,g(PX,PY)=-g(X,Y)
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,
(2)(∇XP)Y=0
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:TpM→Tπ(p)N is onto for all p∈M;(ii)
the fibresπ-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 by𝒱 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 of𝒱, 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
(3)T(E,F)=TEF=h∇vEvF+v∇vEhF,A(E,F)=AEF=v∇hEhF+h∇hEvF
for any E, F∈Γ(TM), where v and h are the vertical and horizontal projections (see [20, 21]). From (3), one can obtain
(4)∇UW=TUW+∇^UW;(5)∇UX=TUX+h(∇UX);(6)∇XU=v(∇XU)+AXU;(7)∇XY=AXY+h(∇XY),
for any X, Y∈Γ((kerπ*)⊥), U, W∈Γ(kerπ*). Moreover, if X is basic then h(∇UX)=h(∇XU)=AXU.We note that forU, V∈Γ(kerπ*), TUV coincides with the second fundamental form of the immersion of the fibre submanifolds and for X, Y∈Γ((kerπ*)⊥), AXY=(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: AXY=-AYX, for X, Y∈Γ((kerπ*)⊥) and T is symmetric on the vertical distribution: TUV=TVU, for U, V∈Γ(kerπ*).We now recall the following result which will be useful for later.Lemma 1 (see [17, 21]).
Ifπ: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
(8)(∇π*)(X,Y)=∇Xππ*Y-π*(∇XY)
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.Letπ:(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
(9)(kerπ*)⊥=P(kerπ*)⊕μ.It is easy to see thatμ is an invariant distribution of (kerπ*)⊥, under the endomorphism P. Thus, for X∈Γ((kerπ*)⊥), we have
(10)PX=BX+CX,
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
(11)TN=π*(P(kerπ*))⊕π*(μ).Note that given a semi-Euclidean spaceR24 with coordinates (x1,…,x4), we can canonically choose an almost para-complex structure P on R24 as follows:
(12)P(a1∂∂x1+a2∂∂x2+a3∂∂x3+a4∂∂x4)=-a2∂∂x1-a1∂∂x2+a4∂∂x3+a3∂∂x4,
where a1,…,a4∈R.Also the neutral metric compatible withP is
(13)g=(dx1)2-(dx2)2+(dx3)2-(dx4)2.We now give an example of an anti-invariant semi-Riemannian submersion.Example 3.
Letπ:R24→R12 be a map defined π(x1,x2,x3,x4)=((x1+x3)/2, (x2+x4)/2). Then, by direct calculations
(14)kerπ*=Span{Z1=-∂∂x1+∂∂x3,Z2=-∂∂x2+∂∂x4},(kerπ*)⊥=Span{X1=∂∂x1+∂∂x3,X2=∂∂x2+∂∂x4}.
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.
Letπ 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
(15)g(CY,PV)=0,(16)g(∇XCY,PV)=-g(CY,PAXV)
for X, Y∈Γ((kerπ*)⊥) and V∈Γ(kerπ*).Proof.
ForY∈Γ((kerπ*)⊥) and V∈Γ(kerπ*), using (1) we have
(17)g(CY,PV)=g(PY-BY,PV)=g(PY,PV)
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
(18)g(∇XCY,PV)=-g(CY,P∇XV)
for X, Y∈Γ((kerπ*)⊥) and V∈Γ(kerπ*). Then using (6) we have
(19)g(∇XCY,PV)=-g(CY,PAXV)-g(CY,Pv∇XV).
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.
Letπ 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,PAXV)+g(CX,PAYV).(iii)
g(PV,AYBX-AXBY)=-g(CY,PAXV)+g(CX,PAYV), for X, Y∈Γ((kerπ*)⊥) and V∈Γ(kerπ*).Proof.
ForY∈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
(20)g([X,Y],V)=-g(∇XPY,PV)+g(∇YPX,PV).
Then from (10) we get
(21)g([X,Y],V)=-g(∇XBY,PV)-g(∇XCY,PV)+g(∇YBX,PV)+g(∇YCX,PV).
Since π is a semi-Riemannian submersion, we have
(22)g([X,Y],V)=-g′(π*∇XBY,π*PV)-g(∇XCY,PV)+g′(π*∇YBX,π*PV)+g(∇YCX,PV).
Thus, from (8) and (16) we obtain
(23)g([X,Y],V)=g′((∇π*)(X,BY)-(∇π*)(Y,BX),π*PV)+g(CY,PAXV)-g(CX,PAYV)
which proves (i)⇔(ii). On the other hand, using (8) we have
(24)(∇π*)(X,BY)-(∇π*)(Y,BX)=-π*(∇XBY-∇YBX).
Then (6) implies that
(25)(∇π*)(X,BY)-(∇π*)(Y,BX)=-π*(AXBY-AYBX).
Since AYBX-AXBY∈Γ((kerπ*)⊥), this shows that (ii)⇔(iii).Theorem 6.
Letπ 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(AXBY,PV)=g(CY,PAXV).(iii)
g′((∇π*)(X,BY),π*PV)=-g(CY,PAXV), for X, Y∈Γ((kerπ*)⊥) and V∈Γ(kerπ*).Proof.
From (1), (2) and (6) we obtain
(26)g(∇XY,V)=-g(AXBY,PV)-g(∇XCY,PV)
for X, Y∈Γ((kerπ*)⊥) and V∈Γ(kerπ*). Then by (16) we have
(27)g(∇XY,V)=-g(AXBY,PV)+g(PAXV,CY)
which shows (i)⇔(ii). On the other hand from (6) and (8) we get
(28)g(AXBY,PV)=g′(-(∇π*)(X,BY),π*PV).
This shows (ii)⇔(iii).Theorem 7.
Letπ 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)
TVBX+ACXV∈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
(29)g(∇VW,X)=-g′((∇π*)(V,PX),π*PW)
which is (i)⇔(ii). By direct calculation, we derive
(30)g(∇VPX,PW)=-g′((∇π*)(V,PX),π*PW).
Using (10) we obtain
(31)g(∇VBX+∇VCX,PW)=-g′((∇π*)(V,PX),π*PW).
Hence we have
(32)g(∇VBX+[V,CX]+∇CXV,PW)=-g′((∇π*)(V,PX),π*PW).
Since [V,CX]∈Γ(kerπ*), using (4) and (6), we get
(33)g(TVBX+ACXV,PW)=-g′((∇π*)(V,PX),π*PW).
This shows (ii)⇔(iii).We say that an anti-invariant semi-Riemannian submersion is a Lagrangian semi-Riemannian submersion ifP(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 Example3 is a Lagrangian semi-Riemannian submersion.Ifπ is a Lagrangian submersion, then (11) implies that TN=π*(P(kerπ*)).From Theorem5 we have the following.Corollary 8.
Letπ 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)
AXPY=AYPX, for X, Y∈Γ((kerπ*)⊥).Theorem 9.
Letπ 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)
TVPW=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
(34)g(∇VPW,PX)=g′((∇π*)(V,PX),π*PW)
for X∈Γ((kerπ*)⊥) and V, W∈Γ(kerπ*). Then using (5) we have
(35)g′((∇π*)(V,PX),π*PW)=g(TVPW,PX).
Since TVPW∈Γ(kerπ*), we get (ii)⇔(iii).We note that a differentiable mapπ between two semi-Riemannian manifolds is called totally geodesic if ∇π*=0.Theorem 10.
Letπ 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
(36)TWPV=0,∀W,V∈Γ(kerπ*),AXPW=0,∀X∈Γ((kerπ*)⊥).Proof.
First of all, we recall that the second fundamental form of a semi-Riemannian submersion satisfies(37)(∇π*)(X,Y)=0,X,Y∈Γ((kerπ*)⊥).
For W, V∈Γ(kerπ*), by using (2), (5), and (8), we get
(38)(∇π*)(W,V)=-π*(PTWPV).
On the other hand, from (1), (2), and (8), we have
(39)(∇π*)(X,W)=-π*(P∇XPW)
for X∈Γ((kerπ*)⊥). Then using (7), we get
(40)(∇π*)(X,W)=-π*(PAXPW).
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.
Letπ 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  PTV=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 ∑i=1r‍Teiei=0. On the other hand, from (2), (4), and (5), we obtain
(41)TVPW=PTVW
for any V, W∈Γ(kerπ*). Using (41), we get
(42)∑i=1r‍g(TeiPei,V)=-∑i=1r‍g(Teiei,PV)
for any V∈Γ(kerπ*). Thus using the properties of the O’Neill tensor T we have
(43)∑i=1r‍g(TeiV,Pei)=∑i=1r‍g(Teiei,PV).
Since T is symmetric, we obtain
(44)∑i=1r‍g(TVei,Pei)=∑i=1r‍g(Teiei,PV).Denote by{Li} the canonical foliations on a product manifold M1×M2 with natural projections pi onto Mi, (i=1,2). Let Pi be the projection of T(M1×M2) onto TLi and also set Pi⊥=I-Pi, (i=1,2).Let(Mi,gi), (i=1,2) be semi-Riemannian manifolds and λi: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
(45)g(X,Y)=λ12g1(P1X,P1Y)+λ22g2(P2X,P2Y)
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).Letg 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 productM1×λM2 if L1 is a totally geodesic foliation and L2 is a totally umbilic foliation,(2)
a warped productM1×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 fL1 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 Mkerπ*×fM(kerπ*)⊥, then M is a usual (locally product) manifold, where M(kerπ*)⊥ and Mkerπ* are integral manifolds of the distribution (kerπ*)⊥ and kerπ*.Proof.
Suppose thatπ:(M,g,P)→(N,g′) is a Lagrangian semi-Riemannian submersion and M is a locally twisted product of the form Mkerπ*×fM(kerπ*)⊥. Then Mkerπ* 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
(46)g(∇XY,V)=g(H,V)g(X,Y),
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
(47)g(∇XY,V)=g(AXPV,PY).
Thus, from (46) and (47), we have
(48)AXPV=-g(H,V)PX.
Hence, we arrive at
(49)g(AXPV,PX)=g(H,V)g(X,X).
Then using (7) we get
(50)g(∇XPV,PX)=g(H,V)g(X,X).
Thus (2) implies that
(51)-g(∇XV,X)=g(H,V)g(X,X).
Hence, we obtain
(52)g(∇XX,V)=g(H,V)g(X,X).
Then, since A is alternating on the horizontal distribution, we have AXX=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.
Letπ 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*)⊥×fMkerπ* if and only if
(53)TVPX=-g(X,TVV)g(V,V)-1PV
for X, Y∈Γ((kerπ*)⊥) and V∈Γ(kerπ*), where M(kerπ*)⊥ and Mkerπ* are integral manifolds of the distribution (kerπ*)⊥ and kerπ*.Proof.
From (2) and (5), we obtain
(54)g(∇VW,X)=-g(TVPW,PX)
for X∈Γ((kerπ*)⊥) and V, W∈Γ(kerπ*). Since TV is skew-symmetric, we get
(55)g(∇VW,X)=g(TVPX,PW).
This implies that kerπ* is totally umbilical if and only if
(56)X(λ)PV=TVPX,
where λ is a function on M. Then by direct computations, it is easy to see that this is equivalent to
(57)TVPX=-g(X,TVV)g(V,V)-1PV.
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 TpM, p∈M, spanned by {X,Y}, is defined by
(58)K(X,Y)=R(X,Y,X,Y)g(X,X)g(Y,Y)-g(X,Y)2.
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.
Letπ 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:
(59)BTUV=TUPV;(60)h∇UPV=CTUV+P∇^UV;(61)PAXY+Ch∇XY=AXBY+h∇XCY;(62)Bh∇XY=v∇XBY+AXCY;(63)BAXU=AXPU.Proof.
From (2) and (5) we have
(64)P∇UV=h∇UPV+TUPV.
Using (4) we get
(65)PTUV+P∇^UV=h∇UPV+TUPV.
Then (10) implies that
(66)BTUV+CTUV+P∇^UV=h∇UPV+TUPV.
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 Lemma14 and pages 13-14 of [21], we obtain the following.Theorem 15.
Letπ 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
(67)K(U∧V)=K*(PU∧PV)+ϵUϵV[3g(APUV,APUV)aaaaaaaaaaaaaaaaaaaaa+3g(CAPUV,CAPUV)]H(U)=-(g((∇PUT)UU,PU)+g(TUU,TUU)+g(CTUU,CTUU)+g(APUU,APUU)),
where ϵU=g(U,U)∈{±1}, ϵV=g(V,V)∈{±1} and H(U)=K(U∧PU) is the para-holomorphic sectional curvature of M.From Theorem15 we have the following result.Corollary 16.
Letπ 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(APUV,APUV)=-g(CAPUV,CAPUV), where K* is the sectional curvature of N.



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*Source: 720623-2013-11-10.xml*