Multiply Twisted Products - Wang

8/22/2019 Multiply Twisted Products - Wang

1/45

arXiv:1207

.0199v1

[math.DG

]1Jul2012

Multiply twisted products

Yong Wang

Abstract

In this paper, we compute the index form of the multiply twisted products.We study the Killing vector fields on the multiply twisted product manifolds anddetermine the Killing vector fields in some cases. We compute the curvatureof the multiply twisted products with a semi-symmetric metric connection andshow that the mixed Ricci-flat multiply twisted products with a semi-symmetricmetric connection can be expressed as multiply warped products. We also studythe Einstein multiply warped products with a semi-symmetric metric connec-tion and the multiply warped products with a semi-symmetric metric connectionwith constant scalar curvature, we apply our results to generalized Robertson-Walker spacetimes with a semi-symmetric metric connection and generalizedKasner spacetimes with a semi-symmetric metric connection and find some newexamples of Einstein affine manifolds and affine manifolds with constant scalarcurvature. We also consider the multiply twisted product Finsler manifolds andwe get some interesting properties of these spaces.

Keywords: Multiply twisted products; index forms; Killing vector fields; semi-symmetric metric connection; Ricci tensor; scalar curvature; Einstein manifolds;

multiply twisted product Finsler manifolds

1 Introduction

The (singly) warped product B b F of two pseudo-Riemannian manifolds (B, gB)and (F, gF) with a smooth function b : B (0, ) is the product manifold B Fwith the metric tensor g = gB b2gF. Here, (B, gB) is called the base manifoldand (F, gF) is called as the fiber manifold and b is called as the warping function.Generalized Robertson-Walker space-times and standard static space-times are twowell-known warped product spaces.The concept of warped products was first intro-

duced by Bishop and ONeil (see [BO]) to construct examples of Riemannian mani-folds with negative curvature. In Riemannian geometry, warped product manifoldsand their generic forms have been used to construct new examples with interestingcurvature properties since then. In [DD], F. Dobarro and E. Dozo had studied fromthe viewpoint of partial differential equations and variational methods, the problemof showing when a Riemannian metric of constant scalar curvature can be producedon a product manifolds by a warped product construction. In [EJK], Ehrlich, Jungand Kim got explicit solutions to warping function to have a constant scalar curvaturefor generalized Robertson-Walker space-times. In [ARS], explicit solutions were also

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obtained for the warping function to make the space-time as Einstein when the fiberis also Einstein.

One can generalize singly warped products to multiply warped products. Briefly,a multiply warped product (M, g) is a product manifold of form M = B b1 F1 b2F2 bm Fm with the metric g = gB b21gF1 b22gF2 b2mgFm , where for eachi {1, , m}, bi : B (0, ) is smooth and (Fi, gFi) is a pseudo-Riemannian man-ifold. In particular, when B = (c, d) with the negative definite metric gB = dt2 and(Fi, gFi) is a Riemannian manifold, we call M as the multiply generalized Robertson-Walker space-time. Geodesic equations and geodesic connectedness of multiply gener-alized Robertson-Walker space-times were studied by Flores and Sanchez in [FS] andthey also noted that the class of multiply generalized warped space-times containsmany well known relativistic space-times. In [U1], necessary and sufficient conditionswere obtained about geodesic completeness of multiply warped space-times. In [DU1],Dobarro and Unal studied Ricci-flat and Einstein-Lorentzian multiply warped prod-ucts and considered the case of having constant scalar curvature for multiply warpedproducts and applied their results to generalized Kasner space-times.

Singly warped products have two natural generalizations. A doubly warped prod-uct (M, g) is a product manifold of form M =f B b F, with smooth functionsb : B (0, ), f : F (0, ) and the metric tensor g = f2gB b2gF. In [U2], Unalstudied geodesic completeness of Riemannian doubly warped products and Lorentziandoubly warped products. A twisted product (M, g) is a product manifold of formM = B b F, with a smooth function b : B F (0, ), and the metric tensorg = gB b2gF. In [FGKU], they showed that mixed Ricci-flat twisted products couldbe expressed as warped products. As a consequence, any Einstein twisted products arewarped products. So it is natural to consider multiply twisted products as generaliza-

tions of multiply warped products and twisted products. A multiply twisted product(M, g) is a product manifold of form M = B b1 F1 b2 F2 bm Fm with the metricg = gBb21gF1b22gF2 b2mgFm, where for each i {1, , m}, bi : BFi (0, )is smooth.

In [EK], Ehrlich and Kim constructed the index form along timelike geodesics ona Lorentzian warped product and applied this index form to generalized Robertson-Walker spacetimes. In [CK], they computed the index form of multiply generalizedRobertson-Walker spacetimes. In the first part of this paper, we compute the indexform of multiply twisted products. In [Sa], the curvature and Killing vector fields ofgeneralized Robertson-Walker spacetimes were studied. The non-trivial Killing vec-tor fields of these space-times were characterized. In [DU2], they provided a global

characterization of the Killing vector fields of a standard static spacetime by a sys-tem of partial differential equations. By studying this system, they determined allthe Killing vector fields when Riemannian part was compact. In the second part ofthis paper, we study the Killing vector fields of multiply twisted products and in somecases, we can determine all Killing vector fields.

The definition of a semi-symmetric metric connection was given by H. Haydenin [Ha]. In 1970, K. Yano [Ya] considered a semi-symmetric metric connection andstudied some of its properties. He proved that a Riemannian manifold admitting

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the semi-symmetric metric connection has vanishing curvature tensor if and onlyif it is conformally flat. Motivated by the Yano result, in [SO], Sular and Ozgur

studied warped product manifolds with a semi-symmetric metric connection, theycomputed curvature of semi-symmetric metric connection and considered Einsteinwarped product manifolds with a semi-symmetric metric connection. In the main partof this paper, we consider multiply twisted products with a semi-symmetric metricconnection and compute the curvature of a semi-symmetric metric connection. Weshow that mixed Ricci-flat multiply twisted products with a semi-symmetric met-ric connection can be expressed as multiply warped products which generalizes theresult in [FGKU]. We also study the Einstein multiply warped products with a semi-symmetric metric connection and multiply warped products with a semi-symmetricmetric connection with constant scalar curvature, we apply our results to generalizedRobertson-Walker spacetimes with a semi-symmetric metric connection and general-ized Kasner spacetimes with a semi-symmetric metric connection and we find somenew examples of Einstein affine manifolds and affine manifolds with constant scalarcurvature. We also classify generalized Einstein Robertson-Walker spacetimes with asemi-symmetric metric connection and generalized Einstein Kasner spacetimes witha semi-symmetric metric connection.

On the other hand, Finsler geometry is a subject studying manifolds whose tan-gent spaces carry a norm varying smoothly with the base point. Indeed, Finslergeometry is just Riemannian geometry without the quadratic restriction. Thus it isnatural to extend the construction of warped product manifolds for Finsler geometry.In the first step, Asanov gave the generalization of the Schwarzschild metric in theFinslerian setting and obtained some models of relativety theory descried throughthe warped product of Finsler metrics [As1,2]. In [KPV], Kozma-Peter-Varga defined

their warped product for Finsler metrics and concluded that completeness of warpedproduct can be related to completeness of its components. In [HR], Using the Cantanconnection for the study of warped product Finsler spaces, they found the necessaryand sufficient conditions for such manifolds to be Riemannian, Landsberg, Berwald,and locally Minkowski, separately. In [PT] and [PTN], they considered the doublywarped product Finsler manifolds and found the necessary and sufficient conditionsfor such manifolds to be Riemannian, Landsberg, Berwald, Douglas, locally duallyflat. They also defined the doubly warped Sasaki-Matsumoto metric for warpedproduct manifolds and found a condition under which the horizontal and verticaltangent bundle were totally geodesic. In this paper, we consider multiply twistedproduct Finsler manifolds. Let (B, FB), (Mi, Fi), 1 i m be Finsler manifoldsand fi : B Mi (0, +) be a smooth function. Let i : T Mi Mi be theprojection map. The product manifold B M1 Mm endowed with the metricF : T B0 T M01 T M0m R is considered,

F(v0, v1, , vm)

=

FB

2(v0) + f21 (0(v0), 1(v1))F1

2(v1) + + f2m(0(v0), m(vm))Fm2(vm), (1.1)

where T B0 = T B {0}, T M0i = T Mi {0}. We find the necessary and sufficientconditions for multiply twisted product Finsler manifolds to be Riemannian, Lands-

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berg, Berwald, locally dually flat, locally Minkowski.We also define the generalizedtwisted Sasaki-Matsumoto metric and get a condition under which the horizontal and

vertical tangent bundle are totally geodesic.This paper is arranged as follows: In Section 2, we compute curvature and theindex form of multiply twisted products. In Section 3, we study the Killing vectorfields of multiply twisted products and in some cases, we can determine all Killingvector fields. In Section 4, we study multiply twisted products with a semi-symmetricmetric connection. In Section 5, we study multiply twisted product Finsler manifolds.

2 The index form of the multiply twisted products

Definition 2.1 A multiply twisted product (M, g) is a product manifold of form

M = B b1 F1 b2 F2 bm Fm with the metric g = gB b2

1gF1 b2

2gF2 b2mgFm,where for each i {1, , m}, bi : B Fi (0, ) is smooth.

Here, (B, gB) is called the base manifold and (Fi, gFi) is called as the fiber mani-fold and bi is called as the twisted function. Obviously, twisted products and multiplywarped products are the special cases of multiply twisted products.

Proposition 2.2 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted productand let X, Y (T B) and U (T Fi), W (T Fj). Then(1) XY = BXY.(2) XU = UX = X(bi)bi U.(3) UW = 0 if i = j.

(4) UW = U(lnbi)W+W(lnbi)UgFi(U,W)

bi gradFibibigFi(U, W)gradBbi+FiUW if i =

j.

Proposition 2.3 Let M = B b1 F1 b2 F2 bm Fm be a multiply twisted prod-uct, then B is a totally geodesic submanifold and Fi is a totally umbilical submanifold.

Define the curvature, Ricci curvature and scalar curvature as follows:

R(X, Y)Z = XY YX [X,Y],

Ric(X, Y) =

kk < R(X, Ek)Y, Ek >, S =

kkRic(Ek, Ek),

where Ek is a orthonormal base of M with < Ek, Ek >= k. The Hessian of f isdefined by Hf(X, Y) = XY f (XY)f.

Proposition 2.4 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted productand let X , Y , Z (T B) and V (T Fi), W (T Fj), U (T Fk). Then(1)R(X, Y)Z = RB(X, Y)Z.

(2)R(V, X)Y = HbiB(X,Y)

biV.

(3)R(X, V)W = R(V, W)X = R(V, X)W = 0 if i = j.

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(4)R(X, Y)V = 0.(5)R(V, W)X = V X(lnbi)W W X(lnbi)V i f i = j.(6)R(V, W)U = 0 if i = j = k or i = j = k.(7)R(U, V)W = g(V, W)gB(gradBbi,gradBbk)bibk U, if i = j = k.(8)R(X, V)W = g(V,W)bi BX(gradBbi)+[W X(lnbi)]VgFi(W, V)gradFi(Xlnbi) if i =j.

(9)R(V, W)U = g(V, U)gradB(W(lnbi)) g(W, U)gradB(V(lnbi)) + RFi(V, W)U |gradBbi|2B

b2i(g(W, U)V g(V, U)W) if i = j = k.

Proposition 2.5 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted productand let X , Y , Z (T B) and V (T Fi), W (T Fj). Then(1)Ric(X, Y) = RicB(X, Y) +

mi=1

libi

HbiB (X, Y).(2)Ric(X, V) = Ric(V, X) = (li

1)[V X(lnbi)].

(3)Ric(V, W) = 0 if i = j.(4)Ric(V, W) = RicFi(V, W) +

Bbibi

+ (li 1) |gradBbi|2B

b2i+

k=i lk

gB(gradBbi,gradBbk)bibk

g(V, W) if i = j.

By Proposition 2.5, similar to the theorem 1 in [FGKU], we get:

Corollary 2.6 Let M = B b1 F1 b2 F2 bm Fm be a multiply twisted productand dimFi > 1, then M is mixed Ricci-flat if and only if M can be expressed as amultiply warped product. In particular, if M is Einstein, then M can be expressed asa multiply warped product.

Similar to the theorem 6 in [BGV], we get:

Corollary 2.7 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted product anddimB > 1, dimFi > 1, if M is locally conformal-flat, then M can be expressed as amultiply warped product.

Proposition 2.8 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted product,then the scalar curvature S has the following expression:

S = SB+2m

i=1li

biBbi+

m

i=1SFi

b2i+

m

i=1li(li1) |gradBbi|

2B

b2i+

m

i=1 k=ililk

gB(gradBbi, gradBbk)

bibk.

(2.1)Similar to the proposition 3.1 in [U2], we have

Proposition 2.9 LetM = Bb1 F1b2 F2 bm Fm be a multiply twisted product. If(B, gB) and (Fi, gFi) are complete Riemannian manifolds and infbi > 0, then (M, g)is a complete Riemannian manifold. Conversely, if (M, g) is a complete Riemannianmanifold and 0 < infbi < supbi < +, then (B, gB) and (Fi, gFi) are complete Rie-mannian manifolds.

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By Proposition 2.2, we have:

Proposition 2.10 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted product.Also let = (, 1, , m) be a curve in M defined on some interval I R. Then is a geodesic in M if and only if for any t I,

1.

=mi=1 bi(, i)gFi(

i,

i)gradBbi.

2.i =2(bi)bi(,i)

i 2i(lnbi)i +gFi(

i,

i)

bi(,i)gradFibi, for any i {1, 2 , m}.

By the proposition 2.2, the formula for the covariant derivative of a smooth vectorfield V = (VB, VF1, , VFm) along the smooth curve = (, 1, , m) may beobtained:

V(t) = VB(t)

m

i=1 bi < i(t), Vi >Fi gradBbi ,

mi=1

[(bi)

bi(, i)Vi +

VB(bi)

bii(t) +

i(lnbi)Vi + Vi(lnbi)

i

gFi(i, Vi)

bigradFibi + Fii Vi]

.

(2.2)By (2.2), we have:

Proposition 2.11 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted product.(a) Let VB be a smooth vector field along the smooth curve B : I (B, gB) and forfixedqi Fi, let (t) = (B(t), q1, , qm) and V(t) = (VB(t), 0, , 0) along . ThenV is parallel along : I (M, g) if and only if VB is parallel along B : I (B, gB).

(b) LetVFi be a smooth vector field along the smooth curve

Fi :

I (

Fi, gFi) andfor fixed b B and qj Fj, j = i, let (t) = (b, q1, , qi1, Fi(t), , qm) and

V(t) = (0, , 0, VFi(t), , 0) along . Then V is parallel along : I (M, g) ifand only if < Fi(t), VFi(t) >Fi gradBbi = 0 and

Fi(t)(lnbi)Vi + Vi(lnbi)Fi

(t) gFi(Fi

(t), Vi)

bigradFibi + FiFi(t)Vi = 0.

Next we compute the index form. We use the variational approach rather thandirect computation of the curvature formula g(R(V, ), V) of the index form.

Now let : [a, b]

(M, g) be a unit timelike curve. Further, let : I

(

, )

(M, g) be a variation of (t). Lets(t) := (t, s) = (B(t, s), 1(t, s), , m(t, s)) (2.3)

and define corresponding variation vector fields

W =

s=

s; WB = (B)

s=

B

s; Wi = (i)

s=

i

s(2.4)

V(t) = W(t, 0), VB(t) = WB(t, 0), Vi(t) = Wi(t, 0) (2.5)

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Also,

s = (B

t

, 1

t

,

, m

t

) = (B

t

,1

t

,

,

m

t

). (2.6)

Since the curve s(t) is timelike, if

h(t, s) = gB( Bt

,B

t)

mi=1

b2i (B, i)(t, s)gFi(i

t,

i

t) = g(

t,

t) (2.7)

and

A(t, s) = gB(Bs

B

t,

B

t) +

1

2

mi=1

s(b2i (B, i))(t, s)gFi(

i

t,

i

t)

+

mi=1

b2i (B, i)(t, s)gFi(

Fis

i

t ,

i

t ) (2.8)

then

h(t, 0) = 1,h

s= 2A(t, s) (2.9)

and

L(s) =

bt=a

g(

t,

t) =

bt=a

(h(t, s)12 dt. (2.10)

Thus

L(s) = bt=a

h(t, s)12 A(t, s)dt; (2.11)

L(s) = bt=a

[h(t, s)32 A(t, s)2 + h(t, s)

12

A(t, s)s

]dt. (2.12)

So

L(0) = bt=a

[A(t, 0)2 +A(t, s)

s(t, 0)]dt. (2.13)

If (t) is a unit timelike geodesic in (M, g), then A(t, 0) = g(V, ). It remains tocalculate A(t,s)

s(t, 0). By (2.8) and commuting the differentiation, we have

A(t, s)

s(t, 0) = gB(BB

s

BBt

B

s,

B

t)|s=0 + gB(VB, VB)

+1

2

mi=1

2

s2(b2i (B, i))(t, s)|s=0gFi(Fi(t), Fi(t))

+2mi=1

s(b2i (B, i))(t, s)|s=0gFi(VFi(t), Fi(t))

+mi=1

b2i (B, Fi)(t)gFi(FiFis

FiFit

Fis

,Fi

t)|s=0 +

mi=1

b2i (B, Fi)(t)gFi(VFi

, VFi)

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= gB(VB, VB) gB(RB

(VB, B)B, VB) + gB(B

Bt

B

Bs

B

s ,

B

t )|s=0

+1

2

mi=1

2


+2mi=1

s(b2i (B, i))(t, s)|s=0gFi(VFi(t), Fi(t))

mi=1

b2i (B, Fi)(t)gFi(RFi(VFi ,

Fi

)Fi , VFi)

+mi=1

b2i (B, Fi)(t)gFi(FiFit

FiFis

Fis

,Fi

t)|s=0 +

mi=1

b2i (B, Fi)(t)gFi(VFi

, VFi).

(2.14)By the proposition 2.10 and preserving the metric, we have

gB(BBt

BBs

B

s,

B

t)|s=0 = d

dt[gB(BVBVB, B)]

1

2

mi=1

gB(BVBVB, gradBb2i )gFi(Fi, Fi),

(2.15)We note that

2

s2(b2i (B, i))(t, s) = (V

2B + 2VBVFi + V

2Fi

)(b2i ), (2.16)

so

gB(BBt

BBs

B

s,

B

t)|s=0 + 1

2

mi=1

2


=d

dt[gB(BVBVB, B)] +

1

2

mi=1

HessB(b2i )(VB , VB)gFi(Fi

, Fi)

+1

2

m

i=1(2VBVFi + V

2Fi

)(b2i )gFi(Fi

, Fi). (2.17)

By the proposition 2.10, similarly we can get

mi=1

b2i (B, Fi)(t)gFi(FiFit

FiFis

Fis

,Fi

t)|s=0

=d

dt[mi=1

b2i (B, Fi)(t)gFi(FiVFiVFi, Fi

)] 12

mi=1

gFi(FiVFiVFi , gradFi(b2i ))gFi(

Fi

, Fi).

(2.18)

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By (2.13) (2.14),(2.17) and (2.18), we get

Proposition 2.12 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted product.Let : [a, b] (M, g) be a unit timelike geodesic and : [a, b] (, ) (M, g) bea smooth variation of (t) with variation vector field V = (VB, VF1, , VFm). Then

L(0) = bt=a

g(V, )2 + gB(VB, V

B) gB(RB(VB, B)B, VB)

+mi=1

b2i (B, Fi)(t)[gFi(VFi

, VFi) gFi(RFi(VFi , Fi)Fi , VFi)]

+1

2

m

i=1[HessB(b2i )(VB, VB) + Hess

Fi(b2i )(VFi , VFi) + 2VBVFi(b2i )]gFi(

Fi

, Fi)

+2mi=1

(VB + VFi)(b2i )gFi(V

Fi

, Fi)

dt

gB(BVBVB, B) +mi=1

b2i (B, Fi)(t)gFi(FiVFiVFi , Fi

)

|ba. (2.19)

In studying the second variation and index form, it suffices to consider vectorfields perpendicular to the given geodesic (t). Let V() denote the vector spaceof piecewise smooth vector fields V along with g(V, ) = 0 and let V0 () =

{V

V()|V(a) = V(b) = 0}. Then guided by the result of Proposition 2.12, the indexform

I : V0 () V0 () Rshould be given by

I(V, V) == bt=a

gB(V

B, V

B) gB(RB(VB, B)B , VB)

+mi=1

b2i (B, Fi)(t)[gFi(VFi

, VFi) gFi(RFi(VFi , Fi)Fi , VFi)]

+ 12

mi=1

[HessB(b2i )(VB, VB) + HessFi(b2i )(VFi , VFi) + 2VBVFi(b2i )]gFi(Fi , Fi)

+2mi=1

(VB + VFi)(b2i )gFi(V

Fi

, Fi)

dt (2.20)

I(V, W) could be obtained from (2.20) by polarization.Now we specialize to the index form to the case that (M, g) = (RRF, du2 +

f21 (u, x)dx2+f22 (u)gF) where (F, gF) is a Riemannian manifold. We call such Lorentzian

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manifolds as static multiply twisted product spacetimes. Let (t) = (B(t), 1(t), 2(t))denote a unit timelike geodesic segment. Consider variation vector fields V = (VB, V1, V2)

along with g(V, ) = 0. We begin with a special case in which the timelike geodesic(t) is of the form (t) = (t, x0, q). By g(V, ) = 0, then VB = 0. By Fi = 0, we get

Proposition 2.13 Let (M, g) = (R R F, du2 + f21 (u, x)dx2 + f22 (u)gF) . Let : [a, b] (M, g) be a unit timelike geodesic having form (t) = (t, x0, q) ThenI : V0 () V0 () R is given by

I(V, V) = bt=a

f21 (t, x0)gR(V

1 , V

1) + f

22 (t)gF(V

2 , V

2)

. (2.21)

By Proposition 2.10, we have

Corollary 2.14 Let (M, g) = (R R F, du2

+ f

2

1 (u, x)dx

2

+ f

2

2 (u)gF) . Then(t) = ((t), 1(t), F(t)) be a geodesic if and only if

(1) (t) 12

1(t)2 f

21

u((t), 1(t)) 1

2

f22u

((t))gF(F,

F) = 0; (2.22)

(2) 1 (t) +1

f21 ((t), 1(t))

f21u

((t), 1(t))(t)1(t) +

1

2

f21x

((t), 1(t))1(t)

2

= 0;

(2.23)

(3) F +1

f22 ((t))

f22u

((t))(t)F(t) = 0. (2.24)

Let VB = B(t) ddu |(t) and V1 = 1(t) ddx |1(t), then direct computation show that

Proposition 2.15 Let (M, g) = (R R F, du2 + f21 (u, x)dx2 + f22 (u)gF) . Let : [a, b] (M, g) be a unit timelike geodesic and V V0 (), then

I(V, V) = bt=a

B(t)2 + f21 ((t), 1(t))1(t)2 +

1

2

2B(t)

2f21u2

((t), 1(t))

+21(t)2f21x2

((t), 1(t)) + 2B(t)1(t)2f21ux

((t), 1(t))

1(t)

2

+2B(t) f21

u((t),

1(t)) +

1(t)

f21

x((t),

1(t)) 1(t)1(t)

+f22 ((t))

gF(V2, V

2) gF(RF(V2, 2)2, V2)

+

1

22B(t)

2f22u2

((t))gF(2(t),

2(t)) + 2B(t)

f22u

((t))gF(V2(t),

2(t))

dt. (2.25)

10


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3 Killing vector fields on the multiply twisted product

manifolds

In this section, we develop the propertied of Killing vector fields, then we focus ourattention on some special cases and characterize Killing vector fields on these spaces.

Lemma 3.1 Let M = B b1 F1 b2 F2 bm Fm be a multiply twisted product. LetX (T B), U (T Fi), then

LMXgM = LBXgB +

mi=1

X(b2i )gFi; LMU gM = b

2iL

FiU gFi + U(b

2i )gFi . (3.1)

Proposition 3.2 LetM = Bb1

F1 b2

F2 bm

Fm

be a multiply twisted product.Let X (T B), Ui (T Fi), then K = X + U1 + Um is a Killing vector fieldsif and only if X is a Killing vector fields on B and Ui is a conformal Killing vector

fields on Fi with the conformal factor X(b2i )+Ui(b

2i )

b2i.

Lemma 3.3 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted product. If Kis a Killing vector field on M, then KB = K(, q1, , qm) is a Killing vector field onB, and KFi = K(p, q1, , qi1, , qi+1, , qm) is Killing vector field on Fi.

Nextly, we assume m = 2 and bi = fi. Let Ka, a {1, , mB} be a basis ofKilling vector fields of (B, gB) and Wb, b {1, , mF1} be a basis of conformalKilling vector fields of (F1, gF1) with L

F1

WbgF1 = 2bgF1 and Gc, c

{1,

, mF2

}be a

basis of conformal Killing vector fields of (F2, gF2) with LF2Gc

gF2 = 2cgF2 . Let x,y,zdenote variables on B, F1, F2 respectively. If K is a Killing vector field on M, thenby Lemma 3.3, there are functions a(y, z), b(x, z), c(x, y) such that

K = KB + KF1 + KF2, KB = aKa, KF1 =

bWb, KF2 = cGc. (3.2)

We note that if, Z and T are respectively a function, a vector field and a 2-covarianttensor field on M, then

LZT(, ) = LZT(, ) + d() T(Z, ) + T(Z, ) d(). (3.3)Let Ka = gB(Ka,

), Wb = gF1(Wb,

) Gc = gF2(Gc,

). By Lemma 3.1 and (3.2), (3.3),

we get

LKgM = 2f21

KB(lnf1) + KF1(lnf1) +

bb

gF1+2f

22 [KB(lnf2) + KF2(lnf2) +

cc] gF2

+da Ka + Ka da + f21 (db Wb + Wb db) + f22 (dc Gc + Gc dc)= 2f21

KB(lnf1) + KF1(lnf1) +

bb

gF1 + 2f

22 [KB(lnf2) + KF2(lnf2) +

cc] gF2

+(dF1a Ka + f21 Wb dBb) + (Ka dF1a + f21 dBb Wb)

11


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+(dF2a

Ka + f

22

Gc dBc) + (

Ka dF2a + f22 dBc

Gc)

+(f21 dF2b

Wb + f22Gc dF1c) + (f21 Wb dF2b + f22dF1c Gc). (3.4)So we obtain the following result.Proposition 3.4 LetM = B f1 F1f2 F2. LetK be a vector field on M as in (3.2),then K is a Killing vector field if and only if the following relations hold:

KB(lnf1) + KF1(lnf1) + bb = 0; KB(lnf2) + KF2(lnf2) +

cc = 0; (3.5)

dF1a Ka + f21 Wb dBb = 0; dF2a Ka + f22Gc dBc = 0; (3.6)

f21 dF2b

Wb + f

22

Gc dF1c = 0. (3.7)

Now we let M = If1F1f2F2 with the metric tensor dt2+f1(t)2gF1+f2(t)2gF2.By (3.2), then K = (y, z)

t+ b(t, z)Wb +

c(t, y)Gc. By Proposition 3.4, we have ifK is a Killing vector field, then

KB(lnf1) + bb = 0; KB(lnf2) +

cc = 0; (3.8)

dF1 dt + f21 Wb dBb = 0; dF2 dt + f22Gc dBc = 0; (3.9)f21 dF2

b Wb + f22Gc dF1c = 0. (3.10)By (3.9), then

dF1(y, z) = f21

b(t, z)

t Wb. (3.11)Since Wb is a base, by separation of variables, we obtain

b(t, z) = b(z)

tt=t0

f21 (u)du + b(z), dF1(y, z) = b(z)Wb. (3.12)Similarly, we obtain

dF2(y, z) = f22

c(t, y)

tGc = c(y)Gc; c(t, y) = c(y)t

t=t0

f22 (u)du + c(y).

(3.13)

One derives (3.8) with respect tot, then

(y, z)

f1f1

+ f21 (t)b(z)b(y) = 0 (3.14)

Case I) = 0, by (3.12) and (3.13),

b(z) = 0, b(t, z) = b(z), c(y) = 0,

c(t, y) = c(y), b(z)b = 0, c(y)c = 0.(3.15)

12


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So we have

Proposition 3.5 LetM = If1 F1 f2 F2 with the metric tensordt2

+ f1(t)2

gF1 +f2(t)2gF2. K is a Killing vector field with = 0, then K = KF1(y, z) + KF2(y, z) and

KF1(y, z0) is a Killing vector field on (F1, gF1) and KF2(y0, z) is a Killing vector fieldon (F2, gF2).

Case II) = 0. By (3.14), thenf1f1

f21 (t) =

b(z)b(y)

(y, z)= C1F; (3.16)

f2f2

f22 (t) =

c(y)c(z)

(y, z)= C2F, (3.17)

where C1F, C2F are constants. By (3.2),(3.12),(3.13),(3.16) and (3.17), we obtain

K = (y, z)

t+ gradF1

tt0

f21 (u)du + gradF2tt0

f22 (u)du + b(z)Wb + c(y)Gc;(3.18)

HessF1 + C1FgF1 = 0; Hess

F2

+ C2FgF2 = 0; (3.19)

We may assume that is not a constant. One can derive (3.8) with respect to t andby (3.12) and (3.16), we get

bb(lnf1)(t0) + C1F

b(z)b(y) = 0. (3.20)

So ifC1

F = 0, then TF1

= b(z)Wb (lnf1)(t0)C1F b(z)Wb is a Killing vector field on F1for fixed z and b(z)Wb = TF1 + (lnf1)(t0)

C1FgradF1; (3.21)

Similarly, if C2F

= 0, then

c(y)Gc = TF2 + (lnf2)(t0)C2F

gradF2. (3.22)

By (3.16),(3.17),(3.18), (3.21) and (3.22), we obtain

Proposition 3.7 If M = If1 F1 f2 F2 with the metric tensor dt2 + f1(t)2gF1 +f2(t)2gF2 admits a Killing vector field with = 0, thenfifi f2i (t) = CiF for i = 1, 2.Proposition 3.8 LetM = If1 F1 f2 F2 with the metric tensordt2 + f1(t)2gF1 +f2(t)

2gF2. andfifi

f2i (t) = C

iF = 0 for i = 1, 2. Then its Killing vector field is

given by

K = (y, z)

t+ gradF1

tt0

f21 (u)du +(lnf1)

(t0)C1F

13


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+gradF2

tt0

f22 (u)du +(lnf2)

(t0)C2F

+ TF1(y, z) + TF2(y, z), (3.23)

where TF1(y, z0) is a Killing vector field on(F1, gF1) and TF2(y0, z) is a Killing vectorfield on (F2, gF2) and satisfies (3.19).

When CiF = 0, we may obtain similar results like Theorem 4.7 in [Sa].Nextly, we consider the Killing vector fields on R2 F with the metric

tensor dt2 + ds2 + f2(t)gF.Direct computations show that vector fields K1 =

t

, K2 =s

, K3 = st

+ t s

are the basis of Killing vector fields on R2 with the metric tensor dt2 + ds2. LetWb, b {1, , mF} be a basis of conformal Killing vector fields of (F, gF) withLFWbgF = 2bgF, then

K = a

Ka + b

Wb; KB(lnf) + b

b = 0; da

Ka + f2Wb db = 0. (3.24)By K1 = dt, K2 = ds, K3 = sdt + tds and (3.24), we have:

d1 + sd3 f2(t) b

tWb = 0; (3.25)

d2 + td3 + f2(t)b

sWb = 0. (3.26)

We derive (3.25) with respect to s, then

b = Cbst

t0

f2(u)du +Cb(s) + Lb(t), (3.27)where Cb is a constant. By (3.25) and (3.27), we have

d3 = CbWb, d1 = dbWb, b = (Cbs + db)tt0

f2(u)du + Cb(s), (3.28)where db is a constant. By (3.26) and (3.28), we obtain

d2 =

Cbt f2(t)Cb

tt0

f2(u)du f2(t)Cb(s) Wb. (3.29)So

d2 = ebWb, Cbt + f2(t)Cb tt0

f2(u)du + f2(t)Cb(s) = eb (3.30)where eb is a constant. Then

Cb(s) = Cbtf2(t) Cb tt0

f2(u)du + f2(t)eb, (3.31)

and Cb(s) = eb, Cb(s) = ebs + eb, (3.32)14


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where =

eb,

eb are constants. By (3.28) and (3.32), we have

b = (Cbs + db)tt0

f2(u)du + ebs + eb. (3.33)We derive (3.31) with respect to t, then

Cbf = f(Cbt eb). (3.34)

Case I) There is a Cb = 0 i.e. 3 is not a constant. So by (3.34), then f =A(t eb

Cb) = At + B f or A = 0 and f is fixed and eb

Cbis a constant independent of b.

So by (3.28) and (3.30),

d2 = d3, 2 = 3 +

. (3.35)

where , are constants. By the second equation in (3.24) and (3.33), we can obtainA

At + B1 +

db

tt0

f2(u)du + ebb = 0, (3.36)A

At + B3 +

Cb

tt0

f2(u)du + ebb = 0. (3.37)Derive (3.36) and (3.37) with respect to t, then

1 1A2

dbb = 0; 3 1A2

Cbb = 0. (3.38)

SoHess1F = A

21gF; Hess3F = A

23gF. (3.39)

By (3.31),(3.32) and f = At + B, then

eb = CbA(At0 + B)

. (3.40)

By (3.24),(3.33),(3.35) and (3.39), we obtain

K = (1 + s3)

t+ [( + t)3 +

]

s s

A(At + B)gradF3

+ 1

A(At + B)+

1

A(At0 + B)

gradF1 + W, (3.41)

where W = ebWb is a conformal killing vector field on (F, gF). In (3.36), we sett = t0 and using (3.38), then

dbb

A(At0 + B)+ ebb = 0. (3.42)

15


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and T = dbWbA(At0+B)

+

ebWb is a Killing vector field on F, So

K = (1 + s3)

t + [( + t)3 + ] s sA(At + B) gradF3 1

A(At + B)gradF1 + T, (3.43)

Theorem 3.9 LetM = R2F with the metric tensordt2 + ds2 + f2(t)gF and K isa Killing vector field given by (3.24) and3 is not a constant, then f = At+B, A = 0and K can be expressed by (3.43) and 1, 3 satisfy (3.39).

Case II) Cb = 0 for any b and there is a eb = 0 i.e. 3 = k0 is a constant and2 is not a constant. By (3.34), f = l0 is a constant. By (3.33), then

b =db

l20(t t0) + ebs + eb. (3.44)

By (3.24) and (3.44), we obtain

dbb = 0, ebb = 0, ebb = 0, (3.45)so bWb = K1t + K2s + K3 where K1, K2, K3 are Killing vector fields on F. We alsoget by (3.31)

Hess1F = Hess2F = 0. (3.46)

Theorem 3.10 LetM = R2 F with the metric tensordt2 + ds2 + f2(t)gF and Kis a Killing vector field given by (3.24) and 3 = k0 and 2 is not a constant, thenf = l0 and K can be expressed by

K = (1 + sk0)

t+ (2 + tk0)

s+ K1t + K2s + K3, (3.47)

where K1, K2, K3 are Killing vector fields on F and 1, 3 satisfy (3.46).

Case III) Cb = eb = 0 for any b i.e. 3 = k0, 2 = k0 are constants. In thiscase,

b = db tt0

f2(u)du + eb. (3.48)By (3.24), then

1f

f+ [db

tt0

f2(u)du + eb]b = 0; k0 ff

= 0. (3.49)

When k0 = 0, similar to the discussions in [Sa], we can obtain

16


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Theorem 3.11 LetM = R2 F with the metric tensordt2 + ds2 + f2(t)gF and Kis a Killing vector field given by (3.24) and 3 = 0 and 2 = k0, then if 1 = 0, K

can be expressed byK = k0

s+ W, (3.50)

where W is a Killing vector field on F. If 1 = 0, thenf

f

f2 = CF and when

CF = 0, then K can be expressed by

K = 1

t+ k0

s+ gradF1

tt0

f2(u)du +(lnf)(t0)

CF

+ T, (3.51)

where T is a Killing vector field on F and Hess1F + 1CFgF = 0. If CF = 0 and 1is a nonzero constant, then K = 1

t + k0

s + W, where W is homothetic.

Theorem 3.12 LetM = R2 F with the metric tensordt2 + ds2 + f2(t)gF and Kis a Killing vector field given by (3.24) and 3 = k0 = 0, 2 = k0, then f = l0 and Kcan be expressed by

K = (1 + sk0)

t+ (k0 + tk0)

s+

gradF1l20

(t t0) + T, (3.52)

where T is a Killing vector field on F and Hess1F = 0.

Remark. Here we can not consider M = R2F as R(RfF) with the metric tensords2+(dt2+f2(t)gF). Although the space of conformal Killing vector fields on (F, gF)is finite dimensional, but the space of Killing vector fields on (RfF, dt

2

+ f2

(t)gF)maybe is infinite dimensional.

Nextly we consider M = IfF with the metric tensor f21 (t)dt2 + f2(t)gF whichgeneralizes the generalized Robertson-Walker spacetime. We note that Ka =

1f1(t)

t

is the base of the Killing vector fields on (I, f21 (t)dt2) and Ka = f1(t)dt. Similarto the discussions in [Sa], we get

Proposition 3.13 LetM = If F with the metric tensor f21 (t)dt2 + f2(t)gF andif M admits a non-trivial Killing vector field then

f

ff1

f2

f1= CF. When CF = 0,

then K can be expressed by

K =(x)

f1(t)

t+

(lnf)(t0)CFf1(t0)

+

tt0

f1f2du

gradF + T, (3.53)

where T is a Killing vector field on F and HessF + CFgF = 0. When CF = 0, if is a constant, then K = 0

f1(t)t + W

where W is homothetic. OtherwiseF is anon-zero parallel field.

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We recall the definition of the curl operator on semi-Riemannian manifolds, namely:if V is a vector field on a seni-Riemannian manifold M, then curlV is the antisym-

metric 2-covariant tensor defined by

curlV(X, Y) := gM(XV, Y) gM(YV, X), (3.54)

where X, Y (T M). A vector field V on a semi-Riemannian manifold M is saidto be non-rotating if curlV(X, Y) = 0 for all X, Y (T M). By the remark 5.1 in[DU], we know that V is non-rotating iff it is parallel. By Proposition 2.2, then forX, Y (T R) and V, W (T F)(1) XY = BXY.(2) XW = WX = X(f)f W.(3) VW = ff

f21gF(V, W) + FVW.

We take a Killing vector field

K =(x)

f1(t)

t+

tt0

f1f2dugradF + W, (3.55)

Then K is non-rotating iff t

K = 0 and VK = 0 for all V (T F). Directcomputations show that

t

(x)

f1(t)

t

= 2

f1(t)f21

(x)

t;

tW =

f

fW; (3.56)

t t

t0

f1f2dugradF = f1f

2gradF + t

t0

f1f2du

f

f

gradF. (3.57)

By t

K = 0 and (3.56), (3.57) and = 0, we obtain iff = 0, f1 = c0 is a constantand t

t0

f1f2dugradF + W =

f1

f fgradF, (3.58)

and

K =(x)

f1(t)

t f1

f fgradF. (3.59)

So

0 = VK = f

f f1V f1

f fFV(gradF). (3.60)

By HessF + CFgF = 0 and (3.60), we have (f)2 = f21 CF, f = f1CFt + r0.

Proposition 3.14 LetM = IfF with the metric tensor f21 (t)dt2 + f2(t)gF and ifK is a non-trivial non-rotating Killing vector field and f = 0, then f1 is a constantand f = f1

CFt + r0 and K can be expressed by

K =(x)

f1(t)

t+

gradF

f1CFt + r0. (3.61)

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4 Multiply twisted products with a semi-symmetric met-

ric connection

4.1 Preliminaries

Let M be a Riemannian manifold with Riemannian metric g. A linear connection on a Riemannian manifold M is called a semi-symmetric connection if the torsiontensor T of the connection

T(X, Y) = XY YX [X, Y] (4.1)

satisfiesT(X, Y) = (Y)X (X)Y, (4.2)

where is a 1-form associated with the vector field P on M defined by (X) =g(X.P). is called a semi-symmetric metric connection if it satisfies g = 0. If is the Levi-Civita connection of M, the semi-symmetric metric connection is givenby

XY = XY + (Y)X g(X, Y)P, (4.3)(see [Ya]). Let R and R be the curvature tensors of and respectively. Then Rand R are related by

R(X, Y)Z = R(X, Y)Z + g(Z, XP)Y g(Z, YP)X

+g(X, Z)YP g(Y, Z)XP + (P)[g(X, Z)Y g(Y, Z)X]

+[g(Y, Z)(X) g(X, Z)(Y)]P + (Z)[(Y)X (X)Y], (4.4)for any vector fields X , Y , Z on M [Ya]. By (4.3) and Proposition 2.2, we have

Proposition 4.1 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted productand let X, Y (T B) and U (T Fi), W (T Fj) and P (T B) . Then(1) XY = BXY.(2) XU = X(bi)bi U.(3) UX = [X(bi)bi + (X)]U.(4) UW = 0 if i = j.(5)

UW = U(lnbi)W+W(lnbi)U

gFi(U,W)

bigradFibi

bigFi(U, W)gradBbi+

FiUW

g(U, W)P if i = j.Proposition 4.2 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted productand let X, Y (T B) and U (T Fi), W (T Fj) and P (T Fk) . Then(1) XY = BXY g(X, Y)P.(2) XU = X(bi)bi U + g(P, U)X.(3) UX = X(bi)bi U.(4) UW = g(W, P)U if i = j.

19


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21/45

|gradBbi|2Bb2i

(g(V, W)U g(U, W)V) + (P)[g(U, W)V g(V, W)U] if i = j = k = l.(12)R(U, V)W = g(U, W)gradB(V(lnbi))

g(V, W)gradB(U(lnbi)) + R

Fi(U, V)W

|gradBbi|2Bb2i

(g(V, W)U g(U, W)V) + g(W,UP)V g(W,VP)U + g(U, W)VP g(V, W)UP+(P)[g(U, W)Vg(V, W)U]+[g(V, W)(U)g(U, W)(V)]P+(W)[(V)U(U)V] if i = j = k = l.

By proposition 4.3 and 4.4, we have

Proposition 4.5 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted productand let X , Y , Z (T B) and V (T Fi), W (T Fj) and P (T B). Then(1)Ric(X, Y) = Ric

B(X, Y) +

mi=1 li

HbiB(X,Y)

bi+ P(bi)

big(X, Y) + (P)g(X, Y)

+ g(Y,

XP)

(X)(Y)] .

(2)Ric(X, V) = Ric(V, X) = (li 1)[V X(lnbi)].(3)Ric(V, W) = 0 if i = j.(4)Ric(V, W) = RicFi(V, W) +

Bbibi

+ (li 1) |gradBbi|2B

b2i+

j=i lj

gB(gradBbi,gradBbj)bibj

+(n 2)(P) + nk=1 k < EkP, Ek > +j=i lj Pbjbj + (n + li 2)Pbibi g(V, W) if i =j,

where Ek, 1 k n is an orthonormal base of B with k = g(Ek, Ek) anddimB = n, dimM = n.

Corollary 4.6 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted product anddimFi > 1 and P (T B), then (M, ) is mixed Ricci-flat if and only if M can beexpressed as a multiply warped product. In particular, if (M, ) is Einstein, then Mcan be expressed as a multiply warped product.

Proposition 4.7 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted productand let X , Y , Z (T B) and V (T Fi), W (T Fj) and P (T Fr). Then(1)Ric(X, Y) = RicB(X, Y) +

mi=1 li

HbiB(X,Y)

bi+ g(X, Y)(P)(n 2)

+ g(X, Y)lrjr=1

jrg(Erjr P, Erjr).(2)Ric(X, V) = (li 1)[V X(lnbi)] + (n 2)X(br)br (V).(3)Ric(V, X) = (li 1)[V X(lnbi)] + (2 n)X(br)br (V).(3)Ric(V, W) = 0 if i = j.(4)Ric(V, W) = RicFi(V, W)+g(V, W) Bbibi + (li 1) |gradBbi|2Bb2i + j=i lj gB(gradBbi,gradBbj)bibj+(n 2)(P)] + (n 2)g(W,VP) + (2 n)(V)(W) + g(V, W)divFrP if i = j,

Corollary 4.8 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted product anddimFi > 1 and P (T Fr), then (M, ) is mixed Ricci-flat if and only if M can beexpressed as a multiply warped product and br is only dependent on Fr. In particular,if (M, ) is Einstein, then M can be expressed as a multiply warped product.

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Proposition 4.9 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted productand P (T B), then the scalar curvature S has the following expression:

S = SB

+ 2mi=1

li

biBbi +

mi=1

SFi

b2i+

mi=1

li(li 1) |gradBbi|2B

b2i

+mi=1

j=i

liljgB(gradBbi, gradBbj)

bibj+

mi=1

li(n + n + li 2)P(bi)bi

+mi=1

j=i

liljP(bj)

bj+

mi=1

li(n + n 3)(P) + 2mi=1

lidivBP. (4.5)

Proposition 4.10 LetM = B b1 F1 b2 F2 bm Fm be a multiply twisted productand P (T Fr), then the scalar curvature S has the following expression:

S = SB + 2mi=1

li

biBbi +

mi=1

SFi

b2i+

mi=1

li(li 1) |gradBbi|2B

b2i

+mi=1

j=i

liljgB(gradBbi, gradBbj)

bibj+ (P)(n 1)(n 2) + 2(n 1)divFrP. (4.6)

4.2 Special multiply warped product with a semi-symmetric connection

Let M = Ib1 F1b2 F2 bm Fm be a multiply warped product with the metrictensor dt2 b21gF1 b2mgFm and I is an open interval in R and bi C(I).

Theorem 4.11 LetM = Ib1 F1b2 F2 bm Fm be a multiply warped product withthe metric tensor dt2 b21gF1 b2mgFm and P = t . Then (M, ) is Einsteinwith the Einstein constant if and only if the following conditions are satisfied forany i {1, , m}(1)(Fi, Fi) is Einstein with the Einstein constant i, i {1, , m}.(2)

mi=1 li

bibi

bibi

= .

(3)i bibi (li 1)b2i + (b2i bibi)j=i ljbjbj

+ (2 n)b2i + (n + li 2)bibi = b2i .

Proof. By Proposition 4.5, we have

Ric

t,

t

=

mi=1

li

bibi

bi

bi

; (4.7)

Ric

t, V

= Ric

V,

t

= 0; (4.8)

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Ric(V, W) = RicFi(V, W) + gFi(V, W)

bibi (li 1)b2i + (b2i bibi)

j=ilj

bjbj

+(2 n)b2i + (n + li 2)bibi

. (4.9)

By (4.7)-(4.9) and the Einstein condition, we get the above theorem.

Theorem 4.12 LetM = Ib1 F1b2 F2 bm Fm be a multiply warped product withthe metric tensor dt2 b21gF1 b2mgFm and P (T Fr) with gFr(P, P) = 1and n > 2. Then (M, ) is Einstein with the Einstein constant if and only if thefollowing conditions are satisfied for any i {1, , m}(1)(Fi, Fi) (i = r) is Einstein with the Einstein constant i, i {1, , m}.(2)br is a constant and

mi=1 li

bibi

= 0; divFrP = 1, 0 1 + = (2 n)b2r, where0, 1 are constants.

(3)RicFr(V, W)+gFr(V, W) = (n2) [(V)(W) g(W,VP)] , for V,W (T Fr).(4)i bibi + (n 2)b2i b2r bibi

j=i lj

bjbj

(li 1)b2i = ( 1)b2i .

Proof. By Proposition 4.7 (2) and gFr(P, P) = 1, we have br is a constant. ByProposition 4.7, then ,

Ric

t,

t

=

mi=1

libibi

+ (2 n)b2r divFrP = ; (4.10)

By variables separation, we have

mi=1

libi

bi = 0, divFrP = 1. 0 1 + = (2 n)b2r, . (4.11)

Ric(V, W) = RicFi(V, W) + b2i gFi(V, W)

bibi

+ (li 1)b2i

b2i+

j=i

ljbibj

bibj

+(n 2)(P)] + (n 2)g(W,VP) + (2 n)(V)(W) + g(V, W)divFrP. (4.12)When i = r, then VP = (V) = 0, so

Ric(V, W) = RicFi(V, W) + b2i

gFi

(V, W)bi

bi+ (l

i 1)

b2i

b2i+ j=i lj

bibjbibj

+(n 2)b2r

+ 1b2i gFi(V, W) = b

2i gFi(V, W). (4.13)

By variables separation, we have (Fi, Fi) (i = r) is Einstein with the Einsteinconstant i and

i bibi + (n 2)b2i b2r bibij=i

ljbjbj

(li 1)b2i = ( 1)b2i . (4.14)

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When i = r and br is a constant, then

RicFi(V, W) + b2r

[(n

2)b2r

+ 1

]gFi

(V, W) = (n

2) [(V)(W)

g(W,V

P)] .(4.15)

So we prove the above theorem.

When M = Ib1 F1 b2 F2 bm Fm be a multiply warped product and P = t ,by Proposition 4.9, we have

S = 2mi=1

libibi

+mi=1

SFi

b2i+

mi=1

li(li 1)b2i

b2i+

mi=1

j=i

liljbibj

bibj

+m

i=1 li(n + li 1)bibi

+m

i=1 j=i liljbjbj

m

i=1 li(n 2). (4.16)The following result just follows from the method of separation of variables and thefact that each SFi is function defined on Fi.

Proposition 4.13 LetM = Ib1 F1 b2 F2 bm Fm be a multiply warped productand P =

t. If (M, ) has constant scalar curvature S, then each (Fi, Fi) has con-

stant scalar curvature SFi.

When P (T Fr), by Proposition 4.10, we have

S = 2mi=1

libibi +

mi=1

SFi

b2i +

mi=1

li(li 1)b2i

b2i

+mi=1

j=i

liljbibj

bibj+ (P)(n 1)(n 2) + 2(n 1)divFrP. (4.17)

Proposition 4.14 LetM = Ib1F1b2F2 bmFm be a multiply warped product andP (T Fr). If (M, ) has constant scalar curvature S, then each (Fi, Fi) (i = r)has constant scalar curvature SFi and if gFr(P, P) and divFrP are constants, thenSFr is also a constant.

4.3 Generalized Robertson-Walker spacetimes with a semi-symmetric met-ric connection

In this section, we study M = I F with the metric tensor dt2 + f(t)2gF. Asa corollary of Theorem 4.11, we obtain:

Corollary 4.15 Let M = I F with the metric tensor dt2 + f(t)2gF and P = t .Then (M, ) is Einstein with the Einstein constant if and only if the following

24


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conditions are satisfied(1)(F, F) is Einstein with the Einstein constant F.(2) l ff ff = .(3)F f f + (1 l)f2 + (1 l )f2 + (2l 1)ff = 0.

Remark. In Theorem 5.1 in [SO], they got the Einstein condition of M = IF witha semi-symmetric metric connection, but they did not consider the above conditions(2) and (3).

Corollary 4.16 Let M = I F with the metric tensor dt2 + f(t)2gF and P = tand dimF = 1. Then (M, ) is Einstein with the Einstein constant if and only iff = f f.

By Corollary 4.15 (2) and (3), we get

Corollary 4.17 Let M = I F with the metric tensor dt2 + f(t)2gF and P = tand dimF > 1. Then (M, ) is Einstein with the Einstein constant if and only ifthe following conditions are satisfied(1)(F, F) is Einstein with the Einstein constant F.(2) f = f

lf.

(3) F1l + f2 + (1 + l )f

2 2f f = 0.

By Corollary 4.16 and elementary methods for ordinary differential equations, weget

Theorem 4.18 Let M = I F with the metric tensor dt2 + f(t)2gF and P = tand dimF = 1. Then (M, ) is Einstein with the Einstein constant if and only if(1) < 14 , f(t) = c1e

1+142

t + c2e1

142

t,

(2) = 14 , f(t) = c1e12t + c2te

12t,

(3) > 14 , f(t) = c1e12tcos

412 t

+ c2e

12tsin

412 t

,

Let l = d0,F1l = d0,

1+14d02 = a0,

114d02 = b0, then a0 + b0 = 1, d0 = a0b0.

When dimF > 1, by Corollary 4.17 (2)

Case i) d0 14 , then f(t) = c1e

12tcos(h0t) + c2e

12tsin(h0t), where h0 =

4d012 . By

Corollary 4.17 (3), then

d0 + et

(

c1

2+ c2h0)cos(h0t) + (

c2

2 c1h0)sin(h0t)

2+(1 + d0)(c1cos(h0t) + c2sin(h0t))

2

2(c1cos(h0t) + c2sin(h0t))

(c1

2+ c2h0)cos(h0t) + (

c2

2 c1h0)sin(h0t)

= 0.

(4.20)Consider the coefficients of cos2(h0t)e

t and sin2(h0t)et, we get

(1

4+ d0)c

21 + c

22h

20 c1c2h0 = 0; (

1

4+ d0)c

22 + c

21h

20 + c1c2h0 = 0. (4.21)

Plusing the above two equalities ,then 14 + d0 + h20 = 0 and d0 = 0. There is a contra-

diction with d0 >14 and in this case we have no solutions. So we obtain the following

theorem.

Theorem 4.19 Let M = I F with the metric tensor dt2 + f(t)2gF and P = tand dimF > 1. Then (M, ) is Einstein with the Einstein constant if and onlyif = 0 and f = c1e

t+c2 and (F, F) is Einstein with the Einstein constant (l1)c22.

By (4.16) and (4.17), we have

Corollary 4.20 Let Let M = IF with the metric tensordt2+f(t)2gF andP = t .If (M, ) has constant scalar curvature S if and only if (F, F) has constant scalarcurvature SF and

S =SF

f2

2l

f

f l(l

1)

f2

f2

+ 2l2f

f

+ (1

l)l. (4.22)

Corollary 4.21 Let Let M = I F with the metric tensor dt2 + f(t)2gF andP (T F) and gF(P, P) = c0, divFrP = c0. If (M, ) has constant scalar curvatureS if and only if (F, F) has constant scalar curvature SF and

S =SF

f2 2l f

f l(l 1)f

2

f2+ c0(l 1)lf2 + 2c0l. (4.23)

26


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by the elementary methods for ordinary differential equations, we prove the abovetheorem.

When dimF = l = 3 and SF = 0, putting v(t) = w(t) 4l+1 , then w(t) satisfies theequation

w lw + (l + 1)4

(l 1 + Sl

)w (l + 1)4

SF

lw1 4

l+1 = 0. (4.27)

4.4 Generalized Kasner spacetimes with a semi-symmetric metric connec-tion

In this section, we consider the scalar and Ricci curvature of generalized Kasnerspacetimes with a semi-symmetric metric connection. We recall the definition of gen-eralized Kasner spacetimes ([DU1]).

Definition 4.24 A generalized Kasner spacetime (M, g) is a Lorentzian multiplywarped product of the form M = I p1 F1 pm Fm with the metric g =dt2 2p1gF1 2pmgFm , where : I (0, ) is smooth and pi R, for anyi {1, , m} and also I = (t1, t2).

We introduce the following parameters =mi=1 lipi and =

mi=1 lip

2i for gen-

eralized Kasner spacetimes. By Theorem 4.11 and direct computations, we get

Proposition 4.25 LetM = I

p1 F1

pm Fm be a generalized Kasner space-

time and P = t . Then (M, ) is Einstein with the Einstein constant if and onlyif the following conditions are satisfied for any i {1, , m}(1)(Fi, Fi) is Einstein with the Einstein constant i, i {1, , m}.(2)

( )2

2= .

(3) i2pi

pi

( 1)pi

2

2+ [ + (n 2)pi]

= n + 2.

By (4.16) we obtain

Proposition 4.26 LetM = Ip1 F1 pm Fm be a generalized Kasner spacetimeandP =

t. Then(M, ) has constant scalar curvatureS if and only if each(Fi, Fi)

has constant scalar curvature SFi and

S =mi=1

SFi

2pi 2

( + 2 2)

2

2+ 2(n 1)

+ (2 n)(n 1). (4.28)

Nextly, we first give a classification of four-dimensional generalized Kasner space-times with a semi-symmetric metric connection and then consider Ricci tensors andscalar curvatures of them.

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Definition 4.27 Let M = Ib1 F1 bm Fm with the metric g = dt2 b21gF1 b2mgFm. (M, g) is said to be of Type (I) if m = 1 and dim(F) = 3. (M, g) is said to be of Type (II) if m = 2 and dim(F1) = 1 and dim(F2) = 2. (M, g) is said to be of Type (III) if m = 3 and dim(F1) = 1, dim(F2) = 1 anddim(F3) = 1.

By Theorem 4.19 and 4.22, we have given a classification of Type (I) Einsteinspaces and Type (I) spaces with the constant scalar curvature.

Classification of Einstein Type (II) generalized Kasner space-times witha semi-symmetric metric connection

Let M = Ip1 F1 p2 F2 be an Einstein type (II) generalized Kasner spacetimeand P =

t

. Then = p1 + 2p2, = p2

1

+ 2p2

2

. By Proposition (4.25), we have

( )

2

2= , (4.29i)

p1

( 1)p1

2

2+ [ + 2p1]

= + 2, (4.29ii)

2

2p2p2

( 1)p2

2

2+ [ + 2p2]

= + 2, (4.29iii)

where 2 is a constant. Consider following two cases:

Case i) = 0In this case, p2 = 12p1, = 32p21. Then by (4.29), we have

2

2= , (4.30i)

p1

+

2

2+ 2

= + 2, (4.30ii)

2

p1 1

2p1

+

2

2+ 2

= + 2, (4.30iii)

Case i a) = 0,

then pi = 0, by (4.30i), = 0. By (4.30ii), + 2 = 0, this is a contradiction.Case i b) = 0,then pi = 0.Case i b)1) 2 = 0by (4.30ii) and (4.30iii), = 2 and

+

2

2+ 2

= 0,

2

2=

2

, (4.31)

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31/45

If p1 = 0, then = 2p2, = 2p22 and b =

(+2)2

, so = 4 and b = 1. By (4.32iii),we get 2 = 0 and 2b2 + 4b = + 2 which is a contradiction.If 2 = 0, by (4.32ii) and b =

(+2)2 , we get = 0 or 2. When = 0, then

b = 22

= 0, this is a contradiction. There is a similar contradiction for = 2.Case ii)(1)(a)3) b = 0,then = c2, by (4.32i), = 0. By (4.32ii), = 2, this is a contradiction.

Case ii)(1)(b) c2 = 0,then

a ( + 2)2

c1e

at =p12

(p2 p1)2 (c1eat)1

2p2 . (4.36)

Case ii)(1)(b)1) p12 = 0,then p2 = 0 and = 2p2, = 2p

22 and a =

1+142

and 2 = + 2

1+142

. By(4.32ii), then a = 1 and = 0, so 2 = 1 and = c1et and satisfies (4.32iii). In

this case, we get p2 = 0, p1 = 0, = c0etp1 , = 0, 2 = 1.

Case ii)(1)(b)2) p12 = 0,

if p1 = 0, then = 2p2, = 2p22 and = c1e

at and a = (+2)2

, so = 0 and

a = 1. By (4.32iii), we get 2 = 0 and satisfies (4.23ii) and (4.32iii). In this case,

p1 = 0, p2 = 0 = 0, 2 = 0, = c0etp2 .

If 2 = 0, by (4.32ii) and a =(+2)2

, then = 0 and a = 22

= 1. By (4.32iii), then

2 = 0 and satisfies (4.23ii) and (4.32iii). In this case, p1 = 0, p2 = 0, = 2 = 0,p1 = 4p2, = c0e

t3p2 .

Case ii)(1)(c) c1 = 0, c2 = 0, b = 0,If p2 = 0, then eat, ebt, (c1eat + c2ebt)1

2p2 are linear independent, by (4.34), then

a ( + 2)2

c1 = 0,

b ( + 2)

2

c2 = 0,

p12

(p2 p1)2 (c1eat)

1 2p2 . (4.37)

So a = b = (+2)2

, this is a contradiction.

If p2 = 0, then by (4.34),

a

( + 2)

2

p12

(p2

p1

)2= 0, b

( + 2)

2

p12

(p2

p1

)2= 0, (4.38)

so a = b and we get a contradiction.Case ii)(1)(d) c1 = 0, c2 = 0, b = 0,When 1 2p2

= 0, we have similar discussions. When 1 2p2

= 0, we have (+2)

2=

1. By b = 0, then = 0 and 2 = 2 = 4p2, so 4p2 = and p1 = 2p2. But = 2p2,then p1 = p2 = 0. This is a contradiction.

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33/45

Then c21 + c22 = 0. This is a contradiction. By the above discussions, we get the

following theorem:

Theorem 4.28 Let M = I p1 F1 p2 F2 be a generalized Kasner spacetime anddimF1 = 1, dimF2 = 2 and P =

t

. Then (M, ) is Einstein with the Einsteinconstant if and only if (F2, F2) is Einstein with the Einstein constant 2, and oneof the following conditions is satisfied

(1) p2 = 0, p1 = 0, = c0etp1 , = 0, 2 = 1.

(2) p1 = 0, p2 = 0 = 0, 2 = 0, = c0etp2 .

(3) p1 = 0, p2 = 0, = 2 = 0, , p1 = 4p2, = c0et

3p2 .

Type (II) generalized Kasner space-times with a semi-symmetric metricconnection with constant scalar curvature

By Proposition 4.26, then (F2, F2) has constant scalar curvature SF2 and

S =SF2

2p2 2

( + 2 2)

2

2+ 6

6. (4.45)

If = 0, when = 0, then p1 = p2 = 0 and S = SF2 6. If = 0, then

2

2=

SF2

2p2 (S+ 6). (4.46)

If = 0, putting = 2

+2 , we get

42

+ 2 +

122

+ 2 (S+ 6) + SF21

4p2

+2 = 0. (4.47)

Type (III) generalized Kasner space-times with a semi-symmetric metricconnection with constant scalar curvature

By Proposition 4.26, then

S = 2

( + 2 2)

2

2+ 6

6. (4.48)

If = = 0, then p1 = p2 = p3 = 0, we get S =

6.

If = 0, = 0, then [(ln)]2 = S+6

, so when S+ 6 > 0, there is no solutions, when

S + 6 = 0, is a constant and when S+ 6 < 0, = c0eS+6

t.

If = 0, then = 0, putting = 2

+2 , then

3 + (S + 6)( + 2)

42 = 0. (4.49)

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35/45

and = c0e

2t. But by (4.50ii), then

+ ( 1)2

2 2 = 0, this is a contradic-

tion.

= 0. If p1 = p2 = p3, we get type (I), so we may let p1 = p2. By (4.50ii) and(4.50iii), we have (

)

= + 2 and (

)

2 ( )

= 0, so = c0e

(+2)t and = 2or 0. When = 2, is a constant, by (4.50i), = 0, this is a contradiction. When = 0, = c0e

2t

2 and = 0, so 22

= 1. In this case, we get when pi = pj forsome i, j {1, 2, 3}, = 0, 2

2= 1, = c0e

2t , . We get the following theorem.

Theorem 4.30 Let M = I p1 F1 p2 F2 p3 F3 be a generalized Kasner space-time for pi = pj for some i, j {1, 2, 3} and dimF1 = dimF2 = dimF3 = 1,and P =

t. Then (M, ) is Einstein with the Einstein constant if and only if

= 0, 22

= 1, = c0e2t .

5 Multiply twisted product Finsler manifolds

In this section, we set (Mi, Fi) is a Finsler manifold for 0 i b and fi :M0 Mi R is a smooth function for 1 i b Let i : T Mi Mi be theprojection map. The product manifold M0 M1 Mb endowed with the metricF : T M0 T M01 T M0b R is considered,

F(v0, v1, , vm)

= F02(v0) + f21 (0(v0), 1(v1))F12(v1) + + f2m(0(v0), m(vm))Fb2(vb), (5.1)where T M0i = T Mi{0}. Let dimMi = mi, for 0 i b and (x1i , , xmii , y1i , , ymiiis the local coordinate on T Mi. Let

gij =1

2

2F2

yiyj, (5.2)

and gij be the inverse of gij . For a Finsler manifold (M, F), a global vector field inintroduced by F on T M0, which in a standard coordinate (xi, yi) for T M0 is givenby G = yi

xi 2Gi(x, y)

yi, where

Gi :=1

4gil

2F2

xky lyk F

2

xl

. (5.3)

Let

Gij =Gi

yj; Gijk =

2Gi

yjyk; Bijkl =

3Gi

yjyky l; Ejk =

1

2Bijki. (5.4)

F is called a Berwald metric and weakly Berwald metric if Bijkl = 0 Ejk = 0 respec-

tively. Let Cijk =12gijyk

. A Finsler metric F is said to be isotropic mean Berwaldmetric if its mean Berwald curvature is in the following form

Eij =1

2(n + 1)cF1hij , (5.5)

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where hij = gij F2yiyj is the angular metric and c = c(x) is a scalar function onM. Let Lijk =

12 ylB

lijk . A Finsler metric is called a Landsgerg metric if Lijk = 0.

A Finsler metric is said to be relatively isotropic Landsberg metric if it satisfiesLijk = cF Cijk , where c = c(x) is a scalar function on M. Let Ji = gjkLijk and

Ii = gjkCijk . A Finsler metric is called a weakly Landsberg metric if Ji = 0. A

Finsler metric is said to be relatively isotropic mean Landsberg metric if Ji = cF Iifor some scalar function c = c(x) on M. A Finsler metric is said to be locally duallyflat if (F2)xkyly

k = 2(F2)xl. A Finsler manifold (M, F) is called a locally Minkowskimanifold if on each coordinate neighborhood ofT M, F is a function of (yi) only. LetIi = g

jkCijk and hij = gij 1F2 gipypgiqyq and Mijk = Cijk 1n+1(Iihjk +Ijhik+Ikhij).A Finsler metric F is said to be C-reducible if Mijk = 0. Direct computations showthat

Gj = (G0)j

1

4

b

i=1m0

k=1 gjk0

f2i

x

k

0

F2i , for 1

j

m0, (5.6)

where (Gl)j is the geodesic coefficient of (Ml, Fl) and (gl)

jk = 122F2lyily

jl

for 0 l b.

Gm0++ml1+j = (Gl)j +1

2f2lyh0y

jl

f2lxh0

+1

2f2lyrl y

jl

f2lxrl

14f2l

(gl)js f

2l

xslF2l , (5.7)

for 1 l b, 1 j ml. In the following, we write l 1 = m0 + + ml1.

Gjk = (G0)

jk

1

4

bi=1

m0k=1

gjs0

yk0

f2ixs0

F2i , (5.8)

Gj

r1+k = 1

2

m0s=1

gjs0

f2rxs0

(gr)ktytr, (5.9)

Gl1+jk =

1

2f2lyjl

f2lxk0

, (5.10)

Gl1+jr1+k =

lr(Gl)

jk +

1

2f2lyh0

f2lxh0

kj lr +

lrLjk, (5.11)

where

Lj

k=

1

2f2lk

jyr

l

f2l

xrl+

1

2f2lyj

l

f2l

xkl 1

4f2l

f2l

xsl (gl)js

yklF2

l+ 2(g

l)js(g

l)kt

yt

l .Gkij = (G0)

kij

1

4

bt=1

m0k=1

2gks0

yi0yj0

f2txs0

F2t , (5.12)

Gkr1+i,j =

1

2

m0s=1

gks0

yj0

f2rxs0

(gr)itytr, (5.13)

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El

1+i,j = Ej,l

1+i =

1

4

m0

s=12gks0

yj

0yk0

f2l

xs

0

(yl)i, (5.36)

El1+i,r1+j = 1

4

(g0)ks

yk0

f2rxs0

lr(gr)ij + lr(El)ij

1

8f2l

f2l

xl

lr3[(gl)

kF2l ]

y ilyjl y

kl

. (5.37)

By (5.35)-(5.37), similar to Theorem in [PTN], we have

Theorem 5.3 M0 f1 M1 fb Mb is a weakly Berwald manifold if and only if(M0, F0) is a weakly Berwald manifold and

(g0)ks

yk0

f2lxs0

= 0 and

(El)ij =1

8f2l

f2

lx

l

3[(gl)kF2

l

]

yilyjl y

kl

, (5.38)

for 1 l b.

By (5.5) and (5.35)-(5.37), then

Lemma 5.4 M0 f1 M1 fb Mb has isotropic mean Berwald curvature if andonly if

(E0)ij 18

b

t=1m0

k=13gks0

yi0yj0y

k0

f2txs0

F2t =1

2(n + 1)cF1[(g0)ij F2(y0)i(y0)j ], (5.39)

14

m0s=1

2gks0

yj0y

k0

f2lxs0

(yl)i = n + 12

cF3f2l (yl)i(y0)j , (5.40)

14

(g0)ks

yk0

f2rxs0

lr(gr)ij + lr(El)ij

1

8f2l

f2l

xl

lr3[(gl)

kF2l ]

yilyjl y

kl

=n + 1

2cF1[f2l

lr(gl)ij F2f2l f2r (yl)i(yr)j ]. (5.41)

By Lemma 5.4, similar to Theorem 3 in [PTN], we get

Theorem 5.5 M0 f1 M1 fb Mb with isotropic mean Berwald curvature is aweakly Berwald manifold.

By (5.29)-(5.34), we have

Lijq = (L0)ijq +1

8

bt=1

m0k=1

3gks0

y i0yj0y

q0

f2txs0

F2t (y0)k, (5.42)

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Lr1+i,j,q =

1

4

m0

s=12gks0

yj

0yq

0

f2r

xs

0

(y0)k(yr)i, (5.43)

Ll1+i,r1+j,q = 0, (5.44)

Ll1+i,r1+j,t1+q =1

2ys0

f2rxs0

lrlt(Cr)ijq + f

2l lrlt(Ll)ijq +

1

8(yl)k

f2l

xl

lrlt

3[(gl)kF2l ]

y ilyjl y

ql

.

(5.45)By (5.42)-(5.45), similar to Theorem 6 in [PTN], we have

Theorem 5.6 If there is a point pi M0 such that f2i

xj0(pi,xi)

= 0 for some j {1, , m0}, then M0 f1 M1 fb Mb is a Landsberg manifold if and only if

(M0, F0) is Landsberg, Mi is Riemannian for 1 i b and3gks0

yi0yj0y

q0

f2rxs0 (y0)k = 0.

Similar to Theorem 6 in [PTN], we have

Theorem 5.7 M0 f1 M1 fb Mb is a relatively isotropic Landsberg manifold,then M is a Landsberg manifold.

By (5.42)-(5.45), then

Ji = (J0)i +1

8

bt=1

m0k=1

3gks0

y i0yj0y

q0

f2txs0

F2t (g0)jq(y0)k, (5.46)

Jl1+i = (Jl)i + 14

m0s=1

2gks0

yj0y

q0

f2lxs0

(y0)k(yl)igjq0

+1

2f2lys0

f2lxs0

(gl)jq (Cl)ijq +

1

8f2l(gl)

jq(yl)kf2l

xl

3[(gl)kF2l ]

yilyjl y

ql

. (5.47)

Similar to Theorem 8 in [PTN], by (5.46) and (5.47) we have

Theorem 5.8 If there is a point pi M0 such that f2i

xj0(pi,xi)

= 0 for some j {1, , m0}, then M0 f1 M1 fb Mb is a weakly Landsberg manifold if and onlyif (M0, F0) is a weakly Landsberg manifold, (Mi, Fi) is Riemannian for 1 i b and

(y0)k(g0)

jq 3(g0)

ks

yi0yj0yq0

f2l

xs0 = 0, 1 l b.Similar to Theorem 9 in [PTN], we have

Theorem 5.9 M0 f1 M1 fb Mb is a relatively isotropic mean Landsberg man-ifold, then (M, F) is a weakly Landsberg manifold.

Similar to Theorem 10 in [PTN], we have

40


41/45

Theorem 5.10 M0 f1 M1 fb Mb is locally dually flat if and only if (M0, F0)is locally dually flat and fl = fl(xl) and (Mi, fiFi) is locally dually flat for 1 l b.

By Lemma 2 and Theorem 4 in [HR] and (5.25), we get

Theorem 5.11 M0 f1 M1 fb Mb is a locally Minkowski manifold if and onlyif fl = fl(xl) and (M0, F0), (Mi, fiFi) are locally Minkowski manifolds for 1 l b.

Similar to Theorem 1 in [PT], we have

Theorem 5.12 M0 f1 M1 fb Mb is C-reducible, then it is a Riemannianmanifold.

Let

Rcab =Gcaxb

Gcbxa

, (5.48)

yi0 = dyi0 + G

ijdx

j0 +

bl=1

Gil1+jdx

jl (5.49)

yir = dyil + G

r1+ij dx

j0 +

bl=1

Gr1+il1+jdx

jl . (5.50)

Then the multiply twisted Miron metric on T M0 can be introduced as follows:

G = (g0)ijdxi0dx

j

0+

bl=1

f2l (gl)ijdx

ildx

jl+(F

2

)(g0)ijyi0y

j

0+(F2

)

bl=1

f2l (gl)ijy

ily

jl .

(5.51)

Proposition 5.13LetM = M0 f1 M1 fb Mb, then the Levi-Civita connection on the Riemannian manifold (T M0, G) is locally expressed as follows:

xi0

xj0

= Fsij

xs0+

1

2Rsij

1

(F2)(C0)

sij

ys0+Fr1+tij

xtr+

1

2Rr1+tij

ytr, (5.52)

xi0

yj0 = (C0)sij + 12(g0)sk(g0)cjRcki(F2) xs0 + Fsij ys0+

1

2f2r(gr)st

f2r G

r1+lij (gr)ls Gli,r1+s(g0)lj

ysr

+1

2f2r(gr)

st(g0)kjRkr1+s,i(F

2)

xsr,

(5.53)

y

j0

xi0=

xi0

yj0

Glij

y l0 Gr1+lij

y lr, (5.54)

41


42/45


43/45


44/45

Proposition 5.14Let M = M0 f1 M1 fb Mb, then V T M0 is totally geodesicif and only if Fcab = G

cab.

Proposition 5.15LetM = M0 f1 M1 fb Mb, then HT M0 is totally geodesicif and only if (Mi, Fi) is Riemannian for 0 i b and Rcab = 0

Acknowledgement. This work was supported by Fok Ying Tong Education Foun-dation No. 121003.

References

[ARS]L. Alas, A. Romero, M. Sanchez, Spacelike hypersurfaces of constant meancurvature and Clabi-Bernstein type problems, Tohoku Math. J. 49(1997) 337-345.[As1]G. Asanov, Finslerian extensions of Schwaraschild metric, Fortschr. Phys. 40(1992)667-693.[As2]G. Asanov, Finslerian metric functions over the product R M and their po-tential applications, Rep. Math. Phys. 41(1998) 117-132.[BO]R. Bishop, B. ONeill, Manifolds of negative curvature, Trans. Am. Math. Soc.145(1969) 1-49.[BGV]M. Brozos-Vazquez, E. Garca-Ro, R. Vazquez-Lorenzo, Some remarks on lo-cally conformally flat static space-times, J. Math. Phys. 46[CK]J. Choi, M. Kim, The index form on the multiply warped spacetime, Bull. Ko-rean Math. Soc. 41(2004) No.4 691-697.[DD]F. Dobarro, E. Dozo, Scalar curvature and warped products of Riemannian man-ifolds, Trans. Am. Math. Soc. 303(1987) 161-168.

[DU1]F. Dobarro, B. Unal, Curvature of multiply warped products, J. Geom. Phys.55(2005) 75-106.[DU2]F. Dobarro, B. Unal, Characterizing Killing vector fields of standard staticspace-time, J. Geom. Phys. 62(2012) 1070-1087.[EJK]P. Ehrlich, Y. Jung, S. Kim, Constant scalar curvatures on warped productmanifolds, Tsukuba J. Math. 20(1996) No.1 239-265.[EK]P. Ehrlich, S. Kim, The index form of a warped product, preprint.[FGKU]M. Fernandez-Lopez, E. Garca-Ro, D. Kupeli, B. Unal, A curvature condi-tion for a twisted product to be a warped product, Manu. math. 106(2001), 213-217.[FS]J. Flores, M. Sanchez, Geodesic connectedness of multiwarped spacetimes, J. Diff.Eqs. 186(1)(2002) 1-30.

[Ha]H. Hayden, Subspace of a space with torsion, Proc. Lond. Math. Soc. 34(1932)27-50.[HR]A. Hushmandi, M. Rezaii, On warped product Finsler spaces of Landsberg type,J. Math. Phys. 52[KPV]L. Kozma, I. Peter, c. Varga, Warped product of Finsler manifolds, Ann. Univ.Sci. Budapest 44(2001)[PT]E. Peyghan, A. Tayebi, On doubly warped product Finsler manifolds, NonlinearAnalysis: Real world Applications 13(2012) 1703-1720.

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[PTN]E. Peyghan, A. Tayebi, B. Najafi, Doubly warped product Finsler manifoldswith some non-Riemannian curvature properties, arXiv:1110.6826.

[Sa]M. Sanchez, On the geometry of generalized Robertson-Walker spacetimes: cur-vature and Killing fields, J. Geom. Phys. 31(1999) 1-15.[SO]S. Sular, C. Ozgur, Warped products with a semi-symmetric metric connection,Taiwanese J. Math. 15(2011) no.4 1701-1719.[U1]B. Unal, Multiply warped products, J. Geom. Phys. 34(2000) 287-301.[U2]B. Unal, Doubly warped products, Diff. Geom. Appl. 15(2001) 253-263.[Ya]K. Yano, On semi-symmetric metric connection, Rev. Roumaine Math. PuresAppl. 15(1970) 1579-1586.

School of Mathematics and Statistics, Northeast Normal University, ChangchunJilin, 130024, China

E-mail: [email protected]

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