On anti-invariant semi-Riemannian submersions from Lorentzian (para)Sasakian manifolds

TL;DR

This paper investigates conditions for semi-Riemannian submersions from Lorentzian (para)Sasakian manifolds, providing decomposition theorems and criteria for the characteristic vector field's orientation.

Contribution

It establishes necessary and sufficient conditions for the characteristic vector field to be vertical or horizontal in such submersions and derives decomposition theorems.

Findings

Criteria for the characteristic vector field orientation

Decomposition theorems for submersions from Lorentzian (para)Sasakian manifolds

Conditions for anti-invariant semi-Riemannian submersions

Abstract

In this paper we study a semi-Riemannian submersion from Lorentzian (para)almost contact manifolds and find necessary and sufficient conditions for the characteristic vector field to be vertical or horizontal. We also obtain decomposition theorems for an anti-invariant semi-Riemannian submersion from Lorentzian (para)Sasakian manifolds onto a Lorentzian manifold.

Equations

$${\varphi }^{2}X=\epsilon X+\eta (X)\xi ,$$

$$g(\varphi X,\varphi Y)=g(X,Y)+\eta (X)\eta (Y),$$

$$\eta (X)=\epsilon g(X,\xi ),$$

$$\eta (\xi )=-\epsilon ,$$

$$g(\varphi X,Y)=\epsilon g(X,\varphi Y).$$

$${\nabla }_X\xi =\epsilon \varphi X,$$

$$({\nabla }_X\varphi )Y=g(\varphi X,\varphi Y)\xi +\eta (Y){\varphi }^{2}X.$$

$${\varphi }_{ϵ}\left(\sum _{i=1}^n(X_i\frac{\partial }{\partial x^i}+Y_i\frac{\partial }{\partial y^i})+Z\frac{\partial }{\partial z}\right)=-ϵ\sum _{i=1}^n{Y}_i\frac{\partial }{\partial x^i}-\sum _{i=1}^n{X}_i\frac{\partial }{\partial y^i}+\sum _{i=1}^n{Y}_i{y}_i\frac{\partial }{\partial z},$$

$$g=-\eta \otimes \eta +\frac{1}{4}\sum _{i=1}^n(d{x}^i\otimes d{x}^i+d{y}^i\otimes d{y}^i),$$

$${\eta }_{ϵ}=-\frac{ϵ}{2}\left(dz-\sum _{i=1}^n{y}^{i}d{x}^i\right),$$

$$\xi =2\frac{\partial }{\partial z}.$$

$$\mathcal{T}(E,F)$$

$$\mathcal{A}(E,F)$$

$$g({\mathcal{T}}_{D}E,G)=-g(E,{\mathcal{T}}_{D}G),$$

$$g({\mathcal{A}}_{D}E,G)=-g(E,{\mathcal{A}}_{D}G),$$

$${\mathcal{T}}_{U}V$$

$${\mathcal{A}}_{X}Y$$

$${\nabla }_{V}W$$

$${\nabla }_{V}X$$

$${\nabla }_{X}V$$

$${\nabla }_{X}Y$$

$$(\nabla {\phi }_{\ast })(X,Y)={\nabla }_X^{\phi }{\phi }_{\ast }(Y)-{\phi }_{\ast }({\nabla }_X^{M}Y)$$

$$(\nabla F_{\ast })(X,Y)=0$$

$$\tau (\phi )=\mathrm{div}{\phi }_{\ast }=\sum _{i=1}^m{ϵ}_i(\nabla {\phi }_{\ast })(e_i,e_i),$$

Full text

1 Introduction

Semi-Riemannian submersions between semi-Riemannian manifolds were studied by O’Neill Oneill:Thefundamentalequationsofsubmersion; oneil:book and Gray Gray:PseudoRiemannianAlmostProductManifoldsSubmersions. Moreover, B. Ṣahin in Shahin:SlantSubmersionHermitian; shahin:AntiInvariantRiemannianSubmersionsFromAlmostHermitianManifolds introduced anti-invariant Riemannian submersions and slant submersions from almost Hermitian manifold onto Riemannian manifolds. Also, anti-invariant Riemannian submersions were studied in Murathan:AntiInvariantRiemannianSubmersionFromCosymplectic; Erken:SlantRiemanniansubmersionsfromSasakianmanifolds2016; Erken:AntiInvariantRiemanniansubmersionsfromSasakianmanifolds; Lee:AntInvariantRiemannianSubmersionsAlmostContact.

The theory of Lorentzian submersion was introduced by Magin and Falcitelli et al in Magid:SubmersionsFromSntiDeSitterspace and Falcitelli:RiemannianSubmersionsandRelatedTopics, respectively. In Kaneyuki:Almostparacontact Kaneyuki and Williams defined the almost paracontact structure on pseudo-Riemannian manifold. Recently, Gündüzalp and Şahin studied paracontact structures in Gunduzlap:SlantsubmersionsfromLorentzian; gunduzlapShahin:ParacontactSemiRiemannianSubmersions; GUNDUZALP:paracontactparacomplexsemiRiemanniansubmersion.

In this paper we studied anti-invariant semi-Riemannian submersion from Lorentzian (para)almost contact manifolds. In Sect. 3, we introduced anti-invariant semi-Riemannian submersion from Lorentzian (para)almost contact manifolds and presented three examples. Also we find necessary and sufficient conditions for the characteristic vector field to be vertical or horizontal. In sect. 4, we studied anti-invariant semi-Riemannian submersion from Lorentzian (para)Sasakian manifolds onto a Riemannian manifold such that the characteristic vector field is vertical and investigated the geometry of leaves of the distributions. In sect. 5, we studied anti-invariant semi-Riemannian submersion from Lorentzian (para)Sasakian manifolds onto a Lorentzian manifold such that the characteristic vector field is horizontal and we obtained decomposition theorems for it.

2 Preliminaries

In this section, we recall some necessary details background on Lorentzian almost contact manifold, Lorentzian almost para contact manifold, semi-Riemannian submersion and harmonic maps.

2.1 Lorentzian almost contact manifold

Let $(M,g)$ be a $(2n+1)$-dimensional Lorentzian manifold with a tensor field $\phi$ of type $(1,1)$, a vector field $\xi$ and a 1-form $\eta$ which satisfy

[TABLE]

for any vector fields $X,Y$ tangent to $M$, it is called Lorentzian almost contact manifold or Lorentzian almost para contact manifold for $\varepsilon=-1$ or $\varepsilon=1$, respectivelyAlegre:Slantsubmanifolds. In this case from (1) and (4) imply that $\phi\xi=0,\eta o\phi=0$, and $rank\phi=2n$. However, for any vector fields $X,Y$ in $\Gamma(TM)$,

[TABLE]

Let $\Phi$ be the 2-form in $M$ given by $\Phi(X,Y)=g(X,\phi Y).$ Then, $M$ is called Lorentzian metric contact manifold if $d\eta(X,Y)=\Phi(X,Y).$ So, $M$ is called almost normal contact Lorentzian manifold if satisfying $[\phi,\phi]+2d\eta\otimes\xi=0.$ If $\xi$ is a Killing tensor vector field, then the (para)contact structure is called K-(para)contact. In such a case, we have

[TABLE]

where $\nabla$ denotes the Levi-Civita connection of $g$. A Lorentzian almost contact manifold or Lorentzian almost para contact manifold $M$ is called Lorentzian Sasakian (LS) or Lorentzian para Sasakian (LPS) if

[TABLE]

Now we will introduce a well known Sasakian manifold example on $\mathbb{R}^{2n+1}.$

Example 1 (blair.Ri:book)

Let $\mathbb{R}^{2n+1}=\{(x^{1},\ldots,x^{n},y^{1},\ldots,y^{n},z)|x^{i},y^{i},z\in\mathbb{R},i=1,\ldots,n\}$. Consider $\mathbb{R}^{2n+1}$ with the following structure:

[TABLE]

Then, $(\mathbb{R}^{2n+1},\phi_{\epsilon},\xi,\eta_{\epsilon},g),$ is a Lorentzian Sasakian manifold if $\epsilon=-1$ and Lorentzian para Sasakian manifold if $\epsilon=1$. The vector fields $E_{i}=2\frac{\partial}{\partial y^{i}},E_{n+i}=2(\frac{\partial}{\partial x^{i}}+y_{i}\frac{\partial}{\partial z})$ and $\xi$ form a $\phi$-basis for the contact metric structure.

2.2 Semi-Riemannian submersion

Let $(M,g_{M})$ and $(N,g_{N})$ be semi-Riemannian manifolds. A semi-Riemannian submersion $F:M\to N$ is a submersion of semi-Riemannian manifolds such that:

The fibers $F^{-1}(q),q\in N,$ are semi-Riemannian submanifolds of $M.$ 2. 2.

$F_{*}$ preserves scalar products of vectors normal to fibers.

For each $q\in N,F^{-1}(q)$ is a submanifold of $M$ of dimension $\dim M-\dim N.$ The submanifolds $F^{-1}(q),q\in N$ are called fibers, and a vector field on $M$ is vertical if it is always tangent to fibers, horizontal if always orthogonal to fibers. A vector field $X$ on $M$ is called basic if $X$ is horizontal and $F$-related to a vector field $X_{*}$ on $N$. Every vector field $X_{*}$ on $N$ has a unique horizontal lift $X$ to $M$, and $X$ is basic. For a semi-Riemannian submersion $F:M\to N$, let $\mathcal{H}$ and $\mathcal{V}$ denote the projections of the tangent spaces of $M$ onto the subspaces of horizontal and vertical vectors, respectively. In the other words, $\mathcal{H}$ and $\mathcal{V}$ are the projection morphisms on the distributions $(\ker F_{*})^{\bot}$ and $\ker F_{*}$, respectively oneil:book.

Lemma 1 (Oneill:Thefundamentalequationsofsubmersion)

Let $F:M\to N$ be semi-Riemannian submersion between Semi-Riemannian manifolds and $X,Y$ be basic vector fields of $M$. Then

1. a)

$g_{M}(X,Y)=g_{N}(X_{*},Y_{*})oF,$** 2. b)

the horizontal part $\mathcal{H}[X,Y]$ of $[X,Y]$ is a basic vector field and corresponds to $[X_{*},Y_{*}],$ i.e., $F_{*}(\mathcal{H}[X,Y])=[X_{*},Y_{*}].$ 3. c)

$[V,X]$* is vertical for any vector field $V$ of $\ker F_{*}$.* 4. d)

$\mathcal{H}(\nabla_{X}^{M}Y)$* is the basic vector field corresponding to $\nabla_{X_{*}}^{N}Y_{*}.$*

The fundamental tensors of a submersion were defined by O’Neill. They are $(1,2)$-tensors on $M$, given by the formula:

[TABLE]

for any vector field $E$ and $F$ on $M$, where $\nabla$ denotes the Levi-Civita connection of $(M,g_{M}).$ It is easy to see that a Riemannian submersion $F:M\to N$ has totally geodesic fibers if and only if $T$ vanishes identically. For any $E\in\Gamma(TM)$, $\mathcal{T}_{E}$ and $\mathcal{A}_{E}$ are skew-symmetric operators on $(\Gamma(TM),g)$ reversing the horizontal and the vertical distributions. In the other words,

[TABLE]

for any $D,E,G\in\Gamma(TM).$ It is also easy to see that $\mathcal{T}$ is vertical, $\mathcal{T}_{E}=\mathcal{T}_{\mathcal{V}E}$ and $\mathcal{A}$ is horizontal, $\mathcal{A}=\mathcal{A}_{\mathcal{H}E}$. For any $U,V$ vertical and $X,Y$ horizontal vector fields $\mathcal{T},\mathcal{A}$ satisfy:

[TABLE]

Moreover, from (12) and (13) we have

[TABLE]

for $X,Y\in\Gamma((\ker F_{*})^{\bot})$ and $V,W\in\Gamma(\ker F_{*}),$ where $\hat{\nabla}_{V}W=\mathcal{V}\nabla_{V}W.$

2.3 Foliations on manifold and decomposition theorem

A foliation $\mathcal{D}$ on a manifold $M$ is an integrable distribution. A foliation $\mathcal{D}$ on a semi-Riemannian manifold $M$ is called totally umbilical, if every leaf of $\mathcal{D}$ is a totally umbilical semi-Riemannian submanifold of $M$. If, in addition, the mean curvature vector of every leaf is parallel in the normal bundle, then $\mathcal{D}$ is called a sphenic foliation, because in this case each leaf of $\mathcal{D}$ is an extrinsic sphere of $M$. If every leaf of $\mathcal{D}$ is a totally geodesic submanifold of $\mathcal{D}$, then $\mathcal{D}$ is called a totally geodesic foliationchen:2011pseudo.The following results were proved inponge:twisted.product.

Let $(M,g)$ be a simply-connected semi-Riemannian manifold which admits two complementary foliations $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ whose leaves intersect perpendicularly.

1. If $\mathcal{D}_{1}$ is totally geodesic and $\mathcal{D}_{2}$ is totally umbilical, then $(M,g)$ is isometric to a twisted product $M_{1}\times_{f}M_{2}$.

2. If $\mathcal{D}_{1}$ is totally geodesic and $\mathcal{D}_{2}$ is spherical, then $(M,g)$ is isometric to a warped product $M_{1}\times_{f}M_{2}$.

3. If $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are totally geodesic, then $(M,g)$ is isometric to a direct product $M_{1}\times M_{2}$, where $M_{1}$ and $M_{2}$ are integral manifolds of distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$.

2.4 Harmonic maps

We now recall the notion of harmonic maps between semi-Riemannian manifolds. Let $(M,g_{M})$ and $(N,g_{N})$ be semi-Riemannian manifolds and suppose that $\varphi:M\to N$ is a smooth mapping between them. Then the differential $\varphi_{*}$ of $\varphi$ can be viewed a section of the bundle $Hom(TM,\varphi^{-1}TN)\to M$, where $\varphi^{-1}TN$ is the pullback bundle which has fibers $\varphi^{-1}(TN_{p})=T_{\varphi(p)}N,p\in M$. $Hom(TM,\varphi^{-1}TN)$ has a connection $\nabla$ induced from the Levi-Civita connection $\nabla^{M}$ and the pullback connection. Then the second fundamental form of $\varphi$ is given by

[TABLE]

for $X,Y\in\Gamma(TM)$, where $\nabla^{\varphi}$ is the pullback connection. It is known that the second fundamental form is symmetric. For a Semi-Riemannian submersion $F$, one can easily obtain

[TABLE]

for $X,Y\in\Gamma((\ker F_{*})^{\bot})$. A smooth map $\varphi:M\to N$ is said to be harmonic if $\operatorname{trace}(\nabla\varphi_{*})=0.$ On the other hand, the tension field of $\varphi$ is the section $\tau(\varphi)$ of $\Gamma(\varphi^{-1}TN)$ defined by

[TABLE]

where $\{e_{1},\ldots,e_{m}\}$ is the orthonormal frame on $M$ and $\epsilon_{i}=g_{M}(e_{i},e_{i})$. Then it follows that $\varphi$ is harmonic if and only if $\tau(\varphi)=0$, for details, see Fuglede:Harmonicmorphismsbetween.

3 Anti-invariant semi-Riemannian submersions

In this section, we study a semi-Riemannian submersion from a Lorentzian almost (para)contact manifold $M(\phi,\xi,\eta,g_{M})$ to a semi-Riemannian manifold $(N,g_{N})$ and give necessary and sufficient conditions for the characteristic vector field to be vertical or horizontal.