On Warped Product Super Generalized Recurrent Manifolds

arXiv:1504.03271v1 [math.DG] 13 Apr 2015

ON WARPED PRODUCT SUPER GENERALIZED RECURRENT MANIFOLDS 1

ABSOS ALI SHAIKH, 1 HARADHAN KUNDU AND 2 MD. SHOWKAT ALI

Abstract. The object of the present paper is to obtain the characterization of a warped product semi-Riemannian manifold with a special type of recurrent like structure, called super generalized recurrent. As consequence of this result we also find out the necessary and sufficient conditions for a warped product manifold to satisfy some other recurrent like structures such as weakly generalized recurrent manifold, hyper generalized recurrent manifold etc. Finally as a support of the main result, we present an example of warped product super generalized recurrent manifold.

1. Introduction To generalize the notion of a manifold of constant curvature Cartan [4] first introduced the notion of local symmetry which can be presented as the curvature restriction ∇R = 0 (i.e., the Riemann-Christoffel curvature tensor R is covariantly constant). But there are many manifolds which does not bear local symmetry and hence to investigate the type of symmetry of such manifolds it is necessary to generalize the notion of local symmetry. During the last eight decades various researchers are working on this area to generalize or extend the notion of local symmetry by weakening its curvature restriction in different directions. Cartan [5] himself first gave a proper generalization of local symmetry and introduced the notion of semisymmetry, which was later classified by Szabó [35]. Then in 1983 Adamów and Deszcz [1] generalized the notion of semisymmetry and introduced the notion of pseudosymmetry, also known as Deszcz pseudosymmetry (see [20]). On the other hand as a direct generalization of local symmetry, Chaki [6] introduced the notion of pseudosymmetry. We note that the interrelation between two types of pseudosymmetry is studied by Shaikh et. al. [20]. In 1989 Tamássy and Binh [36] generalized the Chaki’s notion of pseudosymmetry and introduced the notion of weakly symmetric manifold. We note that Shaikh and his co-authors ([12], [21]-[25]) studied this notion of weak symmetry with various generalized curvature tensors. Again generalizing the results of Binh [2], recently, Shaikh and Kundu [26] obtained the Date: April 14, 2015. 2010 Mathematics Subject Classification. 53C15, 53C25, 53C35. Key words and phrases. recurrent manifold, weakly generalized recurrent manifold, hyper generalized recurrent manifold, super generalized recurrent manifold, warped product. 1

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ABSOS ALI SHAIKH, HARADHAN KUNDU AND MD. SHOWKAT ALI

characterization of warped product weakly symmetric manifold. Again recurrent manifold ([16], [17], [18]) is another type of generalization of local symmetry. In 1979 Dubey defined generalized recurrent manifold (briefly, GKn ). Recently [14] Olszak and Olszak showed that every GKn is concircularly recurrent and every concircularly recurrent manifold is again a Kn and hence every GKn is a Kn . Again as a generalization of Kn , recently, Shaikh and his coauthors introduced three new type of generalized recurrent structures together with their proper existence, namely, quasi generalized recurrent manifold (briefly, QGKn ) [31], hyper-generalized recurrent manifold (briefly, HGKn ) [30] and weakly generalized recurrent manifold (briefly, W GKn ) [32]. For the existence of such structures we refer the reader to see [19], [34]. Very recently Shaikh et. al. ([27], [33]) introduced another generalization of recurrent manifold, called, super generalized recurrent manifold (briefly, SGKn ) which also generalizes the notion of HGKn as well as W GKn . These kinds of generalization of recurrent structures may be called as recurrent like structures. The main object of the present paper is to obtain the necessary and sufficient condition for a warped product manifold to be HGKn and W GKn . For this purpose we first determine the necessary and sufficient condition for a warped product to be SGKn and as consequence of this result we find out the corresponding results for HGKn , W GKn and Kn . We know that decomposable or product manifold is a special case of warped product manifold when the warping function is identically 1. Thus we can present the characterization of a decomposable manifold with various recurrent like structures. The paper is organized as follows: Section 2 deals with rudimentary facts of various recurrent like structures. Section 3 is concerned with basic curvature relations of a warped product manifold. In section 4 we present our main result and, finally, in section 5 a proper example of a warped product SGKn is presented.

2. Preliminaries Let M be a non-flat n-dimensional (n ≥ 3) smooth manifold with semi-Riemannian metric g, Levi-Civita connection be ∇, Riemann-Christoffel curvature tensor R, Ricci tensor S and scalar curvature κ. In this section we define various necessary terms and curvature restricted geometric structures and for this purpose at first we consider some notations: C ∞ (M) = the algebra of all smooth functions on M, χ(M) = the Lie algebra of all smooth vector fields on M, χ∗ (M) = the Lie algebra of all smooth 1-forms on M and Tkr (M) = the space of all smooth tensor fields of type (r, k) on M.

ON WARPED PRODUCT SUPER GENERALIZED RECURRENT MANIFOLDS

3

For A, E ∈ T20 (M) we have their Kulkarni-Nomizu product [27] A ∧ E ∈ T40 (M) as (2.1)

A ∧ Eijkl = Ail Ejk + Ajk Eil − Aik Ejl − Ajl Eik .

In particular we can get g ∧ g, g ∧ S and S ∧ S as follows: (g ∧ g)ijkl = 2(gil gjk − gik gjl ), (g ∧ S)ijkl = gil Sjk + Sil gjk − gik Sjl − Sik gjl and (S ∧ S)ijkl = 2(Sil Sjk − Sik Sjl ). Definition 2.1. The manifold M is said to be recurrent [37] if (2.2)

∇R = Π ⊗ R (⊗ denotes the tensor product) (or locally, Rijkl,m = Πm Rijkl , where ‘,’ denotes the covariant derivative)

holds on {x ∈ M : ∇R 6= 0 at x} for an 1-form Π ∈ χ∗ (M), called the associated 1-form of the recurrent structure. Such an n-dimensional manifold is denoted by Kn . Again M is said to be concircularly recurrent recurrent if its concircular curvature tensor W =R−

κ g 2n(n−1)

∧ g satisfies the condition

(2.3)

∇W = Π ⊗ W

on {x ∈ M : ∇W 6= 0 at x} for an 1-form Π ∈ χ∗ (M), called the associated 1-form. The manifold (M, g) is said to be generalized recurrent [9] if it satisfies (2.4)

∇R = Π ⊗ R + Θ ⊗ g ∧ g (or locally, Rijkl,m = Πm Rijkl + 2Θm (gil gjk − gik gjl )

on {x ∈ M : ∇R 6= ξ ⊗ R at x ∀ ξ ∈ χ∗ (M)} for some 1-forms Π and Θ. The 1-forms Π and Θ are called the associated 1-forms of this structure. Such an n-dimensional manifold is denoted by GKn . In [14] Olszak and Olszak showed that every GKn satisfying (2.4) is concircularly recurrent with (2.3) and every concircularly recurrent manifold is again a Kn with same associated 1-form and thus Θ = 0. Hence the structure GKn reduces to Kn . Consequently the notion of GKn does not exist.

Definition 2.2. The manifold M is said to be quasi generalized recurrent [31], hyper generalized recurrent [30], weakly generalized recurrent [32] and super generalized recurrent respectively if

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the condition (2.5)

∇R = Π ⊗ R + Ψ ⊗ [g ∧ (g + η ⊗ η)] ,

(2.6)

∇R = Π ⊗ R + Ψ ⊗ g ∧ S,

(2.7)

∇R = Π ⊗ R + Φ ⊗ S ∧ S and

(2.8)

∇R = Π ⊗ R + Φ ⊗ S ∧ S + Ψ ⊗ g ∧ S + Θ ⊗ g ∧ g

holds respectively on {x ∈ M : ∇R 6= ξ ⊗ R at x ∀ ξ ∈ χ∗ (M)} ⊂ M for some Π, Φ, Ψ, Θ and η ∈ χ∗ (M), called the associated 1-forms. An n-dimensional manifold satisfying (2.5) is denoted by QGKn with (Π, Θ, η) or simply QGKn . An n-dimensional manifold satisfying (2.6) is denoted by HGKn with (Π, Ψ) or simply HGKn . An n-dimensional manifold satisfying (2.7) is denoted by W GKn with (Π, Φ) or simply W GKn . An n-dimensional manifold satisfying (2.8) is denoted by SGKn with (Π, Φ, Ψ, Θ) or simply SGKn . In terms of local coordinates (2.5)-(2.8) can be respectively written as: (2.9)

Rijkl,m = Πm Rijkl + Θm [2(gil gjk − gik gjl ) + gil ηj ηk + gjk ηi ηl − gik ηj ηl − gjl ηi ηk ],

(2.10)

Rijkl,m = Πm Rijkl + Ψm [gil Sjk + Sil gjk − gik Sjl − Sik gjl ],

(2.11)

Rijkl,m = Πm Rijkl + 2Ψm [Sil Sjk − Sik Sjl ] and

(2.12)

Rijkl,m = Πm Rijkl + 2Φm [Sil Sjk − Sik Sjl ] + Ψm [gil Sjk + Sil gjk − gik Sjl − Sik gjl ] + 2Θ[gil gjk − gik gjl ].

It is obvious that the above structures QGKn , W GKn and HGKn are all generalization of Kn but all three exists independently (see [19], [34]). We also mention that SGKn is a proper generalization of W GKn and HGKn (see Section 5). Definition 2.3. Let D ∈ T40 (M) and A, E, F ∈ T20 (M). Then M is said to be Roter type manifold (briefly, RTn ) with (D; A, E) ([7], [8]) and generalized Roter type manifold (briefly, GRTn ) with (D; A, E, F ) ([20], [28], [29]) respectively if D = N1 A ∧ A − N2 A ∧ E − N3 E ∧ E and


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D = L1 A ∧ A − L2 A ∧ E − L3 E ∧ E − L4 A ∧ F − L5 E ∧ F − L6 F ∧ F respectively, for some Ni , Lj ∈ C ∞ (M), 1 ≤ i ≤ 3 and 1 ≤ j ≤ 6.

3. Warped Product Manifold In 1957 Kru˘ckovi˘c [13] initiated the study of semi-decomposable manifolds which were latter f, e named as warped product manifolds by Bishop and O’Neill [3]. Let (M, g) and (M g ) be two

semi-Riemannian manifolds of dimension p and (n − p) respectively (1 ≤ p < n), and f is f is the product a positive smooth function on M . Then the warped product M = M ×f M

f of dimension n endowed with the metric manifold M × M

(3.1)

g = π ∗ (g) + (f ◦ π)σ ∗ (e g ),

f are the natural projections. Then the local components of where π : M → M and σ : M → M the metric g are given by

(3.2)

gij =

      

g ij

for i = a and j = b,

fe gij

for i = α and j = β,

0

otherwise.

Here a, b ∈ {1, 2, ..., p} and α, β ∈ {p + 1, p + 2, ..., n}. We note that throughout the paper we consider a, b, c, ... ∈ {1, 2, ..., p} and α, β, γ, ... ∈ {p + 1, p + 2, ..., n} and i, j, k, ... ∈ {1, 2, ..., n}. f is called the fiber and f is called warping function of the warped Here M is called the base, M f. It may be mentioned that the warped product metric g can be taken product M = M ×f M

as (see [11], [13], [38])

g ). g = π ∗ (g) + (f ◦ π)2 σ ∗ (e

However throughout the paper we will consider the warped product metric given in (3.1). Again e the we assume that, when Ω is a quantity formed with respect to g, we denote by Ω and Ω,

similar quantities formed with respect to g and e g respectively. By straightforward calculation e for Ω = ∇, R, S and κ one can easily calculate the local components of Ω in terms of Ω and Ω and obtain the following:

The non-zero local components of Levi-Civita connection ∇ of M are given by (3.3)

a

eα , Γabc = Γbc , Γαβγ = Γ βγ

where fa = ∂a f =

∂f . ∂xa

1 Γaβγ = − g ab fb e gβγ , 2

Γαaβ =

1 fa δβα , 2f

The local components of the Riemann-Christoffel curvature tensor R and Ricci tensor S of M

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which may not vanish identically are the following: (3.4) (3.5)

eαβγδ − f 2 P G e αβγδ , gαβ , Rαβγδ = f R Rabcd = Rabcd , Raαbβ = f Tab e Sab = S ab − (n − p)Tab , Sαβ = Seαβ + Qe gαβ ,

where Gijkl = 12 g ∧ gijkl = gil gjk − gik gjl are the components of Gaussian curvature and Tab = −

1 1 (∇b fa − fa fb ), 2f 2f

Q = f ((n − p − 1)P − tr(T )),

tr(T ) = g ab Tab , P =

1 ab g fa fb . 4f 2

The scalar curvature κ of M is given by (3.6)

κ=κ ¯+

κ ˜ − (n − p)[(n − p − 1)P − 2 tr(T )]. f

Again the non-zero local components of ∇R are given by [10]:   (i)Rabcd,e = Rabcd,e ,      (ii)Raαbβ,e = f Tab,e e gαβ ,      (iii)R 2 e e αβγδ,e = −fe Rαβγδ + f Pe Gαβγδ , (3.7) eαβγδ,ǫ ,  (iv)Rαβγδ,ǫ = f R      eαβγǫ + f 2 Pd G eαβγǫ ,  (v)Rαβγd,ǫ = − f2d R  2    (vi)Rabcδ,ǫ = 12 e gǫδ (fa Tbc − fb Tac ) + 12 f d Rabcd e gǫδ . The non-zero components of (g ∧ g), (g ∧ S) and (S ∧ S) are given by     (i)(g ∧ g)abcd = (g ∧ g)abcd , (3.8) (ii)(g ∧ g)aαbβ = −2f g ab e gαβ ,    (iii)(g ∧ g) = f 2 (e g∧e g) , αβγδ

(3.9)

αβγδ

    (i)(g ∧ S)abcd = (g ∧ S)abcd − (n − p)(g ∧ T )abcd , gαβ ) − f e gαβ [S ab − (n − p)Tab ], (ii)(g ∧ S)aαbβ = −g ab (Seαβ + Qe    (iii)(g ∧ S) e e = f (e g ∧ S) + f Q(e g∧e g) , αβγδ

αβγδ

αβγδ

 2    (i)(S ∧ S)abcd = (S ∧ S)abcd − (n − p)(S ∧ T )abcd + (n − p) (T ∧ T )abcd , (3.10) (ii)(S ∧ S)aαbβ = −2(Seαβ + Qe gαβ )[S ab − (n − p)Tab ],    (iii)(S ∧ S) eg ∧ S) e e e αβγδ + Q2 (e g∧e g )αβγδ . αβγδ = (S ∧ S)αβγδ + Q(e

For detailed information about the local components of various tensors on a warped product manifold we refer the reader to see [15], [26], [29] and also references therein.


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4. Warped Product SGKn p fn−p be a warped product manifold. Then M is a SGKn Theorem 4.1. Let M n = M ×f M

with (Π, Φ, Ψ, Θ) if and only if the following conditions hold simultaneously   (i)∇R = Π ⊗ R + Φ ⊗ S ∧ S + Ψ ⊗ g ∧ S + Θ ⊗ g ∧ g      −2(n − p)Φ ⊗ S ∧ T + (n − p)2 Φ ⊗ T ∧ T − (n − p)Ψ ⊗ g ∧ T, (4.1) e ⊗S∧S+Ψ e ⊗g∧S+Θ e ⊗g∧g e ⊗R=Φ  (ii) − Π      e ⊗ S ∧ T + (n − p)2 Φ e ⊗ T ∧ T − (n − p)Ψ e ⊗ g ∧ T, −2(n − p)Φ (4.2)

  e = Φ ⊗ Se ∧ Se + (2QΦ + f Ψ) ⊗ e g ∧ Se (i) − (df + f Π) ⊗ R      +(− 1 f 2 (P Π + dP ) + Q2 Φ + f QΨ + f 2 Θ) ⊗ e g∧e g, 2

eR e = fΠ e ⊗R e+Φ e ⊗ Se ∧ Se + (QΦ e + f Ψ) e ⊗e  (ii)f ∇ g ∧ Se      e + Q2 Φ e + f QΨ e + f 2 Θ) e ⊗e g∧e g, +(− 1 f 2 P Π 2

  ⊗ (S − (n − p)T ) + Ψ ⊗ g ⊗ Se = (i) 2Φ      −[f (∇T − Π ⊗ T ) + (2QΦ + f Ψ) ⊗ (S − (n − p)T ) + (QΨ + 2f Θ) ⊗ g] ⊗ e g, i h (4.3) e ⊗ g ⊗ Se = e ⊗ (S − (n − p)T ) + Ψ  (ii) 2Φ      e ⊗ T + (2QΦ e + f Ψ) e ⊗ (S − (n − p)T ) + (QΨ e + 2f Θ) e ⊗ g] ⊗ e −[−f Π g, (

(4.4)

(i)f d Rabcd = −(fa Tbc − fb Tac ) and e = f 2 ΘP ⊗ G. e (ii)df ⊗ R

Proof: First suppose that M is SGKn . Then in terms of local coordinates the defining condition can be written as (4.5)

Rijkl,m = Πm Rijkl + Φm (S ∧ S)ijkl + Ψm (g ∧ S)ijkl + Θm (g ∧ g)ijkl.

Putting (

(i)i = a, j = b, k = c, l = d, m = e and (ii)i = a, j = b, k = c, l = d, m = ǫ

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respectively in (4.5) and then in view of (3.4)-(3.10) it is easy to check that (4.1) holds. Similarly putting   (iii)i = α, j = β, k = γ, l = δ, m = e;      (iv)i = α, j = β, k = γ, l = δ, m = ǫ;      (v)i = a, j = α, k = b, l = β, m = e;  (vi)i = a, j = α, k = b, l = β, m = ǫ;       (vii)i = a, j = b, k = c, l = α, m = ǫ and     (viii)i = α, j = β, k = γ, l = a, m = ǫ

respectively in (4.5) and then in view of (3.4)-(3.10) we get (4.2)-(4.4) respectively. The converse part is obvious. This proves the theorem. From Theorem 4.1, it follows that the nature of base and fiber of a warped product SGKn is given by the following: p fn−p be a warped product SGKn with (Π, Φ, Ψ, Θ). Then Corollary 4.1. Let M n = M ×f M

(i) M is a SGKp if T can be expressed as a linear combination of S and g. e x 6= 0}. (ii) M is a GRTp with (R; g, S, T ) on {x ∈ M : Π f is a SGKn−p with (Π, e 1 Φ, e QΦ e + Ψ, e −1fP Π e + Q2 Φ e + QΨ e + f Θ). e (iii) M f f 2 f f is a RTn−p with (R; e e e on {x ∈ M : (df + f Π)x 6= 0}. (iv) M g , S)

f satisfies Einstein metric condition on {x ∈ M : (2(κ − (n − p)tr(T ))Φ + pΨ)x 6= 0} ∪ (v) M e + pΨ) e x 6= 0} = {x ∈ M : (2(κ − (n − p)tr(T ))Φ + pΨ)x 6= 0}. {x ∈ M : (2(κ − (n − p)tr(T ))Φ f is of constant curvature on {x ∈ M : dfx 6= 0}. (vi) M

From Theorem 4.1, the characterization of a decomposable or product SGKn is given by the

following: p fn−p be a product manifold. Then M is a SGKn with Corollary 4.2. Let M n = M × M

(Π, Φ, Ψ, Θ) if and only if

1.(i) ∇R = Π ⊗ R + Φ ⊗ S ∧ S + Ψ ⊗ g ∧ S + Θ ⊗ g ∧ g, e ⊗S∧S+Ψ e ⊗g∧S+Θ e ⊗ g ∧ g, e ⊗R=Φ (ii) −Π eR e=Π e ⊗R e+Φ e ⊗ Se ∧ Se + Ψ e ⊗e e ⊗e 2.(i) ∇ g ∧ Se + Θ g∧e g, e = Φ ⊗ Se ∧ Se + Ψ ⊗ ge ∧ Se + Θ ⊗ e (ii) −Π ⊗ R g∧e g, 3.(i) 2Φ ⊗ S + Ψ ⊗ g ⊗ Se = −[Ψ ⊗ S + 2Θ ⊗ g] ⊗ e g, h i e ⊗ Se + 2Θ e ⊗e e ⊗ Se + Ψ e ⊗e g ] ⊗ g. (ii) 2Φ g ⊗ S = −[Ψ

From the above, the nature of each factor of a product SGKn is given by the following: p

fn−p be a product SGKn with (Π, Φ, Ψ, Θ). Then Corollary 4.3. Let M n = M × M (i) base and fiber are both super generalized recurrent manifolds.


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e x 6= 0}. (ii) M is a RTp with (R; g, S) on {x ∈ M : Π f is a RTn−p with (R; e e e on {x ∈ M : Πx 6= 0}. (iii) M g , S)

e + (n − p)Ψ) e x 6= 0}. κΦ (iv) M satisfies Einstein metric condition on {x ∈ M : (2e f satisfies Einstein metric condition on {x ∈ M : (2κΦ + pΨ)x 6= 0}. (v) M

p fn−p be a warped product manifold. Then M is a Corollary 4.4. [26] Let M n = M ×f M

recurrent manifold with

∇R = Π ⊗ R

if and only if the following conditions hold simultaneously e ⊗ R = 0, 1.(i) ∇R = Π ⊗ R, (ii) Π

e = 1 f 2 (P Π − dP ) ⊗ ge ∧ ge, 2.(i) −(df + f Π) ⊗ R 2 e ⊗ T = 0, 3.(i) ∇T = Π ⊗ T, (ii) Π

eR e=Π e ⊗R e and P Π e = 0, (ii) ∇

e = f 2 dP ⊗ G. e 4.(i) f d Rabcd = −(fa Tbc − fb Tac ) and (ii) df ⊗ R

p fn−p be a warped product recurrent manifold satisfying Corollary 4.5. Let M n = M ×f M

∇R = Π ⊗ R. Then f are both recurrent. Also T is recurrent with associated 1-form Π. (i) M and M e 6= 0}. (ii) R = 0, T = 0 and P = 0 on the set {x ∈ M : Π

f is of constant curvature on {x ∈ M : dfx 6= 0} ∪ {x ∈ M : (df + f Π)x 6= 0}. (iii) M

p fn−p be a product manifold. Then M is recurrent satisfying Corollary 4.6. Let M n = M × M

∇R = Π ⊗ R

if and only if (i) ∇R = Π ⊗ R,

e ⊗ R = 0, Π

e = 0, (ii) Π ⊗ R

eR e=Π e ⊗ R. e ∇

p fn−p be a warped product manifold. Then M is a HGKn Corollary 4.7. Let M n = M ×f M

with (Π, Ψ) if and only if the following conditions hold simultaneously

1.(i) ∇R = Π ⊗ R + Ψ ⊗ g ∧ S − (n − p)Ψ ⊗ g ∧ T, e ⊗R=Ψ e ⊗ g ∧ S − (n − p)Ψ e ⊗ g ∧ T, (ii) −Π

e = f Ψ ⊗ ge ∧ Se + (− 1 f 2 (P Π + dP ) + f QΨ) ⊗ ge ∧ ge, 2.(i) −(df + f Π) ⊗ R 2 e e e e e e e + f QΨ) e ⊗e (ii) f ∇R = f Π ⊗ R + f Ψ ⊗ e g ∧ S + (− 1 f 2 P Π g∧e g, 2

g, 3.(i) Ψ ⊗ g ⊗ Se = −[f (∇T − Π ⊗ T ) + f Ψ ⊗ (S − (n − p)T ) + QΨ ⊗ g] ⊗ e e ⊗ g ⊗ Se = −[−f Π e ⊗ T + fΨ e ⊗ (S − (n − p)T ) + QΨ e ⊗ g] ⊗ e (ii) Ψ g, 4.(i) f d Rabcd = −(fa Tbc − fb Tac ) and e = f 2 dP ⊗ G. e (ii) df ⊗ R

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p fn−p be a warped product HGKn with (Π, Ψ). Then Corollary 4.8. Let M n = M ×f M

(i) M is hyper generalized recurrent manifold if T and S are linearly dependent. f is a SGKn−p with (Π, e 0, Ψ, e −1fP Π e + QΨ). e (ii) M 2

f is of vanishing conformal curvature tensor on {x ∈ M : (df + f Π)x 6= 0}. (iii) M f satisfies Einstein metric condition on {x ∈ M : Ψx 6= 0} and M f is of constant curvature (iv) M on {x ∈ M : dfx 6= 0}.

Proof: Results of (i), (ii) and (iv) are obvious from Corollary 4.7. By virtue of 2.(i) of Corollary e can be exressed as a linear combination of e g ∧ Se and 4.7, on {x ∈ M : (df + f Π)x 6= 0}, R

f vanishes on g ∧ ge. Hence in view of Corollary 6.1 of [27], the conformal curvature tensor of M e

this set, which proves (iii).

p

fn−p be a product manifold. Then M is a HGKn with (Π, Ψ) Corollary 4.9. Let M n = M × M if and only if the following conditions hold simultaneously e ⊗R+Ψ e ⊗ g ∧ S = 0, 1.(i) ∇R = Π ⊗ R + Ψ ⊗ g ∧ S, (ii) Π e+Ψ⊗e eR e=Π e ⊗R e+Ψ e ⊗e e 2.(i) Π ⊗ R g ∧ Se = 0, (ii) ∇ g ∧ S,

g, 3.(i) Ψ ⊗ g ⊗ Se = −Ψ ⊗ S ⊗ e

e ⊗ g ⊗ Se = −Ψ e ⊗ S ⊗ ge. (ii) Ψ

p fn−p be a warped product manifold. Then M is a W GKn Corollary 4.10. Let M n = M ×f M

with (Π, Φ) if and only if the following conditions hold simultaneously

1.(i) ∇R = Π ⊗ R + Φ ⊗ S ∧ S − 2(n − p)Φ ⊗ (S ∧ T ) + (n − p)2 Φ ⊗ T ∧ T, e ⊗ S ∧ S − 2(n − p)Φ e ⊗ (S ∧ T ) + (n − p)2 Φ e ⊗ T ∧ T, e ⊗R=Φ (ii) −Π

e = Φ ⊗ Se ∧ Se + 2QΦ ⊗ e 2.(i) −(df + f Π) ⊗ R g ∧ Se + (− 12 f 2 (P Π − dP ) + Q2 Φ) ⊗ e g∧e g, eR e = fΠ e ⊗R e+Φ e ⊗ Se ∧ Se + QΦ e ⊗ ge ∧ Se + (− 1 f 2 P Π e + Q2 Φ) e ⊗e (ii) f ∇ g∧e g, 2

g, 3.(i) 2Φ ⊗ (S − (n − p)T ) ⊗ Se = −[f (∇T − Π ⊗ T ) + 2QΦ ⊗ (S − (n − p)T )] ⊗ e e ⊗ T + 2QΦ e ⊗ (Se − (n − p)T )] ⊗ e e ⊗ (S − (n − p)T ) ⊗ Se = −[−f Π g, (ii) 2Φ 4.(i) f d Rabcd = −(fa Tbc − fb Tac ) and e = f 2 dP ⊗ G. e (ii) df ⊗ R

p fn−p be a warped product W GKn with (Π, Φ). Then Corollary 4.11. Let M n = M ×f M

(i) M is a W GKp if T and S are linearly dependent. e −1fP Π e + Q2 Φ). e f is a SGKn−p with (Π, e Φ, Q Ψ, (ii) M f 2 f f is a RTn−p with (R; e e e on {x ∈ M : (df + f Π)x = (iii) M g , S) 6 0}. f satisfies Einstein metric condition on {x ∈ M : ((κ − (n − p)tr(T ))Φ)x 6= 0} and is of (iv) M constant curvature on {x ∈ M : dfx 6= 0}. p

fn−p be a product manifold. Then M is a W GKn with Corollary 4.12. Let M n = M × M

(Π, Φ) if and only if the following conditions hold simultaneously


11

e ⊗R+Φ e ⊗ S ∧ S = 0, 1.(i) ∇R = Π ⊗ R + Φ ⊗ S ∧ S, (ii) Π e + Φ ⊗ Se ∧ Se = 0, (ii) ∇ eR e=Π e ⊗R e+Φ e ⊗ Se ∧ S, e 2.(i) Π ⊗ R e ⊗ S ⊗ Se = 0. 3.(i) Φ ⊗ S ⊗ Se = 0, (ii) Φ 5. An example of warped product SGK4 f, where M is a 3-dimensional Example 1: Consider the warped product M = M ×f M

manifold equipped with the metric 2

2

1

ds = ex (dx1 )2 + ex (dx2 )2 + (dx3 )2 f is an open interval of R with local coordinate x4 and in local coordinates (x1 , x2 , x3 ) and M 3

the warping function f = ex . The non-zero components of the Riemann-Christoffel curvature tensor R and Ricci tensor S of M are given by 1 x1 1 1 x1 −x2 x2 x2 −x2 , S 11 = , S 22 = . 1+e e R1212 = − e + e 4 4 4

Again the non-zero components of ∇R are

2

1

ex ex R1212,1 = , R1212,2 = . 4 4 x2 x1 If we consider the 1-form Π = − ex1e+ex2 , − ex1e+ex2 , 0 , then we can easily check that the manifold M is recurrent satisfying ∇R = Π ⊗ R. f is given by Now the metric of M = M ×f M 2

1

3

ds2 = ex (dx1 )2 + ex (dx2 )2 + (dx3 )2 + ex (dx4 )2 .

The non-zero local components of the Riemann-Christoffel curvature tensor R and Ricci tensor S of M are given by

3 1 x1 ex x2 R1212 = − e + e , R3434 = − , 4 4 3 1 x1 −x2 1 ex 1 x2 −x1 e + 1 , S22 = e + 1 , S33 = , S44 = . S11 = 4 4 4 4 Again the non-zero local components of ∇R are given by 2

R1212,1 =

1

ex ex , R1212,2 = . 4 4

Again the non-zero components of g ∧ g, g ∧ S and S ∧ S given as follows: g ∧ g1212 = −2ex

1 +x2

, g ∧ g1313 = −2ex , g ∧ g1414 = −2ex

2

2 +x3

1 +x3

3

g ∧ g2424 = −2ex

, g ∧ g3434 = −2ex ,

1

, g ∧ g2323 = −2ex ,

12


1 x2 −x1 x1 1 x1 2 e +1 −1 , −e − ex , g ∧ S1313 = −e 2 4 1 x1 −x2 x2 1 x3 −x1 x2 x1 x1 , g ∧ S2323 = e +1 −1 , e +1 +e −e e g ∧ S1414 = − e 4 4 3 1 x3 −x2 x2 x1 ex x1 g ∧ S2424 = − e , g ∧ S3434 = − , e +1 +e e 4 2 1 1 2 1 −ex −x − 1 , − cosh x1 − x2 − 1 , S ∧ S1313 = S ∧ S1212 = 4 8 1 x3 x2 −x1 1 x1 −x2 S ∧ S1414 = − e −e −1 , e + 1 , S ∧ S2323 = 8 8 3 1 x3 x1 −x2 ex S ∧ S2424 = − e e + 1 , S ∧ S3434 = − . 8 8 If we consider the 1-forms Π, Φ, Ψ and Θ as:  2 1 2 Ψ1 ex ex −2 +2Ψ1 cosh(x1 −x2 )−(2Ψ1 +1)ex +2Ψ1    for i = 1  2(ex1 +ex2 )  1 2  1   Ψ2 ex −2 ex +2Ψ2 cosh(x1 −x2 )−(2Ψ2 +1)ex +2Ψ2   for i = 2  2(ex1 +ex2 ) 2 (5.1) Πi = 1 2 1 2 1 2 Ψ3 e−x −x −ex +x +ex +ex    for i = 3  2(ex1 +ex2 )   2  1 2 1 2 1 2  Ψ4 e−x −x −ex +x +ex +ex    for i = 4, 2(ex1 +ex2 ) g ∧ S1212 =

(5.2)

(5.3)

Φi =

Θi =

                  

              

1 2 2 Ψ1 −ex +x +2Ψ1 cosh(x1 −x2 )+2Ψ1 +ex

for i = 1

−2 cosh(x 1 −x2 )+sinh(x1 )+cosh(x1 )+sinh(x2 )+cosh(x 2 )−2

ex

1

1

Ψ2 ex +1 ex1 +ex2

− Ψ2 +

− − −Ψ1 ex 1 16

1

1

− Ψ2

(ex1 −1)ex2−ex1

for i = 2

1 2 1 2 Ψ3 ex +x +ex +ex

for i = 3

1 2 1ex 2+ex 1 2 x +x x Ψ4 e +e +ex

for i = 4,

ex1 +ex2

1 −x2

2

+2Ψ1 ex sinh(x1 )−2Ψ1 +ex 16(−ex1 +x2 +ex1 +ex2 ) 2

2

x1

−Ψ2 e−x − Ψ2 e−x − Ψ2 + −ex1 +x2e+ex1 +ex2 1 2 1 2 1 2 1 Ψ3 e−x −x ex +x + ex + ex − 16 1 x1 +x2 x1 x2 −x1 −x2 − 16 Ψ4 e e +e +e

for i = 1 for i = 2 for i = 3 for i = 4

then we can check that M is a SGK4 with (Π, Φ, Ψ, Θ), which is neither a HGK4 nor a W GK4 . Acknowledgment: The second named author gratefully acknowledges to CSIR, New Delhi (File No. 09/025 (0194)/2010-EMR-I) for the financial assistance. All the algebraic computations of Section 5 are performed by a program in Wolfram Mathematica.


13

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Department of Mathematics,

University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India E-mail address: [email protected] E-mail address: [email protected] 2

Department of Mathematics,

Dhaka University, Dhaka-1000, Bangladesh. E-mail address: [email protected]