Some results on D-homothetic deformation of (LCS)_2n+1-manifolds

Matsumoto et al.([1], [2]) introduced the idea of a Lorentzian para Sasakian manifold (briefly LP-Sasakian manifold) in 1988. Shaikh in 2003, gave the notion of a Lorentzian concircular structure manifolds (briefly LCS-manifold) [3], which is the generalization of an LP-Sasakian manifold. Since then, many geometers studied the properties of this manifold, for instance ([4], [5], [6], [7], [8]). The notion of local symmetry of a Riemannian manifold has been studied by many author in several ways to a different structures. As a weaker version of local symmetry, Takahashi [22] introduced the notion of a local ϕ-symmetry on a Sasakian manifold. Generalizing the notion of a local ϕ-symmetry of Takahashi [22], De et al. [10] introduced the idea of ϕ-recurrent for the Sasakian manifolds. Locally symmetric and ϕ-symmetric LP-Sasakian manifolds were studied by Shaikh and Baishya [21]. The properties of the locally ϕ-symmetric and the locally ϕ-recurrent (LCS)_n-manifolds were, respectively, studied in [4] and [5]. The notion of a generalized recurrent manifold has been introduced by Dubey et al. [12] and then studied by others. Again, the notion of a generalized Ricci-recurrent manifold has been introduced and studied by De et al. [11].

A Riemannian manifold (Mⁿ, g), (n > 2), is called a generalized recurrent manifold [12], if its non-vanishing curvature tensor R satisfies

∇R=A⊗R+B⊗G, $$\begin{array}{} \displaystyle \nabla{R}=A\otimes R+B \otimes G, \end{array}$$(1.1)

where A and B are non-vanishing 1-forms such that A(.) = g(., ρ₁), B(.) = g(., ρ₂) and the tensor G is defined by

G(X,Y)Z=g(Y,Z)X−g(X,Z)Y, $$\begin{array}{} \displaystyle G(X, Y)Z=g(Y, Z)X-g(X, Z)Y, \end{array}$$(1.2)

for X, Y, Z ∈ χ(M), where χ(M) is the collection of all smooth vector fields of M and ∇ denotes the operator of covariant differentiation with respect to the metric g. The 1-forms A and B are called the associated 1-forms of M.

A Riemannian manifold (Mⁿ, g), (n > 2), is said to be a generalized Ricci-recurrent manifold [11], if its Ricci tensor S of type (0, 2) satisfies

∇S=A⊗S+B⊗g, $$\begin{array}{} \displaystyle \nabla{S}=A\otimes S+ B \otimes g, \end{array}$$(1.3)

where A and B are non vanishing 1-forms defined as (1.1).

In 2007, Özgür [15] studied generalized recurrent Kenmotsu manifolds. Generalizing the notion of Özgür [15], Basari and Murathan [9] introduced the notion of the generalized ϕ-recurrent Kenmotsu manifolds. In addition, the properties of the generalized ϕ-recurrent Sasakian, LP-Sasakian, Lorentzian α-Sasakian, Kenmotsu manifolds, generalized Sasakian space-forms and (LCS)_2n+1-manifolds are, respectively, studied in [7], [16], [17], [19]. The properties of the extended generalized ϕ-recurrent β-Kenmotsu, Sasakian and (LCS)_2n+1-manifolds have been studied in [20], [18] and [7], respectively. As a continuation of above studies, we characterize the (LCS)_2n+1-manifolds under D-homothetic deformation. The outline of this paper is as follows:

After introduction in Section 1, we brief the known results of the (LCS)_2n+1-manifolds in Section 2. In Section 3, we prove our main results in the form of theorems and corollaries. It is proved that the structure tensor of the manifold commutes with the Ricci tensor under the D-homothetic deformation. This section also covers the properties of extended generalized ϕ-recurrent, ϕ-sectional curvature tensor, locally ϕ-Ricci symmetric, η-parallel Ricci tensor and extended generalized concircularly ϕ-recurrent (LCS)_2n+1-manifolds. In the last section, we give a non-trivial example of an extended generalized ϕ-recurrent (LCS)_2n+1-manifold under D-homothetic deformation and validate our results.

A Lorentzian manifold M of dimension (2n + 1) is a smooth connected para-contact Hausdorff manifold with the Lorentzian metric g, that is, M admits a smooth symmetric tensor field g of type (0, 2) such that for each point p ∈ M, the tensor g_p : T_pM × T_pM → ℜ is a non degenerate inner product of signature (−, +, …, +), where T_pM denotes the tangent space of M at p and ℜ are the real number space. A non-zero vector field V ∈ T_pM is said to be time like (respectively, non-space like, null, and space like) if it satisfies g_p(V, V) < 0 (respectively, ≤ 0, = 0, > 0) ([1], [2]).

Let M is a Lorentzian manifold admitting a unit time like concircular vector field ξ, which is called the generator of the manifold. Then we have

g(ξ,ξ)=−1. $$\begin{array}{} \displaystyle g(\xi, \xi)=-1. \end{array}$$(2.1)

Since ξ is a unit concircular vector field on M and therefore there exists a non-zero 1-form η such that

g(X,ξ)=η(X), $$\begin{array}{} \displaystyle g(X, \xi)=\eta(X), \end{array}$$(2.2)

which satisfies

(∇Xη)(Y)=α{g(X,Y)+η(X)η(Y)}(α≠0), $$\begin{array}{} \displaystyle (\nabla_X\eta)(Y)=\alpha\{g(X, Y)+\eta(X)\eta(Y)\} \,\, (\alpha \neq {0}), \end{array}$$(2.3)

for all the vector fields, X and Y, where α is the non-zero scalar function that satisfies

(∇Xα)=Xα=dα(X)=ρη(X). $$\begin{array}{} \displaystyle (\nabla_{X}\alpha)=X\alpha=d \alpha(X)=\rho \eta(X). \end{array}$$(2.4)

Here, ρ is the certain scalar function such that ρ = −(ξ α). If we put

ϕX=1α∇Xξ, $$\begin{array}{} \displaystyle \phi{X}=\frac{1}{\alpha}\nabla_{X}\xi, \end{array}$$(2.5)

then from (2.3) and (2.5), we have

ϕX=X+η(X)ξ, $$\begin{array}{} \displaystyle \phi{X}=X+\eta(X)\xi, \end{array}$$(2.6)

from which it follows that ϕ is a tensor field of type (1, 1), which is called the structure tensor of M. Thus M together with the unit timelike concircular vector field ξ, its associated 1-form η and (1, 1)-tensor field ϕ is said to be a Lorentzian concircular structure manifold (briefly (LCS)-manifold) [3]. Especially, if we take α = 1, then we can obtain the LP-Sasakian structure of Matsumoto [2]. Thus, we can say that the (LCS)-manifold is the generalization of the LP-Sasakian manifold. In the present paper, we consider the LCS-manifold of dimension (2n + 1). We have the following basic results of (LCS)_2n+1-manifold as:

η(ξ)=−1,ϕξ=0,ϕ2X=X+η(X)ξ,η(ϕX)=0andg(ϕX,ϕY)=g(X,Y)+η(X)η(Y), $$\begin{array}{} \displaystyle \eta(\xi)=-1, \quad \phi{\xi}=0,\quad \phi^{2}X=X+\eta(X)\xi,\quad \eta(\phi{X})=0 \\\displaystyle ~and~~~ g(\phi{X}, \phi{Y})=g(X, Y)+\eta(X)\eta(Y), \end{array}$$(2.7)

η(R(X,Y)Z)=(α2−ρ){η(X)g(Y,Z)−η(Y)g(X,Z)}, $$\begin{array}{} \displaystyle \eta(R(X, Y)Z)=(\alpha^2-\rho)\{\eta(X)g(Y, Z)-\eta(Y)g(X, Z)\}, \end{array}$$(2.8)

R(X,Y)ξ=(α2−ρ){η(Y)X−η(X)Y}, $$\begin{array}{} \displaystyle R(X, Y)\xi=(\alpha^2-\rho)\{\eta(Y)X-\eta(X)Y\}, \end{array}$$(2.9)

R(ξ,X)Y=(α2−ρ){g(X,Y)ξ−η(Y)X}, $$\begin{array}{} \displaystyle R(\xi, X)Y=(\alpha^2-\rho)\{g(X, Y)\xi-\eta(Y)X\}, \end{array}$$(2.10)

(∇Xϕ)(Y)=α{g(X,Y)ξ+2η(X)η(Y)ξ+η(Y)X}, $$\begin{array}{} \displaystyle (\nabla_{X}\phi)(Y)=\alpha\{g(X, Y)\xi+2\eta(X)\eta(Y)\xi+\eta(Y)X\}, \end{array}$$(2.11)

S(X,ξ)=2n(α2−ρ)η(X), $$\begin{array}{} \displaystyle S(X, \xi)=2n(\alpha^2-\rho)\eta(X), \end{array}$$(2.12)

S(ϕX,ϕY)=S(X,Y)+2n(α2−ρ)η(X)η(Y), $$\begin{array}{} \displaystyle S(\phi{X}, \phi{Y})=S(X, Y)+2n(\alpha^2-\rho)\eta(X)\eta(Y), \end{array}$$(2.13)

Xρ=dρ(X)=βη(X) $$\begin{array}{} \displaystyle X\rho=d \rho(X)=\beta \eta(X) \end{array}$$(2.14)

for all the vector fields X, Y, Z on M [3].

The notion of η-parallelism on a Sasakian manifold was introduced by Kon [13]. An (LCS)_2n+1-manifold is said to be η-parallel if its Ricci tensor S satisfies

(∇XS)(ϕY,ϕZ)=0, $$\begin{array}{} \displaystyle (\nabla_{X}S)(\phi{Y}, \phi{Z})=0, \end{array}$$(2.16)

for X, Y, Z ∈ χ(M²ⁿ⁺¹).

If M(ϕ, ξ, η, g) is an almost contact metric manifold of dimension (2n + 1) (i. e., dim M = m = 2n + 1), then the equation η = 0 defines an (m − 1)-dimensional distribution D on M [24], and if we change the structure tensors of an almost contact metric manifold by

η¯=aη,ξ¯=1aξ,ϕ¯=ϕ,g¯=ag+a(a−1)η⊗η, $$\begin{array}{} \displaystyle \bar{\eta}=a\eta, \quad \bar{\xi}=\frac{1}{a}\xi, \quad \bar{\phi}=\phi, \quad \bar{g}=ag+a(a-1)\eta \otimes \eta, \end{array}$$

where a is the non-zero positive constant. Then such transformation is known as the (m − 1)-homothetic deformation or D-homothetic deformation [23]. The study of D-homothetic deformation has been noticed in ([26], [27]). If M(ϕ, ξ, η, g) is an almost contact metric structure with contact form η, then M(ϕ̄, ξ̄, η̄, ḡ) is also an almost contact metric structure [23]. If we denote the difference Γ¯jki−Γjki $\begin{array}{} \displaystyle \bar{\Gamma}^{i}_{jk}-\Gamma^i_{jk} \end{array}$ of Christoffel symbols by Vjki $\begin{array}{} \displaystyle V^{i}_{jk} \end{array}$, then we have

V(X,Y)=(1−a){η(Y)ϕX+η(X)ϕY}+12(1−1a){(∇Xη)(Y)+(∇Yη)(X)}ξ $$\begin{array}{} \displaystyle V(X, Y)=(1-a)\{\eta(Y)\phi{X}+\eta(X)\phi{Y}\}+\frac{1}{2}(1-\frac{1}{a})\{(\nabla_{X}\eta)(Y)+(\nabla_{Y}\eta)(X)\}\xi \end{array}$$(2.17)

for X, Y ∈ χ(M) [23]. If R and R̄ denote, respectively, the curvature tensors of the manifolds M(ϕ, ξ, η, g) and M(ϕ̄, ξ̄, η̄, ḡ), then it is related to the expression

R¯(X,Y)Z=R(X,Y)Z+(∇XV)(Z,Y)−(∇YV)(Z,X)+V(V(Z,Y),X)−V(V(Z,X),Y), $$\begin{array}{} \displaystyle \bar{R}(X, Y)Z=R(X, Y)Z+(\nabla_{X}V)(Z, Y)-(\nabla_{Y}V)(Z, X) +V(V(Z, Y), X)-V(V(Z, X), Y), \end{array}$$(2.18)

for arbitrary vector fields X, Y, Z [23].

A plane section in the tangent space T_p(M) is called a ϕ-section if there exists a unit vector X in T_p(M) orthogonal to ξ such that {X, ϕX} is an orthonormal basis of the plane section. A sectional curvature of the form

K(X,ϕX)=g(R(X,ϕX)X,ϕX) $$\begin{array}{} \displaystyle K(X, \phi{X})=g(R(X, \phi{X})X, \phi{X}) \end{array}$$

is known as a ϕ-sectional curvature in T_p(M). A para contact metric manifold M(ϕ, ξ, η, g) is said to be of constant ϕ-sectional curvature if at each point of the manifold, the sectional curvature K(X, ϕX) is independent of the choice of non-zero vector X ∈ D_p, where D denotes the contact distribution of the para contact metric manifold defined by the equation η = 0.

In this section, we study the extended generalized ϕ-recurrent, ϕ-sectional curvature, locally ϕ-Ricci symmetric, η-parallel Ricci tensor and extended generalized concircularly ϕ-recurrent (LCS)_2n+1-manifolds under D-homothetic deformation. In consequence of (2.3) and (2.17), we get

V(X,Y)=(1−a){η(Y)ϕX+η(X)ϕY}+α(1−1a){g(X,Y)+η(X)η(Y)}ξ, $$\begin{array}{} \displaystyle V(X, Y)=(1-a)\{\eta(Y)\phi{X}+\eta(X)\phi{Y}\}+\alpha(1-\frac{1}{a})\{g(X, Y)+\eta(X)\eta(Y)\}\xi, \end{array}$$(3.1)

In view of (2.3), (2.4), (2.7) and (2.11), (3.1) it yields

(∇ZV)(X,Y)=α(1−a){g(Y,Z)ϕX+g(Z,X)ϕY+η(Y)η(Z)ϕX+η(X)η(Z)ϕY+g(X,Z)η(Y)ξ+g(Z,Y)η(X)ξ+2η(X)η(Y)Z+4η(X)η(Y)η(Z)ξ}+α2(1−1a)[{g(Z,X)η(Y)+η(Z)g(X,Y)+g(Z,Y)η(X)+3η(X)η(Y)η(Z)}ξ+g(ϕX,ϕY)Z]−(a−1a)(ξα){g(X,Y)+η(X)η(Y)}η(Z)ξ. $$\begin{array}{} \displaystyle (\nabla_{Z}V)(X, Y)=\alpha(1-a)\{g(Y, Z)\phi{X}+g(Z, X)\phi{Y}+\eta(Y)\eta(Z)\phi{X}\\\displaystyle\qquad\qquad\qquad\quad +\eta(X)\eta(Z)\phi{Y}+g(X, Z)\eta(Y)\xi+g(Z, Y)\eta(X)\xi \\\displaystyle\qquad\qquad\qquad\quad +2\eta(X)\eta(Y)Z+4\eta(X)\eta(Y)\eta(Z)\xi\}\\\displaystyle\qquad\qquad\qquad\quad +\alpha^2(1-\frac{1}{a})[\{g(Z, X)\eta(Y)+\eta(Z)g(X, Y)\\\displaystyle\qquad\qquad\qquad\quad +g(Z, Y)\eta(X)+3\eta(X)\eta(Y)\eta(Z)\}\xi+g(\phi{X}, \phi{Y})Z]\\\displaystyle\qquad\qquad\qquad\quad -(\frac{a-1}{a})(\xi \alpha)\{g(X, Y)+\eta(X)\eta(Y)\}\eta(Z)\xi. \end{array}$$(3.2)

Using (3.1) and (3.2) in (2.18) and then by virtue of (2.3) and (2.9), we obtain

R¯(X,Y)Z=R(X,Y)Z−α(1−a){g(Y,Z)ϕX+η(Y)η(Z)ϕX−g(X,Z)ϕY−η(X)η(Z)ϕY+g(Y,Z)η(X)ξ+2η(Z)η(X)Y−η(Y)g(Z,X)ξ−2η(Z)η(Y)X}+(a−1a)(ξα){η(Y)g(Z,X)−η(X)g(Z,Y)}ξ+(1−a)2{η(X)η(Z)ϕ2Y−η(Y)η(Z)ϕ2X}−α2(a−1a){g(Z,X)Y+η(Z)η(X)Y−g(Z,Y)X−η(Z)η(Y)X}−α(1−a)2a{g(Z,Y)ϕX+η(Z)η(Y)ϕX−g(Z,X)ϕY−η(Z)η(X)ϕY+[η(X)g(ϕZ,Y)−g(ϕZ,X)η(Y)]ξ}. $$\begin{array}{} \displaystyle \bar{R}(X, Y)Z=R(X, Y)Z-\alpha(1-a)\{g(Y, Z)\phi{X}+\eta(Y)\eta(Z)\phi{X}-g(X, Z)\phi{Y}\\\displaystyle\qquad\qquad\quad~\, -\eta(X)\eta(Z)\phi{Y}+g(Y, Z)\eta(X)\xi+2\eta(Z)\eta(X)Y-\eta(Y)g(Z, X)\xi\\\displaystyle\qquad\qquad\quad~\, -2\eta(Z)\eta(Y)X\}+(\frac{a-1}{a})(\xi \alpha)\{\eta(Y)g(Z, X)-\eta(X)g(Z, Y)\}\xi \\\displaystyle\qquad\qquad\quad~\, +(1-a)^2\{\eta(X)\eta(Z)\phi^2{Y}-\eta(Y)\eta(Z)\phi^2{X}\}\\\displaystyle\qquad\qquad\quad~\, -\alpha^2(\frac{a-1}{a})\{g(Z, X)Y +\eta(Z)\eta(X)Y-g(Z, Y)X-\eta(Z)\eta(Y)X\}\\\displaystyle\qquad\qquad\quad~\, -\alpha \frac{(1-a)^2}{a}\{g(Z, Y)\phi{X}+\eta(Z)\eta(Y)\phi{X} -g(Z, X)\phi{Y}\\\displaystyle\qquad\qquad\quad~\, -\eta(Z)\eta(X)\phi{Y}+[\eta(X)g(\phi{Z}, Y) -g(\phi{Z}, X)\eta(Y)]\xi\}. \end{array}$$(3.3)

Taking Y = Z = ξ in (3.3) and then use of (2.7) it gives

R¯(X,ξ)ξ=R(X,ξ)ξ+(1−a)(2α+a−1)ϕ2X. $$\begin{array}{} \displaystyle \bar{R}(X, \xi)\xi=R(X, \xi)\xi+(1-a)(2 \alpha +a-1) \phi^2{X}. \end{array}$$(3.4)

Let {e_i, ϕe_i, ξ}, i = 1, 2, …, n, be an orthonormal frame at any point of the tangent space T(M) of the manifold M. Then replacing Y = Z = e_i in (3.3), taking summation over i, 1 ≤ i ≤ n, and using η(e_i) = 0, we obtain

∑i=1nεiR¯(X,ei)ei=∑i=1nεiR(X,ei)ei−n(a−1a)(ξα)η(X)ξ+α(a−1)a[(a+1)n−1]η(X)ξ+α(a−1)(α+1)a(n−1)X, $$\begin{array}{} \displaystyle \sum_{i=1}^{n}\varepsilon_{i}\overline{R}(X, e_i)e_i=\sum_{i=1}^{n}\varepsilon_{i}{R}(X, e_i)e_i-n(\frac{a-1}{a})(\xi \alpha)\eta(X)\xi\\\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{\alpha(a-1)}{a}[(a+1)n-1]\eta(X)\xi +\frac{\alpha (a-1)(\alpha+1)}{a}(n-1)X, \end{array}$$(3.5)

where ε_i = g(e_i, e_i). Again, taking Y = Z = ϕ e_i in (3.3) and taking summation over i, 1 ≤ i ≤ n, and using the fact that η ∘ ϕ = 0, we get

∑i=1nεiR¯(X,ϕei)ϕei=∑i=1nεiR(X,ϕei)ϕei−n(a−1a)(ξα)η(X)ξ+α(a−1)a[(a+1)n−1]η(X)ξ+α(a−1)(α+1)a(n−1)X. $$\begin{array}{} \displaystyle \sum_{i=1}^{n}\varepsilon_{i}\overline{R}(X, \phi e_i)\phi e_i=\sum_{i=1}^{n}\varepsilon_{i}{R}(X, \phi e_i)\phi e_i-n(\frac{a-1}{a})(\xi \alpha)\eta(X)\xi\\\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{\alpha(a-1)}{a}[(a+1)n-1]\eta(X)\xi +\frac{\alpha (a-1)(\alpha+1)}{a}(n-1)X. \end{array}$$(3.6)

In view of (3.5) and (3.6), we have

Q¯X−R¯(X,ξ)ξ=QX−R(X,ξ)ξ−2n(a−1a)(ξα)η(X)ξ+2α(a−1)a[(a+1)n−1]η(X)ξ+2α(a−1)(α+1)a(n−1)X. $$\begin{array}{} \displaystyle \overline{Q}X-\overline{R}(X, \xi)\xi=QX-R(X, \xi)\xi-2n(\frac{a-1}{a})(\xi \alpha)\eta(X)\xi\\\displaystyle \,\,\,\,\,\,\,\,\,\,+\frac{2\alpha(a-1)}{a}[(a+1)n-1]\eta(X)\xi +\frac{2\alpha (a-1)(\alpha+1)}{a}(n-1)X. \end{array}$$(3.7)

In consequence of (2.7) and (3.4), (3.7) it yields

S¯(X,Y)=S(X,Y)+(a−1a){2α(α+1)(n−1)−a(2α+a−1)}g(X,Y)+(a−1a){2α(a+1)(n−1)−a(a−1)−2n(ξα)}η(X)η(Y), $$\begin{array}{} \displaystyle \overline{S}(X, Y)=S(X, Y)+(\frac{a-1}{a})\{2\alpha(\alpha+1)(n-1)-a(2\alpha+a-1)\}g(X, Y)\\\displaystyle \,\,\,\,\,\,\,\,\,\,+(\frac{a-1}{a})\{2\alpha(a+1)(n-1)-a(a-1)-2n (\xi \alpha)\}\eta(X)\eta(Y), \end{array}$$(3.8)

which implies that

Q¯X=QX+(a−1a){2α(α+1)(n−1)−a(2α+a−1)}X+(a−1a){2α(a+1)(n−1)−a(a−1)−2n(ξα)}η(X)ξ. $$\begin{array}{} \displaystyle \overline{Q} X =QX+(\frac{a-1}{a})\{2\alpha(\alpha+1)(n-1)-a(2\alpha+a-1)\}X\\\displaystyle \,\,\,\,\,\,\,\,\,\,+(\frac{a-1}{a})\{2\alpha(a+1)(n-1)-a(a-1)-2n (\xi \alpha)\}\eta(X)\xi. \end{array}$$(3.9)

Applying ϕ = ϕ on both sides of (3.9) and using (2.7), we have

ϕ¯Q¯X=ϕQX+(a−1a){2α(α+1)(n−1)−a(2α+a−1)}ϕX. $$\begin{array}{} \displaystyle {\overline{\phi}}{\hspace{0.10cm}} {\overline{Q}}X=\phi QX+(\frac{a-1}{a})\{2\alpha(\alpha+1)(n-1)-a(2\alpha+a-1)\}\phi{X}. \end{array}$$(3.10)

Again applying ϕX = ϕ X in (3.9) and using (2.7) give

Q¯ϕ¯X=QϕX+(a−1a){2α(α+1)(n−1)−a(2α+a−1)}ϕX. $$\begin{array}{} \displaystyle \overline{Q}{\hspace{0.10cm}}\overline{\phi}X=Q \phi X+(\frac{a-1}{a})\{2\alpha(\alpha+1)(n-1)-a(2\alpha+a-1)\}\phi{X}. \end{array}$$(3.11)

By virtue of (3.10) and (3.11), we obtain

{ϕ¯Q¯−Q¯ϕ¯}X={ϕQ−Qϕ}X. $$\begin{array}{} \displaystyle {\{\overline{\phi} {\hspace{0.10cm}}\overline{Q}-\overline{Q}{\hspace{0.10cm}} \overline{\phi}\}X}=\{\phi Q-Q \phi\}X. \end{array}$$(3.12)

The Ricci operator Q commutes with the structure tensor ϕ in an (LCS)_2n+1-manifold [6]. Thus we can state the following theorem as:

We consider a 3-dimensional manifold M = {(x, y, z) ∈ ℜ³ : z ≠ 0}, where (x, y, z) are the standard coordinates in ℜ³. Let {E₁, E₂, E₃} be linearly independent global frame on M given by

E1=ezx∂∂x+y∂∂y,E2=ez∂∂y,E3=e2z∂∂z. $$\begin{array}{} \displaystyle E_{1}=e^{z}\left( x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}\right), {\hspace{0.5cm}} E_{2}=e^{z}\frac{\partial}{\partial y},{\hspace{0.5cm}} E_{3}=e^{2z}\frac{\partial}{\partial z}. \end{array}$$

Let g be the Lorentzian metric defined by

g(E1,E3)=g(E2,E3)=g(E1,E2)=0,g(E1,E1)=g(E2,E2)=1,g(E3,E3)=−1. $$\begin{array}{} \displaystyle g(E_{1}, E_{3})=g(E_{2}, E_{3})=g(E_{1}, E_{2})=0,{\hspace{0.15cm}} g(E_{1}, E_{1})=g(E_{2}, E_{2})=1, {\hspace{0.15cm}}g(E_{3}, E_{3})=-1. \end{array}$$

Let η be the 1-form defined by η(V) = g(V, E₃) for any V ∈ χ(M). Let ϕ be the (1, 1)-tensor field defined by ϕE₁ = E₁, ϕE₂ = E₂, ϕE₃ = 0. Then using the linearity of ϕ and g, we have

η(E3)=−1,ϕ2V=V+η(V)E3,g(ϕV,ϕW)=g(V,W)+η(V)η(W), $$\begin{array}{} \displaystyle \eta(E_3)=-1,{\hspace{0.15cm}} \phi^{2}V=V+\eta(V)E_3, {\hspace{0.15cm}}g(\phi{V}, \phi{W})=g(V, W)+\eta(V)\eta(W), \end{array}$$

for any V, W ∈ χ(M). Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g and R be the curvature tensor of g. Then we have

[E1,E2]=−ezE2,[E1,E3]=−e2zE1,[E2,E3]=−e2zE2. $$\begin{array}{} \displaystyle [E_{1}, E_{2}]=-e^{z}E_{2},{\hspace{0.5cm}}[E_{1}, E_{3}]=-e^{2z}E_{1},{\hspace{0.5cm}}[E_{2}, E_{3}]=-e^{2z}E_{2}. \end{array}$$

For the Levi-Civita connection ∇ of the metric g, we have

2g(∇XY,Z)=Xg(Y,Z)+Yg(X,Z)−Zg(X,Y)−g(X,[Y,Z])−g(Y,[X,Z])+g(Z,[X,Y]). $$\begin{array}{} \displaystyle 2g(\nabla_{X}Y, Z)=Xg(Y, Z)+Yg(X, Z)-Zg(X, Y)-g(X, [Y, Z])-g(Y, [X, Z])+g(Z, [X, Y]). \end{array}$$

This expression is known as Koszul’s formula. Taking E₃ = ξ and using Koszula’s formula for the Lorentzian metric g, we can easily calculate the following:

∇E1E3=−e2zE1,∇E1E1=−e2zE3,∇E1E2=0,∇E2E3=−e2zE2,∇E3E2=0,∇E2E1=ezE2,∇E3E3=0,∇E2E2=−e2zE3−ezE1,∇E3E1=0. $$\begin{array}{} \displaystyle \nabla_{E_1}{E_{3}}=-e^{2z}E_1,{\hspace{0.5cm}}\nabla_{E_1}{E_{1}}=-e^{2z}E_3,{\hspace{0.5cm}}\nabla_{E_1}{E_{2}}=0,\\\displaystyle \nabla_{E_2}{E_{3}}=-e^{2z}E_2,{\hspace{0.5cm}}\nabla_{E_3}{E_{2}}=0,{\hspace{0.5cm}}\nabla_{E_2}{E_{1}}=e^{z}E_2,\\\displaystyle \nabla_{E_3}{E_{3}}=0,{\hspace{0.5cm}}\nabla_{E_2}{E_{2}}=-e^{2z}E_3-e^{z}E_1,{\hspace{0.5cm}}\nabla_{E_3}{E_{1}}=0. \end{array}$$