Semi-Riemannian Generalized Sasakian Space Forms

Abstract

In this paper, we extend the notion of generalized Sasakian space form to the semi-Riemannian setting. We consider several interesting cases and we give examples of them all. We also study their structures.

1 Introduction

The study of generalized space forms, i.e., Riemannian manifolds with constant pointwise sectional-like curvatures, has proved to be a very interesting topic in modern Differential Geometry.

Classically, sectional curvatures were considered. Given a Riemannian manifold (M, g), for any point $p\in M$ and any plane section $\Pi \subseteq T_pM$, the sectional curvature $K(\Pi )$ is defined by $K(\Pi )=g(R(X,Y)Y,X)(p)$, where X, Y are orthonormal vector fields in $\Pi$. It is well known that the curvature tensor of a Riemannian manifold with constant sectional curvature c is given by

$$\begin{aligned} R(X,Y)Z = c\{g(Y,Z)X-g(X,Z)Y\}. \end{aligned}$$

These spaces are called real space forms, and they can be modeled depending on the value of the constant c. Therefore, we find the Euclidean spaces ($c=0$), the spheres ($c>0$) and the hyperbolic spaces ($c<0$).

This situation was translated to the study of complex manifolds by considering holomorphic sectional curvatures, i.e., the curvature of sections spanned by a unit vector field X and JX, where J is the almost complex structure on the manifold. A Kaehlerian manifold with constant holomorphic sectional curvature c is said to be a complex space form, and its curvature tensor is given by

$$\begin{aligned} \begin{aligned} R(X,Y)Z =&\, \frac{c}{4}\{ g(Y,Z)X-g(X,Z)Y\\&+g(X,JZ)JY-g(Y,JZ)JX+2g(X,JY)JZ\}. \end{aligned} \end{aligned}$$

Now, the models are $\mathbf{C}^n$, $\mathbf{C}\mathbf{P}^n$, and $\mathbf{C}\mathbf{H}^n$, depending on $c=0$, $c>0$ or $c<0$. More generally, if the curvature tensor of an almost Hermitian manifold (M, J, g) satisfies

$$\begin{aligned} \begin{aligned} R(X,Y)Z =&\, F_1\{g(Y,Z)X-g(X,Z)Y\}\\&+F_2\{g(X,JZ)JY-g(Y,JZ)JX+2g(X,JY)JZ\}, \end{aligned} \end{aligned}$$

$F_1,F_2$ being differentiable functions on M, then M is said to be a generalized complex space form (see [17, 19]), and it is denoted by $M(F_1,F_2)$.

In the odd-dimensional case, Sasakian space forms play a similar role to that of complex space forms (see the preliminaries section for more details). For such a manifold, the curvature tensor is given by

$$\begin{aligned} R(X,Y)Z= & {} \frac{c+3}{4}\{g(Y,Z)X-g(X,Z)Y\}\nonumber \\&+\frac{c-1}{4}\{g(X,\phi Z)\phi Y-g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi Z\}\nonumber \\&+\frac{c-1}{4}\{\eta (X)\eta (Z)Y-\eta (Y)\eta (Z)X\nonumber \\&+g(X,Z)\eta (Y)\xi -g(Y ,Z)\eta (X)\xi \}. \end{aligned}$$

(1)

These spaces can also be modeled, depending on $c>-3$, $c=-3$ or $c<-3$.

In [1], we defined (jointly with David E. Blair) a generalized Sasakian space form as an almost contact metric manifold satisfying a similar equation, in which the constant quantities $(c+3)/4$ and $(c-1)/4$ are replaced by differentiable functions. Therefore, we can write

$$\begin{aligned} R=f_1R_1+f_2R_2+f_3R_3, \end{aligned}$$

where $R_1,R_2,R_3$ are the tensors appearing in (1). We used the notation $M(f_1,f_2,f_3)$. We will give more details about generalized Sasakian space forms (as, for example, a method to construct interesting examples in any dimension) in the preliminaries section.

Now, our objective is to define similar spaces in the semi-Riemannian setting. In the first section, we give some background on almost contact metric geometry and semi-Riemannian geometry. We also remind one result of [1], where we showed a method to construct generalized Sasakian space forms. Next, we present three sections devoted to the study of different semi-Riemannian manifolds with pointwise constant $\phi$-sectional curvature: $\varepsilon$-almost contact, $\varepsilon$-almost para-contact, and hyperbolic almost contact manifolds. All sections have the same structure: we present the ambient space, we construct an example of generalized space form, and we study its structure under certain conditions. Sometimes, we can also give a theorem with some restrictions over the functions for high-dimensional cases.

2 Preliminaries

In this section, we recall some definitions and basic formulas which we will use later. Anyway, we will also recall some appropriate definitions at the beginning of every section, concerning its content.

An odd-dimensional Riemannian manifold (M, g) is said to be an almost contact metric manifold if there exist on M a (1, 1) tensor field $\phi$, a vector field $\xi$ (called the structure vector field) and a 1-form $\eta$ such that $\eta (\xi )=1$, $\phi ^{2}(X)=-X+\eta (X)\xi$ and $g(\phi X,\phi Y)=g(X,Y)-\eta (X)\eta (Y)$, for any vector fields X, Y on M. In particular, in an almost contact metric manifold we also have $\phi \xi =0$ and $\eta \circ \phi =0$.

Such a manifold is said to be a contact metric manifold if $\mathrm{d}\eta =\Phi$, where $\Phi (X,Y)=g(X,\phi Y)$ is called the fundamental 2-form of M. On the other hand, the almost contact metric structure of M is said to be normal if $[\phi ,\phi ](X,Y)+2d\eta (X,Y)\xi =0$, for any X, Y, where $[\phi ,\phi ]$ denotes the Nijenhuis torsion of $\phi$, given by $[\phi ,\phi ](X,Y)=\phi ^{2}[X,Y]+[\phi X,\phi Y]-\phi [\phi X,Y]-\phi [X,\phi Y]$. A normal contact metric manifold is called a Sasakian manifold. It can be proved that an almost contact metric manifold is Sasakian if and only if

$$\begin{aligned} (\nabla _X\phi )Y \ = \ g(X,Y)\xi -\eta (Y)X, \end{aligned}$$

for any X, Y.

An almost contact metric manifold $(M,\phi ,\xi ,\eta ,g)$ is said to have an $(\alpha ,\beta )$ trans-Sasakian structure if

$$\begin{aligned} (\nabla _X\phi )Y \ = \ \alpha \{g(X,Y)\xi -\eta (Y)X\}+\beta \{g(\phi X,Y)\xi -\eta (Y)\phi X\}, \end{aligned}$$

where $\alpha ,\beta$ are differentiable functions on M. Particular cases of trans-Sasakian manifolds are Sasakian ($\alpha =1,\beta =0$), cosymplectic ($\alpha =\beta =0$), or Kenmotsu ($\alpha =0,\beta =1$) manifolds.

Given an almost contact metric manifold $(M,\phi ,\xi ,\eta ,g)$, a $\phi$-section of M at $p\in M$ is a section $\Pi \subseteq T_{p} M$ spanned by a unit vector $X_{p}$ orthogonal to $\xi _p$, and $\phi X_{p}$. The $\phi$-sectional curvature of $\Pi$ is defined by $K(\Pi )=R(X,\phi X,\phi X,X)$. A Sasakian manifold with constant $\phi$-sectional curvature c is called a Sasakian space form. In such a case, its Riemann curvature tensor is given by Eq. (1). For more background on almost contact metric manifolds, we recommend the reference [3].

In [1], we introduced (jointly with David E. Blair) generalized Sasakian space forms, as we already pointed out in Introduction. We presented different examples of such spaces. One of the most useful constructions was made by using warped products. Given an almost Hermitian manifold (N, J, G), the warped product $M=\mathbf{R}\times _{f}N$, where $f>0$ is a function on $\mathbf{R}$, can be endowed with an almost contact metric structure $(\phi ,\xi ,\eta , g_f)$. In fact,

$$\begin{aligned} g_{f}=\pi ^{*}(g_{\mathbf{R}})+(f\circ \pi )^{2}\sigma ^{*}(G) \end{aligned}$$

(2)

is the warped product metric (where $\pi$ and $\sigma$ are the projections from $\mathbf{R}\times N$ on $\mathbf{R}$ and N, respectively), $\phi (X)=(J\sigma _{*}X)^{*}$, for any vector field X on M, and $\xi =\partial /\partial t$, where t denotes the coordinate of $\mathbf{R}$. We proved:

Theorem 1

([1]) Let $N(F_{1},F_{2})$ be a generalized complex space form. Then, the warped product $M=\mathbf{R}\times _{f}N$, endowed with the almost contact metric structure $(\phi ,\xi ,\eta , g_f)$, is a generalized Sasakian space form $M(f_{1},f_{2},f_{3})$ with functions:

$$\begin{aligned} f_{1} = \frac{(F_{1}\circ \pi )-f'^{2}}{f^{2}}, \quad f_{2} = \displaystyle {\frac{F_{2}\circ \pi }{f^{2}}}, \quad f_{3}\ = \ \displaystyle {\frac{(F_{1}\circ \pi )-f'^{2}}{f^{2}}}+\displaystyle {\frac{f''}{f}}. \end{aligned}$$

Also in [1] we proved that such a warped product is a $(0,\beta )$ trans-Sasakian manifold, with $\beta =f'/f$, if and only if N is a Kaehlerian manifold. Moreover, we also had:

Theorem 2

([1]) Let $M(f_1,f_2,f_3)$ be a $\beta$-Kenmotsu generalized Sasakian space form, with dimension greater than or equal to 5. Then, $f_1,f_2,f_3$ depend only on the direction of $\xi$ and the following equations hold

$$\begin{aligned} \xi (f_1)+2\beta f_3= & {} 0, \end{aligned}$$

(3)

$$\begin{aligned} \xi (f_2)+2\beta f_2= & {} 0. \end{aligned}$$

(4)

We gave some more results about the structure of generalized Sasakian space forms in [2].

Now, we are interested in studying odd-dimensional generalized space forms, that is, manifolds with pointwise $\phi$-sectional curvature. We will consider them not in a Riemannian setting, but in a Lorentzian or indefinite one. We will construct different examples using warped products, but we need to start from other semi-Riemannian manifolds.

Let us remember that a metric $(J^4=1)$-manifold, [6], is a semi-Riemannian manifold $(M^n,g)$ together with a (1, 1) tensor field J such that $J^4=1$. The metric g and the structure tensor J are related by one of the following conditions:

• (i) g is semi-Riemannian and $g(JX,Y)+g(X,JY)=0$. In this case, it is said that g is an adapted in the electromagnetic sense metric (aem).

• (ii) g is Riemannian and $g(JX,JY)=g(X,Y)$. In this case it is said that g is an adapted Riemannian metric (arm).

We focus our interest in some special cases. Firstly, if $J^2=-1$, there is no distinction between aem and arm; in such a case the manifold is called an almost Hermitian manifold. Secondly, if $J^2=1$, we distinguish between an almost para-Hermitian manifold, if it has an aem, and a Riemannian almost product manifold, if it has an arm.

Finally, let us point out that all the manifolds we will refer to during this paper will be simply connected and all the functions will be differentiable functions on the corresponding manifolds.

3 Generalized Indefinite Sasakian Space Forms

Our purpose in this section is to generalize the results given in [1] for semi-Riemannian manifolds with an almost contact structure and index greater than or equal to 1.

A $\varepsilon$ -almost contact metric manifold [5] or almost contact pseudo-metric manifold [4] is an odd-dimensional semi-Riemannian manifold $(M^{2n+1}_{2s,2s+1},g)$, with a structure $(\phi ,\xi ,\eta ,g)$ satisfying

$$\begin{aligned} \begin{array}{ll} \phi ^2 X=-X+\eta (X)\xi , &{}\quad g(\phi X,\phi Y)=g(X,Y)-\varepsilon \eta (X)\eta (Y),\\ \eta (\xi )=1, &{}\quad \eta (X)=\varepsilon g(X,\xi ), \end{array} \end{aligned}$$

for any vectors fields X, Y on M, where $\varepsilon =\pm 1$. It follows from the above conditions that $g(\xi ,\xi )=\varepsilon$, i.e., $\varepsilon$ indicates the causal character of $\xi$: $\varepsilon =1$ (resp. $\varepsilon =-1$) if $\xi$ is a spacelike (resp. timelike) vector field. Given that X and $\phi X$ have the same causal character, we get $M^{2n+1}_{2s}$ and $M^{2n+1}_{2s+1}$ for $\varepsilon =1$ and $\varepsilon =-1$, respectively. We will use the name indefinite almost contact metric manifold. For index $s=0$ and $\varepsilon =1$ we obtain almost contact metric manifolds and for $s=0$ and $\varepsilon =-1$, almost contact Lorentzian manifolds. Indefinite contact metric and Sasakian manifolds are defined similarly to the Riemannian case.

The expression of the curvature tensor of an indefinite Sasakian manifold with constant $\phi$-sectional curvature has been given in [8]:

$$\begin{aligned} R(X,Y)Z= & {} \frac{c+3\varepsilon }{4}\{g(Y,Z)X-g(X,Z)Y\}\\&+\frac{c-\varepsilon }{4}\{g(X,\phi Z)\phi Y-g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi Z\}\\&+\frac{c-\varepsilon }{4}\{\eta (X)\eta (Z)Y-\eta (Y)\eta (Z)X+\varepsilon g(X,Z)\eta (Y)\xi -\varepsilon g(Y ,Z)\eta (X)\xi \}. \end{aligned}$$

Later, Lee [11] considers generalized indefinite Sasakian space forms whose curvature tensors take this appearance with three functions instead of constants, although he gave no examples. For them, we will write $R=f_1R_1+f_2R_2+f_3R_{3,\varepsilon }$. Notice that $R_{3,1}=R_3$ and $R_{3,-1}$ coincides with the expression given by Ikawa [9], for the curvature tensor of a contact Lorentzian manifold with constant $\phi$-sectional curvature c.

We are now constructing an example of such a space. Given an indefinite almost complex manifold $(N^{2n}_{2s},J,G)$ and a positive smooth function f on $\mathbf{R}$, we consider the warped product $M=\mathbf{R}\times _f N$, with metric given by

$$\begin{aligned} g_{f}=\varepsilon \pi ^*(g_{\mathbf{R}})+(f\circ \pi )^2\sigma ^*(G), \end{aligned}$$

(5)

where $f>0$ is a function on $\mathbf{R}$ and $\pi$ and $\sigma$ are the projections from $\mathbf{R}\times N$ on $\mathbf{R}$ and N, respectively. Then

$$\begin{aligned} \phi (X)=(J\sigma _*X)^*,\quad \xi =\displaystyle {\frac{\partial }{\partial t}},\quad \eta (X)=\varepsilon g_{f}(X,\xi ) \end{aligned}$$

is an indefinite almost contact structure on the warped product M. Notice that this metric is the one used to construct the Robertson–Walker spaces (see [14]).

We need the following two lemmas from [14] to compute the curvature tensor of M:

Lemma 3

Let us consider $M=B\times _{f} F$ and denote by $\nabla$, $\nabla ^{B}$, and $\nabla ^{F}$ the Riemannian connections on M, B, and F. If X, Y are vector fields on B and V, W are vector fields on F, then

• (1) $\nabla _{X}Y$ is the lift of $\nabla ^{B}_{X}Y$.

• (2) $\nabla _{X}V=\nabla _{V}X=(Xf/f)V.$

• (3) The component of $\nabla _{V}W$ normal to the fibers is $-(g_{f}(V,W)/f) \text{ grad } f$

• (4) The component of $\nabla _{V}W$ tangent to the fibers is the lift of $\nabla ^{F} _{V}W$.

Lemma 4

Let $M=B\times _{f}F$ be a warped product, with Riemann curvature tensor R. Given fields X, Y, Z on B and U, V, W on F, then

• (1) R(X, Y)Z is the lift of $R^{B}(X,Y)Z$.

• (2) $R(V,X)Y=-(H^{f}(X,Y)/f)V$, where $H^{f}$ is the Hessian of f.

• (3) $R(X,Y)V=R(V,W)X=0.$

• (4) $R(X,V)W=-(g_{f}(V,W)/f)\nabla _{X}(\text{ grad } f).$

• (5) $R(V,W)U=R^{F}(V,W)U+(g_{f}$(grad f, grad f)$/f^{2})\{g_{f}(V,U)W -g_{f} (W,U) V\}.$

We obtain the following result:

Theorem 5

Let $N^{2n}_{2s}(F_{1},F_{2})$ be an indefinite generalized complex space form. Then the warped product $M^{2n+1}=\mathbf{R}\times _f N$ endowed with the indefinite almost contact structure $(\phi ,\xi ,\eta ,g_f)$ is a generalized indefinite Sasakian space form, $M(f_1,f_2,f_3)$, with functions:

$$\begin{aligned} f_1=\frac{(F_1\circ \pi )-\varepsilon f'^2}{f^2},\quad f_2=\frac{(F_2\circ \pi )}{f^2},\quad f_3=\varepsilon \frac{(F_1\circ \pi )-\varepsilon f'^2}{f^2}+\frac{f''}{f}. \end{aligned}$$

(6)

Proof

For any vector fields X, Y, Z on M, we can write $X=\eta (X)\xi +U$, $Y=\eta (Y)\xi +V$, $Z=\eta (Z)\xi +W$, where U, V, W are vector fields on N. Then, by virtue of Lemma 4, and by taking into account that $\mathbf{R}$ is flat, we have

$$\begin{aligned} \begin{aligned} R(X,Y)Z =&\,\eta (X)\eta (Z)\frac{H^{f}(\xi ,\xi )}{f}V-\eta (Y)\eta (Z)\frac{H^{f}(\xi ,\xi )}{f}U\\&-\frac{g_{f}(V,W)}{f}\eta (X)\nabla _{\xi }\mathrm{grad}(f)+\frac{g_{f}(U,W)}{f}\eta (Y)\nabla _{\xi }\mathrm{grad}(f)\\&+R^{N}(U,V)W+\frac{g_{f}(\mathrm{grad}(f),\mathrm{grad}(f))}{f^2}\{g_{f}(U,W)V-g_{f}(V,W)U\}. \end{aligned} \end{aligned}$$

(7)

Let us first notice that

$$\begin{aligned} \text{ grad }(f)=\sum _{i,j}g_f^{i,j}\displaystyle {\frac{\partial f}{\partial x_i}}\partial _j=\varepsilon f'\xi , \end{aligned}$$

since $f=f(t)$. Therefore,

$$\begin{aligned} \nabla _{\xi }\mathrm{grad}(f) = \varepsilon f''\xi , \end{aligned}$$

(8)

since, from Lemma 4, we know that $\nabla _{\xi }\xi =0$. Moreover,

$$\begin{aligned}&H^f(\xi ,\xi ) = g_f(\nabla _{\xi }\mathrm{grad}(f),\xi ) = g_f(\varepsilon f''\xi ,\xi )= f'', \end{aligned}$$

(9)

$$\begin{aligned}&g_{f}(\mathrm{grad}(f),\mathrm{grad}(f)) = g_f(f'\xi , f'\xi ) = \varepsilon f'^2. \end{aligned}$$

(10)

Now, from (5) and (7)–(10), and by using that N is a generalized complex space form, we have

$$\begin{aligned} \begin{aligned} R(X ,Y)Z\,=\,&\, \frac{f''}{f}\{\eta (X)\eta (Z)V-\eta (Y)\eta (Z)U\\&+\varepsilon f^2g(U,W)\eta (Y)\xi -\varepsilon f^2 g(V,W)\eta (X)\xi \}\\&+(F_{1}\circ \pi )\{g(V,W)U-g(U,W)V\}\\&+(F_{2}\circ \pi )\{ g(U,JW)JV-g(V,JW)JU+2g(U,JV)JW\}\\&+\varepsilon \left( \displaystyle {\frac{f'}{f}}\right) ^{2}\{f^{2}g(U,W)V-f^{2}g(V,W)U\}. \end{aligned} \end{aligned}$$

(11)

Then, by taking into account (5) and the relationship between X, Y, Z and U, V, W,

$$\begin{aligned} \begin{aligned} R(X,Y)Z =&\displaystyle {\frac{(F_{1}\circ \pi ) -\varepsilon f'^2}{f^2}}\{g_f(Y,Z)X-g_f(X,Z)Y\}\\&+\displaystyle {\frac{F_2\circ \pi }{f^2}}\{g_f(X,\phi Z)\phi Y-g_f(Y,\phi Z)\phi X+2g_f(X,\phi Y)\phi Z\}\\&+\left( \varepsilon \displaystyle {\frac{(F_1\circ \pi )-\varepsilon f'^2}{f^2}}+\displaystyle {\frac{f''}{f}}\right) \{\eta (X)\eta (Z)Y-\eta (Y)\eta (Z)X\\&\qquad -g_f(X,Z)\eta (Y)\xi +g_f(Y,Z)\eta (X)\xi \}, \end{aligned} \end{aligned}$$

which finishes the proof. $\square$

Remark 6

In particular, if N(c) is a complex space form, the functions $f_1,f_2,f_3$ of (6) are given by

$$\begin{aligned} f_{1}=\displaystyle {\frac{c+4f'^{2}}{4f^{2}}},\quad f_{2}=\displaystyle {\frac{c}{f^{2}}},\quad f_{3}=\displaystyle {\frac{c+4f'^{2}}{4f^{2}}}-\displaystyle {\frac{f''}{f}}. \end{aligned}$$

Therefore, we obtain examples with non-constant functions in any dimension.

Now, as we did with generalized Sasakian space forms in [1], we must study the structure of this warped product.

Let us remember that a semi-Riemannian almost contact metric manifold is called an $(\varepsilon ,\delta )$-trans-Sasakian manifold [13] if

$$\begin{aligned} (\nabla _X \phi )Y=\alpha \{g(X,Y)\xi -\varepsilon \eta (Y)X\}+\beta \{g(\phi X,Y)\xi -\delta \eta (Y)\phi X\}, \end{aligned}$$

(12)

for some functions $\alpha$ and $\beta$, where $\varepsilon =g(\xi ,\xi )=\pm 1$ and $\delta =\pm 1$. If $\varepsilon =\delta$, then the manifold is simply called $(\varepsilon )$-trans-Sasakian, [16]. They include both $(\varepsilon )$-Sasakian and $(\varepsilon )$-Kenmotsu manifolds.

Proposition 7

Let N be an indefinite almost Hermitian manifold. Then, $(\mathbf{R}\times _f N,\phi ,\xi ,\eta ,g_{f})$ is an $(\varepsilon )$-trans-Sasakian manifold with $\alpha =0$ and $\beta =\varepsilon f'/f$, if and only if N is an indefinite Kaehlerian manifold.

Proof

Let $X=\eta (X)\xi +U$ and $Y=\eta (Y)\xi +V$ be, with U, V vector fields on N. Then, taking into account Lemma 3,

$$\begin{aligned} (\nabla _X\phi )Y= & {} \nabla _{\eta (X)\xi }JV+\nabla _UJV- \phi (\nabla _{\eta (X)\xi }\eta (Y)\xi +\nabla _U\eta (Y)\xi +\nabla _{\eta (X)\xi }U+\nabla _UV)\\= & {} \frac{\eta (X)\xi (f)}{f}JV-\frac{g_f(U,JV)}{f}\mathrm{grad}(f)+\nabla _{U}^{N}JV\\&-\frac{\eta (Y)\xi (f)}{f}JU-\frac{X(f)}{f}JV-J\nabla _UV\\= & {} \varepsilon \displaystyle {\frac{f'}{f}}\{g_f(\phi X, Y)\xi +\eta (Y)\phi X\}+(\nabla _{U}^{N}J)V.\\ \end{aligned}$$

The result holds if and only if $(\nabla _{U}^{N}J)V=0$, that is, N is Kaehler. $\square$

Therefore, for the case $s=0$, Proposition 7 shows that if N is a Kaehler manifold, then $M=\mathbf{R}\times _f N$ is a $(\varepsilon )$-trans-Sasakian manifold with $\varepsilon =-1$, $\alpha =0$ and $\beta =-f'/f$.

But, just like in the generalized Sasakian space forms case, there are some constrains over the functions if the Sasakian Lorentzian space form is endowed with such a structure:

Theorem 8

Let $M(f_1,f_2,f_3)$ be a generalized Sasakian Lorentzian space form with an $(\varepsilon )$-trans-Sasakian structure and dimension greater than or equal to 5. Then, $f_1,f_2,f_3$ depend only on the direction of $\xi$ and the following equations hold

$$\begin{aligned}&\xi (f_1)-2f_3\beta =0, \end{aligned}$$

(13)

$$\begin{aligned}&\xi (f_2)-2f_2\beta =0. \end{aligned}$$

(14)

Proof

We consider Bianchi’s second identity,

where

represents the cyclic sum on W, X, Y. For a generalized Sasakian Lorentzian space form it looks like

(15)

since $(\nabla _{W}R_1)(X,Y)Z=0$.

If we first take $W=\phi X$ and $Z=\phi Y$ in (15), multiplying by X we get

$$\begin{aligned} X(f_2)+3f_2g(X,(\nabla _Y \phi )\phi Y)=0, \end{aligned}$$

(16)

and multiplying by $\phi X$:

$$\begin{aligned} -X(f_1)+3f_2g((\nabla _Y\phi )Y,\phi X)=0. \end{aligned}$$

(17)

For $W=\xi$ and multiplying by X, taking $Z=\xi$ and $Z=Y$ we get, respectively,

$$\begin{aligned}&-f_3g(Y,\nabla _\xi \xi )+Y(f_1)-Y(f_3)=0, \end{aligned}$$

(18)

$$\begin{aligned}&\xi (f_1)+f_3\{g(\nabla _X\xi ,X)+g(Y,\nabla _Y\xi )\}=0. \end{aligned}$$

(19)

Finally, multiplying by $\phi X$, and taking $Z=\phi Y$ we get

$$\begin{aligned} \xi (f_2)+f_2\{-g((\nabla _X \phi )\xi ,\phi X)+g(\xi ,(\nabla _Y\phi )\phi Y\}. \end{aligned}$$

(20)

Since M is trans-Sasakian with $\delta =-1$, the three first equations imply $X(f_i)=0$ for $i=1,2,3$ and every X orthogonal to $\xi$, that is, $f_i$ only depends on $\xi$. And Eqs. (19) and (20) turn into (13) and (14). $\square$

4 Generalized Indefinite Para-Sasakian Space Forms

In the recent paper [18], $\varepsilon$-almost para-contact metric manifolds have been introduced as those semi-Riemannian manifolds $(M^{2n+1}_{2s,2s+1},g)$, endowed with an almost para-contact structure $(\phi ,\xi ,\eta )$ satisfying

$$\begin{aligned} \begin{array}{ll} \phi ^2 X=X+\eta (X)\xi ,&{} \quad g(\phi X,\phi Y)=g(X,Y)-\varepsilon \eta (X)\eta (Y),\\ \eta (\xi )=-1,&{} \quad \eta (X)=-\varepsilon g(X,\xi ), \end{array} \end{aligned}$$

for any vector fields X, Y on M, where $\varepsilon =\pm 1$. It follows from the above conditions that $g(\xi ,\xi )=\varepsilon$, i.e., $\varepsilon$ indicates the causal character of $\xi$: $\varepsilon =1$ (resp. $\varepsilon =-1$) if $\xi$ is a spacelike (resp. timelike) vector field. For index 0 and 1 we have both almost para-contact and Lorentzian almost para-contact manifolds, [12].

Moreover, if

$$\begin{aligned} (\nabla _X\phi )Y=g(X,Y)\xi -\varepsilon \eta (Y)X, \end{aligned}$$

(21)

then it is said that M has an $(\varepsilon )$-para-Sasakian structure.

Remark 9

In the original definition given in [18], $\phi ^2=I-\eta \otimes \xi$ and $\eta (\xi )=1$, because those authors use Sato’s definition for an almost para-contact structure (see [15]). Nevertheless, we are changing $\xi$ by $-\xi$, because we follow Matsumoto’s definition (see [12]).

Given an indefinite almost product manifold $(N^{2n}_{2s},J,G)$ and a positive smooth function f on $\mathbf{R}$, we consider two different warped products $M=\mathbf{R}\times _f N$ depending on the desired causal character for $\xi$, with metric given by

$$\begin{aligned} g_{f}=\varepsilon \pi ^*(g_\mathbf{R})+(f\circ \pi )^2\sigma ^*(G). \end{aligned}$$

(22)

Then,

$$\begin{aligned} \phi (X)=(J_{\sigma ^*}X)^*,\quad \xi =\displaystyle {\frac{\partial }{\partial t}},\quad \eta (X)=-\varepsilon g_{f}(X,\xi ) \end{aligned}$$

is an indefinite almost para-contact structure on M. We get $M_{2s}^{2n+1}$ and $M_{2s+1}^{2n+1}$ for $\varepsilon =1$ and $\varepsilon =-1$, respectively. Under certain assumptions, we also find a special writing for the curvature tensor of these warped products:

Theorem 10

Let $N^{2n}_{2s}$ be an indefinite almost product manifold with

$$\begin{aligned} \begin{aligned} R^N(U,V)W&=F_1\{G(V,W)U-G(U,W)V\}\\&\quad \quad +F_2\{G(U,JW)JV-G(V,JW)JU\}, \end{aligned} \end{aligned}$$

where $F_1,F_2$ are differentiable functions on N. Then the curvature tensor of the warped product $M^{2n+1}=\mathbf{R}\times _f N$ can be written as

$$\begin{aligned} R=f_1R_1+f_2\widetilde{R}_2+f_3 \widetilde{R}_{3,\varepsilon }, \end{aligned}$$

(23)

where the functions $f_1,f_2,f_3$ are given by

$$\begin{aligned}&\displaystyle f_1=\frac{(F_1\circ \pi )-\varepsilon f'^2}{f^2},\quad f_2=\frac{(F_2\circ \pi )}{f^2},\quad f_3=-\frac{(F_1\circ \pi )-\varepsilon f'^2}{f^2}-\varepsilon \frac{f''}{f},\\&\displaystyle \widetilde{R}_2(X,Y)Z=g_f(X,\phi Z)\phi Y-g_f(Y,\phi Z)\phi X \end{aligned}$$

and

$$\begin{aligned} \widetilde{R}_{3,\varepsilon }(X,Y)Z\!=\!-\varepsilon \eta (X)\eta (Z)Y+\varepsilon \eta (Y)\eta (Z)X+g_f(X,Z)\eta (Y)\xi -g_f(Y ,Z)\eta (X)\xi . \end{aligned}$$

Proof

The proof is quite similar to the one of Theorem 5. Again, for any vector fields on M, we write $X=-\eta (X)\xi +U$, $Y=-\eta (Y)\xi +V$ and $Z=-\eta (Z)\xi +W$. Proceeding in a similar way we arrive to the given expression of the Riemannian curvature tensor. $\square$

An indefinite almost para-contact metric manifold with curvature tensor given by (23) can be called a generalized para-Sasakian space form.

Again, if the initial almost product structure is parallel, the warped product has also a special structure.

Proposition 11

Let N be an almost product manifold. Then, $(\mathbf{R}\times _f N,\phi ,\xi ,\eta , g_{f})$ satisfies

$$\begin{aligned} (\nabla _X\phi )Y=\beta \{g_f(\phi X,Y)\xi -\varepsilon \eta (Y)\phi X\}, \end{aligned}$$

with $\beta =-\varepsilon f'/f$, if and only if N satisfies $\nabla ^N J=0$.

Proof

It is similar to that of Proposition 7. $\square$

This fact aims us to define an $(\varepsilon ,\delta )$-trans-para-Sasakian manifold as an indefinite almost para-contact metric manifold satisfying

$$\begin{aligned} (\nabla _X\phi )Y=\alpha \{g(X,Y)\xi -\varepsilon \eta (Y)X\} +\beta \{g(\phi X,Y)\xi -\delta \eta (Y)\phi X\}, \end{aligned}$$

(24)

for some functions $\alpha ,\beta$.

In this case, we cannot give a theorem for dimensions over five just studying Bianchi’s second identity, because, in general, X and $\phi X$ are not orthogonal.

5 Generalized Hyperbolic Almost Contact Space Forms

In this last section, we study odd-dimensional manifolds with an indefinite metric, (M, g). Such a manifold is said to be an hyperbolic almost contact metric manifold if there exits on M a (1, 1) tensor field $\phi$, a vector field $\xi$ and a 1-form $\eta$ such that

$$\begin{aligned} \begin{array}{ll} \phi ^2 X=X+\eta (X)\xi ,&{} \quad g(\phi X,\phi Y)=-g(X,Y)-\eta (X)\eta (Y),\\ \eta (\xi )=-1,&{} \quad \eta (X)=g(X,\xi ), \end{array} \end{aligned}$$

for any vector fields X, Y on M.

Given an almost para-Hermitian manifold (N, J, G), if we consider a warped product $M=\mathbf{R}\times _{f} N$, $f>0$, in the Robertson-Walker way, with

$$\begin{aligned} g_{f}=-\pi ^{*}(g_{\mathbf{R}})+(f\circ \pi )^{2}\sigma ^{*}(G), \end{aligned}$$

and we define

$$\begin{aligned} \phi (X)=(J\sigma _{*}X)^{*}, \qquad \xi =\frac{\partial }{\partial t},\qquad \eta (X)=g_{f}(X,\xi ), \end{aligned}$$

then we obtain an hyperbolic almost contact metric manifold $(M,\phi ,\xi ,\eta ,g_f)$.

In [6], it is proved that the curvature tensor of a para-Hermitian manifold with constant para-holomorphic sectional curvature is given by

$$\begin{aligned} R^N(U,V)W= & {} \, \frac{c}{4}\left\{ G(V,W)U-G(U,W)V\right\} \nonumber \\&-\displaystyle {\frac{c}{4}}\left\{ G(U,JW)JV-G(V,JW)JU+2G(U,JV)JW\right\} .\qquad \quad \end{aligned}$$

(25)

These spaces have also been studied in [10] with pointwise constant holomorphic curvature.

We can now state the following result:

Theorem 12

Let $N^{2n}(F_1,F_2)$ be a para-Hermitian manifold with pointwise constant J-sectional curvature. Then the curvature tensor of the warped product $M ^{2n+1}=\mathbf{R}\times _{f} N$ can be written as

$$\begin{aligned} R={f_{1}}R_1+{f_{2}}R_2+{f_{3}}R_3, \end{aligned}$$

where the functions $f_1,f_2,f_3$ are given by

$$\begin{aligned} f_1=\displaystyle {\frac{F_1+f'^{2}}{f^{2}}},\quad f_2=-\displaystyle {\frac{F_2}{f^{2}}},\quad f_3=-\displaystyle {\frac{F_1+f'^{2}}{f^{2}}}+\displaystyle {\frac{f''}{f}}. \end{aligned}$$

Proof

The proof is similar to those of Theorems 5 and 10, but now for N being a para-Hermitian manifold with pointwise constant J-sectional curvature. In this case, the curvature tensor of such a manifold is given by (25), which explains the change of signs. $\square$

Let us notice that here the tensors $R_i$, $i=1,2,3$, are the same as in (1). Therefore, we find an example of a generalized hyperbolic Sasakian space form which is related to a para-Hermitian space form in the same way that a generalized Sasakian one is related to a complex space form.

We can also study the structure of the above warped product. Trans-hyperbolic Sasakian manifolds were introduced in [7] as those manifolds satisfying

$$\begin{aligned} (\nabla _X\phi )Y=\alpha \left\{ g(X,Y)\xi -\eta (Y)X\right\} +\beta \left\{ g(\phi X,Y)\xi -\eta (Y)\phi X\right\} \end{aligned}$$

for any vector fields X, Y.

Proposition 13

Let N be a para-Hermitian manifold. Then, $(\mathbf{R}\times _f N,\phi ,\xi ,\eta ,g_f)$ is a trans-hyperbolic Sasakian manifold with $\alpha =0$ and $\beta ={-f'}/{f}$, if and only if N is a para-Kaehlerian manifold.

Proof

Let $X=U-\eta (X)\xi$ and $Y=V-\eta (Y)\xi$ be. Taking into account Lemma 3, it is just a direct computation that

$$\begin{aligned} (\nabla _X\phi )Y=-\displaystyle {\frac{f'}{f}}\{g_f(\phi X,Y)\xi -\eta (Y)\phi X\}+(\nabla _{U}^{N}J)V. \end{aligned}$$

Therefore, the statement is true if and only if $(\nabla _{U}^{N}J)V=0$, that is, N is para-Kaehler. $\square$

Let us remember that para-Kaehler manifolds are also called hyperbolic Kaehler manifolds (see [6]).

Theorem 14

Let $M(f_1,f_2,f_3)$ be a generalized hyperbolic Sasakian space form with a $(0,\beta )$ trans-hyperbolic Sasakian structure and dimension greater than or equal to 5. Then, $f_1,f_2,f_3$ depend only on the direction of $\xi$ and the following equations hold

$$\begin{aligned} \xi (f_1)+2f_3\beta= & {} 0, \end{aligned}$$

(26)

$$\begin{aligned} \xi (f_2)-2f_2\beta= & {} 0. \end{aligned}$$

(27)

Proof

It is similar to the proof of Theorem 8, but considering an orthonormal frame $\{X,Y,\phi X,\phi Y,\xi \}$, where X, Y are spacelike and $\phi X, \phi Y,\xi$ are timelike vector fields. $\square$