# Constancy ofϕ¯-Holomorphic Sectional Curvature for an Indefinite Generalized g·f·f-Space Form

Authors: Jae Won Lee
Journal: Advances in Mathematical Physics
 (2011)

---

## Abstract

Bonome et al., 1997, provided an algebraic characterization for an indefinite Sasakian manifold to reduce to a space of constantϕ-holomorphic sectional curvature. In this present paper, we generalize the same characterization for indefinite g·f·f-space forms.


---

## Body

## 1. Introduction

For an almost Hermitian manifold(M2n,g,J) with dim(M)=2n>4, Tanno [1] has proved the following.Theorem 1.1.
Letdim(M)=2n>4, and assume that almost Hermitian manifold (M2n,g,J) satisfies
(1.1)R(JX,JY,JZ,JX)=R(X,Y,Z,X)
for every tangent vector X, Y, and Z. Then (M2n,g,J) has a constant holomorphic sectional curvature at x if and only if
(1.2)R(X,JX)XisproportionaltoJX
for every tangent vector X at x∈M.Tanno [1] has also proved an analogous theorem for Sasakian manifolds as follows.Theorem 1.2.
A Sasakian manifold ≥5 has a constantϕ-sectional curvature if and only if
(1.3)R(X,ϕX)XisproportionaltoϕX
for every tangent vector X such that g(X,ξ)=0.Nagaich [2] has proved the generalized version of Theorem 1.1, for indefinite almost Hermitian manifolds as follows.Theorem 1.3.
Let(M2n,g,J)(n>2) be an indefinite almost Hermitian manifold that satisfies (1.1), then (M2n,g,J) has a constant holomorphic sectional curvature at x if and only if
(1.4)R(X,JX)XisproportionaltoJX
for every tangent vector X at x∈M.Bonome et al. [3] generalized Theorem 1.2 for an indefinite Sasakian manifold as follows.Theorem 1.4.
Let(M2n+1,ϕ,η,ξ,g)(n≥2) be an indefinite Sasakian manifold. Then M2n+1 has a constant ϕ-sectional curvature if and only if
(1.5)R(X,ϕX)XisproportionaltoϕX
for every vector field X such that g(X,ξ)=0.In this paper, we generalize Theorem1.4 for an indefinite generalized g·f·f-space form by proving the following.Theorem 1.5.
Let(M¯2n+r,F1,F2,ℱ) be an indefinite generalized g·f·f-space form. Then M¯2n+r is of constant ϕ-sectional curvature if and only if
(1.6)R¯(X,ϕ¯X)Xisproportionaltoϕ¯X
for every vector field X such that g¯(X,ξα¯)=0, for any α∈{1,…,r}.


## 2. Preliminaries

A manifoldM¯ is called a globally framed f-manifold (org·f·f-manifold) if it is endowed with a nonnull (1,1)-tensor field ϕ¯ of constant rank, such that kerϕ¯ is parallelizable; that is, there exist global vector fields ξα¯, α∈{1,…,r}, with their dual 1-forms η¯α, satisfying ϕ¯2=-I+∑α=1rη¯α⊗ξα¯ and η¯α(ξβ¯)=δβα.Theg·f·f-manifold (M¯2n+r,ϕ¯,ξα¯,η¯α), α∈{1,…,r}, is said to be an indefinite metric g·f·f-manifold if g¯ is a semi-Riemannian metric with index ν(0<ν<2n+r) satisfying the following compatibility condition: (2.1)g¯(ϕ¯X,ϕ¯Y)=g¯(X,Y)-∑α=1rϵαη¯α(X)η¯α(Y),
for any X,Y∈Γ(TM¯), being ϵα=±1 according to whether ξ¯α is spacelike or timelike. Then, for any α∈{1,…,r}, one has η¯α(X)=ϵαg¯(X,ξ¯α). Following the notations in [4, 5], we adopt the curvature tensor R, and thus we have R(X,Y,Z)=∇X∇YZ-∇Y∇XZ-∇[X,Y]Z and R¯(X,Y,Z,W)=g¯(R¯(Z,W,Y),X), for any X, Y, Z, W∈Γ(TM).We recall that, as proved in [6], the Levi-Civita connection ∇¯ of an indefinite g·f·f-manifold satisfies the following formula:(2.2)2g¯((∇¯Xϕ¯)Y,Z)=3dΦ(X,ϕ¯Y,ϕ¯Z)-3dΦ(X,Y,Z)+g¯(N(Y,Z),ϕ¯X)+ϵαNαϕ¯(Y,Z)η¯α(X)+2ϵαdη¯α(ϕ¯Y,X)η¯α(Z)-2ϵαdη¯α(ϕ¯Z,X)η¯α(Y),
where Nαϕ¯ is given by Nαϕ¯(X,Y)=2dη¯α(ϕ¯X,Y)-2dη¯α(ϕ¯Y,X).An indefinite metricg·f·f-manifold is called an indefinite𝒮-manifold if it is normal and dη¯α=Φ, for any α∈{1,…,r}, where Φ(X,Y)=g¯(X,ϕ¯Y) for any X,Y∈Γ(TM¯). The normality condition is expressed by the vanishing of the tensor field N=Nϕ¯+∑α=1r2dη¯α⊗ξ¯α, Nϕ¯ being the Nijenhuis torsion of ϕ¯.Furthermore, the Levi-Civita connection of an indefinite𝒮-manifold satisfies(2.3)(∇¯Xϕ¯)Y=g¯(ϕ¯X,ϕ¯Y)ξ¯+η¯(Y)ϕ¯2(X),
where ξ¯=∑α=1rξ¯α and η¯=∑α=1rϵαη¯α. We recall that ∇¯Xξ¯α=-ϵαϕ¯X and kerϕ¯ is an integrable flat distribution since ∇¯ξ¯αξ¯β=0 (see more details in [6]).A plane section inTpM¯ is a ϕ¯-holomorphic section if there exists a vector X∈TpM¯ orthogonal to ξ¯1,…,ξ¯r such that {X,ϕ¯X} span the section. The sectional curvature of a ϕ¯-holomorphic section, denoted by c(X)=R(X,ϕ¯X,ϕ¯X,X), is called a ϕ¯-holomorphic sectional curvature.Proposition 2.1 (see [7]).
An indefinite Sasakian manifold(M¯2n+1,ϕ¯,ξ¯,η¯,g¯) has ϕ¯-sectional curvature c if and only if its curvature tensor verifies
(2.4)R¯(X,Y)Z=(c+3)4{g¯(Y,Z)X-g¯(X,Z)Y}+(c-1)4{Φ(X,Z)ϕ¯Y-Φ(Y,Z)ϕ¯X+2Φ(X,Y)ϕ¯Z-g¯(Z,Y)η¯(X)ξ¯+g¯(Z,X)η¯(Y)ξ¯-η¯(Y)η¯(Z)X+η¯(Z)η¯(X)Y}
for any vector fields X,Y,Z,W∈Γ(TM¯).
A Sasakian manifoldM¯2n+1 with constant ϕ¯-sectional curvature c∈ℝ is called a Sasakian space form, denoted by M¯2n+1(c).Definition 2.2.
An almost contact metric manifold(M¯2n+1,ϕ¯,ξ¯,η¯,g¯) is an indefinite generalized Sasakian space form, denoted by M¯2n+1(f1,f2,andf3), if it admits three smooth functions f1, f2, f3 such that its curvature tensor field verifies
(2.5)R¯(X,Y)Z=f1{g¯(Y,Z)X-g¯(X,Z)Y}+f2{Φ(X,Z)ϕ¯Y-Φ(Y,Z)ϕ¯X+2Φ(X,Y)ϕ¯Z}+f3{-g¯(Z,Y)η¯(X)ξ¯+g¯(Z,X)η¯(Y)ξ¯-η¯(Y)η¯(Z)X+η¯(Z)η¯(X)Y-g¯(Z,Y)η¯(X)ξ¯+g¯(Z,X)η¯(Y)ξ¯}
for any vector fields X,Y,Z,W∈Γ(TM¯).Remark 2.3.
Any indefinite generalized Sasakian space form hasϕ¯-sectional curvature c=f1+3f2. Indeed, f1=(c+3)/4 and f2=f3=(c-1)/4.Proposition 2.4 (see [6]).
An indefinite𝒮-manifold M¯2n+r has ϕ¯-sectional curvature c if and only if its curvature tensor verifies
(2.6)R¯(X,Y)Z=(c+3ϵ)4{g¯(ϕ¯X,ϕ¯Z)ϕ¯2Y-g¯(ϕ¯Y,ϕ¯Z)ϕ¯2X}+(c-ϵ)4{Φ(Z,Y)ϕ¯X-Φ(Z,X)ϕ¯Y+2Φ(X,Y)ϕ¯Z}+{η¯(Z)η¯(X)ϕ¯2Y-η¯(Y)η¯(Z)ϕ¯2X+g¯(ϕ¯Z,ϕ¯Y)η¯(X)ξ¯-g¯(ϕ¯Z,ϕ¯X)η¯(Y)ξ¯}
for any vector fields X,Y,Z,W∈Γ(TM¯) and ϵ=∑ϵα.
An indefinite𝒮-manifold M¯2n+r with constant ϕ¯-sectional curvature c∈ℝ is called a 𝒮-space form, denoted by M¯2n+r(c). One remarks that for r=1 (2.6) reduces to (2.4).


## 3. An Indefinite Generalizedg·f·f-Manifold

Letℱ denote any set of smooth functions Fij on M¯2n+r such that Fij=Fji for any i,j∈{1,…,r}.Definition 3.1.
Anindefinite generalizedg·f·f-space-form, denoted by (M¯2n+r,F1,F2,ℱ), is an indefinite g·f·f-manifold (M¯2n+r,ϕ¯,ξα¯,η¯α,g¯) which admits smooth function F1,F2,ℱ such that its curvature tensor field verifies
(3.1)R¯(X,Y)Z=F1{g¯(ϕ¯X,ϕ¯Z)ϕ¯2Y-g¯(ϕ¯Y,ϕ¯Z)ϕ¯2X}+F2{Φ(Z,Y)ϕ¯X-Φ(Z,X)ϕ¯Y+2Φ(X,Y)ϕ¯Z}+∑α,β=1rFαβ{η¯α(X)η¯β(Z)ϕ¯2Y-η¯α(Y)η¯β(Z)ϕ¯2X+g¯(ϕ¯Z,ϕ¯Y)η¯α(X)ξβ¯-g¯(ϕ¯Z,ϕ¯X)η¯α(Y)ξβ¯η¯α(X)η¯β(Z)ϕ¯2Y-η¯α(Y)η¯β(Z)ϕ¯2X}
for any vector fields X,Y,Z,W∈Γ(TM¯).
Forr=1, we obtain an indefinite Sasakian space form M¯2n+1(f1,f2,f3) with f1=F1, f2=F2, and f3=F1-F11. In particular, if the given structure is Sasakian, (3.1) holds with F11=1, F1=(c+3)/4, F3=(c-1)/4, and f3=F1-F11=(c-1)/4=f2.Theorem 3.2.
Let(M¯2n+r,F1,F2,ℱ) be an indefinite generalized g·f·f-space form. Then M¯2n+r is of constant ϕ-sectional curvature if and only if
(3.2)R¯(X,ϕ¯X)Xisproportionaltoϕ¯X
for every vector field X such that g¯(X,ξα¯)=0, for any α∈{1,…,r}.Proof.
Let(M¯2n+r,F1,F2,ℱ) be an indefinite generalized g·f·f-space form. To prove the theorem for n≥2, we will consider cases when n=2 and when n>2, that is, when n≥3.
Case 1 (g¯(X,X)=g¯(Y,Y)).
The proof is similar as given by Lee and Jin [8], so we drop the proof.Case 2 (g¯(X,X)=-g¯(Y,Y)).
Here, ifX is spacelike, then Y is timelike or vice versa. First of all, assume that M¯ is of constant ϕ¯-holomorphic sectional curvature. Then (3.1) gives
(3.3)R¯(X,ϕ¯X)X={F1+3F2}ϕ¯X=cϕ¯X.
Conversely, let {X,Y} be an orthonormal pair of tangent vectors such that g¯(ϕ¯X,Y)=g¯(X,Y)=g¯(Y,ξα¯)=0, α∈{1,…,r}, and n≥3. Then Ẍ=(X+iY)/2 and Ÿ=(iϕ¯X+ϕ¯Y)/2 also form an orthonormal pair of tangent vectors such that g¯(ϕ¯Ẍ,Ÿ)=0. Then (3.1) and curvature properties give
(3.4)0=R¯(Ẍ,ϕ¯Ẍ,Ÿ,Ẍ)=g¯(R¯(X,ϕ¯X)X,ϕ¯X)-g¯(R¯(Y,ϕ¯Y)Y,ϕ¯Y)-2g¯(R¯(X,ϕ¯Y)Y,ϕ¯Y)+2g¯(R¯(X,ϕ¯X)Y,ϕ¯X).
From the assumption, we see that the last two terms of the right-hand side vanish. Therefore, we get c(X)=c(Y).
Now, ifspan{U,V} is ϕ¯-holomorphic, then for ϕ¯U=aU+bV, where a and b are constant, we have
(3.5)span⁡{U,ϕ¯U}=span⁡{U,aU+bV}=span⁡{U,V}.
Similarly,(3.6)span⁡{V,ϕ¯V}=span⁡{U,V},span⁡{U,ϕ¯U}=span⁡{V,ϕ¯V}.
These imply
(3.7)R¯(U,ϕ¯U,U,ϕ¯U)=R¯(V,ϕ¯V,V,ϕ¯V),orc(U)=c(V).
If span{U,V} is not ϕ¯-holomorphic section, then we can choose unit vectors X∈span{U,ϕ¯U}⊥ and Y∈span{V,ϕ¯V}⊥ such that span{X,Y} is ϕ¯-holomorphic. Thus we get
(3.8)c(U)=c(X)=c(Y)=c(V),
which shows that any ϕ¯-holomorphic section has the same ϕ¯-holomorphic sectional curvature.
Now, letn=2, and let {X,Y} be a set of orthonormal vectors such that g¯(X,X)=-g¯(Y,Y) and g¯(X,ϕ¯X)=0, and we have c(X)=c(Y) as before. Using the property (3.2), we get
(3.9)R¯(X,ϕ¯X)X=-{F1+3F2}ϕ¯X=-c(X)ϕ¯X,R¯(X,ϕ¯X)Y=-2F2ϕ¯Y,R¯(X,ϕ¯Y)X=-F1ϕ¯Y,R¯(X,ϕ¯Y)Y=F2ϕ¯X,R¯(Y,ϕ¯X)Y=F1ϕ¯X,R¯(Y,ϕ¯X)X=-F2ϕ¯Y,R¯(Y,ϕ¯Y)X=2F2ϕ¯X,R¯(Y,ϕ¯Y)Y={F1+3F2}ϕ¯=c(Y)ϕ¯Y=c(X)ϕ¯Y.
Now, define X̂=aX+bY such that a2-b2=1 and a2≠b2. Using the above relations, we get
(3.10)R(X̂,ϕ¯X̂)X̂=C1ϕ¯X+C2ϕ¯Y.
Therefore, we have(3.11)C1=-a3c(X)+ab2c(X),C2=b3c(X)-a2bc(X).
On the other hand,
(3.12)R¯(X̂,ϕ¯X̂)X̂=c(X̂)ϕ¯X̂=c(X̂){aϕ¯X+bϕ¯Y}.
Comparing (3.11) and (3.12), we get
(3.13)-a2c(X)+b2c(X)=c(X̂),b2c(X)-a2c(X)=c(X̂).
On solving (3.13), we have
(3.14)c(X)=c(X̂).
Similary, we can prove
(3.15)c(Y)=c(Ŷ).
Therefore, M¯ has constant ϕ¯-holomorphic sectional curvature.Case 3 (g¯(U,U)=0).
It is enough to show a sufficient condition. LetYα be a unit vector tangent to ξ¯α, for any α∈{1,…,r}, such that g¯(Yα,Yα)=-g¯(ξα,ξα)=-ϵα, and consider the null vector Uα=ξα+Y. From (3.2),
(3.16)c(Uα)ϕ¯Uα=c(Uα)ϕ¯(ξα+Yα)=R¯(ξα+Yα,ϕ¯(ξα+Yα))(ξα+Yα).
Therefore,
(3.17)c(Uα)=g¯(c(Uα)ϕ¯(ξα+Yα),ϵαϕ¯Yα)=ϵαg¯(R¯(ξα+Yα,ϕ¯(ξα+Yα))(ξα+Yα),ϕ¯Yα)=ϵαg¯(R¯(ξα,ϕ¯ξα)ξα,ϕ¯Yα)+ϵαg¯(R¯(ξα,ϕ¯Yα)ξα,ϕ¯Yα)+ϵαg¯(R¯(ξα,ϕ¯ξα)Yα,ϕ¯Yα)+ϵαg¯(R¯(ξα,ϕ¯Yα)Yα,ϕ¯Yα)+ϵαg¯(R¯(Yα,ϕ¯ξα)ξα,ϕ¯Yα)+ϵαg¯(R¯(Yα,ϕ¯Yα)ξα,ϕ¯Yα)+ϵαg¯(R¯(Yα,ϕ¯ξα)Yα,ϕ¯Yα)+ϵαg¯(R¯(Yα,ϕ¯Yα)Yα,ϕ¯Yα)+ϵαg¯(R¯(ξα,ϕ¯Yα)ξα,ϕ¯Yα)+2ϵαg¯(R¯(Yα,ϕ¯Yα)ξα,ϕ¯Yα)+ϵαg¯(R¯(Yα,ϕ¯Yα)Yα,ϕ¯Yα)=ϵαg¯(R¯(Yα,ϕ¯Yα)Yα,ϕ¯Yα).
From Cases 1 and 2, depending on the sign of ϵα, g¯(R¯(Yα,ϕ¯Yα)Yα,ϕ¯Yα)=ϵαc(Yα) is constant, and hence c(Uα)=c(Yα) is constant.Theorem 3.3 (see [9]).
Let(M¯2n+r,ϕ¯,η¯α,ξα¯,g¯)(n≥2) be an indefinite 𝒮-manifold. Then M2n+r is of constant ϕ-sectional curvature if and only if
(3.18)R(X,ϕ¯X)Xisproportionaltoϕ¯X
for every vector field X such that g(X,ξα¯)=0, for any α∈{1,…,r}.Proof.
An𝒮-space form is a special case of g·f·f-space form, and hence the proof follows from Theorem 3.2 and (2.6).Theorem 3.4 (cf. Bonome et al. [3]).
Let(M2n+1,ϕ,η,ξ,g)(n≥2) be an indefinite Sasakian manifold. Then M2n+1 is of constant ϕ-sectional curvature if and only if
(3.19)R(X,ϕX)XisproportionaltoϕX
for every vector field X such that g(X,ξ)=0.Proof.
Whenr=1, an indefinite 𝒮-space form M2n+1(c) reduces to a Sasakian space form. The proof follows from (2.4) and Theorem 3.3.



---

*Source: 527434-2011-11-09.xml*