S. L. Drut¸˘
a-Romaniuc and V. Oproiu
Abstract.We study some almost contact metric structures on the tangent sphere bundles, induced from some almost Hermitian structures of natural diagonal lift type on the tangent bundle of a Riemannian manifold (M, g). The above almost contact metric structures are not automatically contact metric structures. In order to get such properties we made some rescalings of the metric, of the fundamental vector field, and of the 1-form. Then we gave the characterization of the Sasakian structures on the tangent sphere bundles. In this case, the base manifold must be of constant sectional curvature. For the obtained Sasakian manifolds we got the condition under which they areη - Einstein.
M.S.C. 2010: 53C05, 53C15, 53C55.
Key words: natural lift, tangent sphere bundle, almost contact structure, Sasakian structure,η−Einstein manifold.
1
Introduction
In the last years, many researchers have been exhibiting a great interest in investi-gating the geometry of tangent sphere bundles of constant radius (see e. g. the papers [1], [2], [4], [5], [7], [10], [13], [15]).
The tangent sphere bundles TrM of constant radius r are hypersurfaces of the
tangent bundles, obtained by considering only the tangent vectors which have the norm equal tor. Every almost Hermitian structure from the tangent bundle induces an almost contact structure on the tangent sphere bundle of constant radius r. In papers such as [4], [5] and [15], the metric considered on the tangent bundle T M
was the Sasaki metric, but then E. Boeckx remarked that the unit tangent bundle equipped with the induced Cheeger-Gromoll metric is isometric to the tangent sphere bundle of radius √1
2, endowed with the metric induced by the Sasaki metric. This suggested to O. Kowalski and M. Sekizawa the idea that the tangent sphere bundles with different constant radii and with the metrics induced from the Sasaki metric might possess different geometrical properties, and they showed how the geometry of the tangent sphere bundles depends on the radius. All the important results obtained by the two authors in the field of the Riemannian geometry of the tangent sphere
∗
Balkan Journal of Geometry and Its Applications, Vol.16, No.2, 2011, pp. 48-61. c
bundles with arbitrary constant radius, done since 2000, may be found in the survey [10] from 2008.
In the present paper we study some geometric properties of the tangent sphere bundle of constant radiusTrM, by considering on the tangent bundleT M of a
Rie-mannian manifold (M, g), a natural almost complex structureJ and a natural metric
G, both of them obtained by the second author in [14] as diagonal lifts of the Rieman-nian metric g from the base manifold. We determine the almost contact structure on the tangent sphere bundles of constant radiusr, induced by the almost Hermitian structure of natural diagonal lift type from the tangent bundle, we get the conditions under which these structures are Sasakian, then we find the conditions under which the determined Sasakian tangent sphere bundles areη- Einstein. Extensive literature concerning Einstein equations can be mentioned from different perspectives (e. g. see [3], [8], [11]).
The manifolds, tensor fields and other geometric objects considered in this paper are assumed to be differentiable of classC∞ (i.e. smooth). The Einstein summation convention is used throughout this paper, the range of the indices h, i, j, k, l, m, r,
being always{1, . . . , n}.
2
Preliminary results
Let (M, g) be a smooth n-dimensional Riemannian manifold and denote its tangent bundle byτ : T M → M. The total spaceT M has a structure of a 2n-dimensional smooth manifold, induced from the smooth manifold structure ofM. This structure is obtained by using local charts on T M induced from usual local charts on M. If (U, ϕ) = (U, x1, . . . , xn) is a local chart on M, then the corresponding induced
local chart onT M is (τ−1(U),Φ) = (τ−1(U), x1, . . . , xn, y1, . . . , yn), where the local
coordinatesxi, yj, i, j= 1, . . . , n, are defined as follows. The firstnlocal coordinates
of a tangent vectory ∈τ−1(U) are the local coordinates in the local chart (U, ϕ) of its base point, i.e. xi=xi◦τ, by an abuse of notation. The lastnlocal coordinates
yj, j = 1, . . . , n, ofy ∈ τ−1(U) are the vector space coordinates of y with respect to the natural basis inTτ(y)M defined by the local chart (U, ϕ). Due to this special structure of differentiable manifold forT M, it is possible to introduce the concept of
M-tensor field on it (see [12]).
Denote by ˙∇the Levi Civita connection of the Riemannian metricgonM. Then we have the direct sum decomposition
(2.1) T T M =V T M⊕HT M
of the tangent bundle toT M into the vertical distributionV T M = Ker τ∗ and the horizontal distributionHT M defined by ˙∇(see [20]). The vertical and horizontal lifts of a vector fieldX onM will be denoted byXV andXHrespectively. The set of vector
fields{ ∂ ∂y1, . . . ,
∂
∂yn} onτ−
1(U) defines a local frame field forV T M, and for HT M
we have the local frame field{ δ δx1, . . . ,
δ
δxn}, where
δ δxi =
∂
∂xi−Γh0i∂y∂h, Γh0i=ykΓhki,
and Γh
ki(x) are the Christoffel symbols ofg.
The set { ∂
∂yi,δxδj}i,j=1,n,denoted also by{∂i, δj}i,j=1,n,defines a local frame on
Consider the energy density of the tangent vectorywith respect to the Riemannian metricg
(2.2) t= 1
2kyk 2=1
2gτ(y)(y, y) = 1 2gik(x)y
iyk, y∈τ−1(
U).
Obviously, we havet∈[0,∞) for everyy∈T M.
There are many types of lifts of a metric from the base manifold to the tangent, cotangent, or jet bundles (e.g. see [16]-[19]), the most general being the natural lifts (in the sense of [9]), studied in a few recent papers, such as [1], [2], [6], [13].
The second author considered in [14] an (1,1)-tensor fieldJ on the tangent bundle
T M, obtained as natural 1-st order lift of the metricgfrom the base manifold to the tangent bundleT M:
J XH
y =a1(t)XyV +b1(t)gτ(y)(X, y)yVy,
J XV
y =−a2(t)XyH−b2(t)gτ(y)(X, y)yyH,
(2.3)
∀X ∈ T1
0(M), ∀y∈T M, a1, a2, b1, b2 being smooth functions of the energy density. The above (1,1)-tensor fieldJ defines an almost complex structure on the tangent bundle if and only if
a2= 1
a1
, b2=−
b1
a1(a1+ 2tb1)
.
(2.4)
Then it was considered onT M a Riemannian metricGe of natural diagonal lift type:
e G(XH
y , YyH) =c1(t)gτ(y)(X, Y) +d1(t)gτ(y)(X, y)gτ(y)(Y, y),
e G(XV
y , YyV) =c2(t)gτ(y)(X, Y) +d2(t)gτ(y)(X, y)gτ(y)(Y, y),
e G(XV
y , YyH) =Ge(XyH, XyV) = 0,
(2.5)
∀X, Y ∈ T1
0(T M), ∀y∈T M,wherec1, c2, d1, d2are smooth functions of the energy density onT M. The conditions for Geto be a Riemannian metric on T M (i.e. to be positive definite) arec1>0, c2>0, c1+ 2td1>0, c2+ 2td2>0 for everyt≥0.
The Riemannian metricGeis almost Hermitian with respect to the almost complex structureJ if and only if
c1
a1 = c2
a2
=λ, c1+ 2td1 a1+ 2tb1
= c2+ 2td2
a2+ 2tb2
=λ+ 2tµ,
(2.6)
whereλ >0, µ >0 are functions oft. The symmetric matrix of type 2n×2n
à e G(1)ij 0
0 Ge(2)ij !
=
µ
c1(t)gij+d1(t)g0ig0j 0
0 c2(t)gij+d2(t)g0ig0j
¶ ,
associated to the metricGe in the adapted frame{δj, ∂i}i,j=1,n, has the inverse
à e Hkl
(1) 0 0 Hekl
(2)
!
=
µ
p1(t)gkl+q1(t)ykyl 0
0 p2(t)gkl+q2(t)ykyl
where gkl are the entries of the inverse matrix of (g
ij)i,j=1,n, and p1, q1, p2, q2, are some real smooth functions of the energy density. More precisely, they may be expressed as rational functions ofc1, d1, c2, d2:
Proposition 2.1. The Levi-Civita connection∇e associated to the Riemannian metric
e
Gfrom the tangent bundleT M has the form
where theM-tensor fieldsQ, P, S,have the following components with respect to the adapted frame{∂i, δj}i,j=1,n:
kij being the components of the curvature tensor field of the Levi Civita connection
˙
∇ from the base manifold (M, g) and Rh0ij =ykRhkij. The vector fields Q(XV, YV)
andS(XH, YH)are vertically valued, while the vector fieldP(YV, XH)is horizontally
valued.
Using the relations (2.8), we may easily prove that theM-tensor fieldsQ, P, S,
have invariant expressions of the forms
Q(XV, YV) = c′2
Since in the following sections we shall work on the subsetTrM ofT M consisting
become simpler: (2.9)
Q(XV, YV) = d2
c2+r2d2g(X, Y)y V,
P(XV, YH) = d1
2c1g(Y, y)X
H+ d1
2c1(c1+r2d1)[c1g(X, Y)−d1g(X, y)g(Y, y)]y H
−c2
2c1(R(X, y)Y)
H− c2d1
2c1(c1+r2d1)g(X, R(Y, y)y)y H,
S(XH, YH) =−d1
2c2[g(X, y)Y
V +g(Y, y)XV]
+ d1d2
c2(c2+2td2)g(y, X)g(y, Y)y V −1
2(R(X, Y)y)
V.
3
Sasakian tangent sphere bundles of natural
diag-onal lifted type
Let TrM = {y ∈ T M : gτ(y)(y, y) = r2}, with r ∈ [0,∞), and the projection τ :
TrM →M, τ=τ◦i,wherei is the inclusion map.
The horizontal lift of any vector field on M is tangent to TrM, but the vertical
lift is not always tangent toTrM. The tangential lift of a vector X to (p, y)∈TrM
is tangent toTrM and is defined by
XyT =XyV − 1
r2gτ(y)(X, y)y
V y.
Remark that the tangential lift of the tangent vector y ∈ TrM, vanishes, i.e.
yT y = 0.
The tangent bundle toTrMis spanned byδiand∂jT =∂j−r12g0jyk∂k, i, j, k= 1, n, although{∂T
j}j=1,nare not independent. They fulfill the relation
(3.1) yj∂jT = 0,
and in any pointy∈TrM they span an (n−1)-dimensional subspace ofTyTrM.
Denote by G′ the metric on T
rM induced by the metric Ge from T M. Remark
that the functions c1, c2, d1, d2 become constant, since in the case of the tangent sphere bundle of constant radius r, the energy densityt becomes a constat equal to
r2
2. It follows
G′(XH
y , YyH) =c1gτ(y)(X, Y) +d1gτ(y)(X, y)gτ(y)(Y, y),
G′(XT
y, YyT) =c2[gτ(y)(X, Y)−r12gτ(y)(X, y)gτ(y)(Y, y)],
G′(XH
y , YyT) =G′(YyT, XyH) = 0,
∀X, Y ∈ T1
0(M), ∀y ∈TrM, wherec1, d1, c2 are constants. The conditions forG′ to be positive arec1>0, c2>0, c1+r2d1>0.
When the tangent bundleT M endowed with the metricGeand the almost complex structureJ defined respectively by the relations (2.3) and (2.5) is almost Hermitian, we may construct an almost contact metric structure on the tangent sphere bundle
To this aim, we choose an appropriate normal (non unitary) vector field N to
TrM, given by
(3.2) N = (a1+r2b1)yV.
Using the almost complex structureJ onT M, we define a vector fieldξ′,and a 1-formη′, onTrM as follows:
ξ′=−J N, η′(X) =G′(X, ξ′), ∀X ∈ T01(TrM).
We may easily prove that
(3.3) ξ′ =yH, η′(XT) = 0, η′(XH) = (c1+r2d1)g(X, y),∀X∈ T01(M), y∈TrM.
The local coordinate expressions ofξ′ andη′ are
ξ′=yiδi, η′= (c1+r2d1)g0idxi, ∀i= 1, n.
Sinceη′(ξ′) =r2(c1+r2d1)6= 1,we shall define another 1-form, η and an appro-priate vector fieldξ,given by
(3.4) η=βη′, ξ= 1
βr2(c
1+r2d1)
ξ′,
whereβ is an appropriate constant, whose value will be determined later. Obviously, we haveη(ξ) = 1.
Taking into account the relations (3.3) and (3.4), we obtain the invariant expres-sions ofη andξ:
η(XT) = 0, η(XH) =β(c1+r2d1)g(X, y), ξ=
1
βr2(c
1+r2d1)
yH,
for every vector fieldX tangent toM, and every tangent vectory from TrM.
The (1,1)-tensor fieldϕonTrM, obtained by eliminating the normal component
of the almost complex structureJ fromT M, will be expressed by
(3.5) ϕXH =a1XT, ϕXT =−a2XH+
a2
r2g(X, y)y
H, ∀X, Y ∈ T1
0(M), y∈TrM.
Since the (1,1)-tensor fieldϕgiven by (3.5) andξ, η from (3.4) verify the relations
ϕ2=−I+η⊗ξ, ϕξ= 0, η◦ϕ= 0, η(ξ) = 1,
we have that (ϕ, ξ, η) is an almost contact structure on TrM.
The aim of this section is to find the Sasakian structures (ϕ, ξ, η, G) of natural diagonal lifted type on TrM, i.e. the conditions under which the almost contact
structure (ϕ, ξ, η) is normal contact metric with respect to a certain metric G on
TrM, which we shall determine.
The obtained almost contact structure is not almost contact metric with respect to the metricG′, since the relation G′(ϕX, ϕY) =G′(X, Y)−η(X)η(Y), ∀X, Y ∈
T1
0(TrM) is not satisfied, i.e. the metricG′is not compatible with the almost contact
Using (3.3) and (3.4) we obtain the exterior differential ofη:
for every tangent vector fieldsX, Y onM and every tangent vectory∈TrM.
We have too that the condition G′(X, ϕY) = dη(X, Y), ∀X, Y ∈ T1
0(TrM), is
not satisfied, so the almost contact metric structure (ϕ, ξ, η) is not a contact metric structure with respect toG′.
We shall change the metricG′, multiplying it by a scalarα. Thus, the new metric
GonTrM will have the form
G=αG′.
The scalarsα, βmay be determined from the conditions
G(ξ, ξ) = 1, G(XT, ϕYH) =dη(XT, YH), G(XH, ϕYT) =dη(XH, YT),
which, due to the proportionality relation (2.6), are equivalent to a simple system in
αandβ: (
The solution of this system is
α= c1+r
Thus, the final expressions forϕ, ξ, ηandGare:
(3.6) ϕXH=a1XT, ϕXT =−a2XH+
for every tangent vector fieldsX, Y ∈ T1
In the computations we shall use the following shorter equivalent expressions for
ξ, η, G:
(3.10) ξ= 1
2λr2αy
H, η(XT) = 0, η(XH) = 2αλg(X, y), G=αG′,
whereα=c1+r 2
d1 4r2λ2 .
Now we may prove the following result.
Theorem 3.1. The almost contact metric structure(ϕ, ξ, η, G)onTrM, withϕ,ξ, η,
andG given by(3.6) and (3.10), is a contact metric structure, and it is Sasakian if and only if the base manifold has constant sectional curvaturec= a
2 1 r2. Proof: Since the metricity and the contact conditions
G(ϕX, ϕY) =G(X, Y)−η(X)η(Y), G(X, ϕY) =dη(X, Y), ∀X, Y ∈ T01(TrM),
are fulfilled, it follows that the almost contact structure (ϕ, ξ, η, G) on TrM is a
contact metric structure.
The normality condition for the contact metric structure found onTrM is
(3.11) Nϕ(X, Y) + 2dη(X, Y)ξ= 0, ∀X, Y ∈ T01(TrM),
whereNϕis the Nijenhuis tensor field of the (1,1)−tensor fieldϕ, and it is given by
(3.12) Nϕ(X, Y) =ϕ2[X, Y] + [ϕX, ϕY]−ϕ[ϕX, Y]−ϕ[X, ϕY].
When bothX and Y are horizontal vector fields, δi, δj, the mentioned relation
becomes
(3.13) Rh0ij+a
2 1
r2(gi0δ
h
j −gj0δhi) = 0,
whereRh
0ij =Rhkijyk, gi0=gikyk.
By differentiating (3.13) with respect to the tangential coordinatesyk,we obtain
that the base manifoldM has constant sectional curvature:
(3.14) Rh
kij=
a2 1
r2(δ
h
igjk−δhjgik).
Next the relation (3.11) is identically fulfilled when we replace at least one of the vector fields X, Y by ∂T
i . Thus (3.14) is the only condition for the contact metric
structure (ϕ, ξ, η, G) on TrM to be Sasakian (normal contact metric). Thus the
theorem is proved.
In the sequel we shall see that the Levi-Civita connection associated to the metric
Gsatisfies the necessary Sasaki condition
(3.15) (∇Xϕ)Y =G(X, Y)ξ−η(Y)X, ∀X, Y ∈ T01(TrM),
for the almost contact metric manifold (TrM, ϕ, ξ, η, G).
Proposition 3.2. The Levi-Civita connection∇, associated to the Riemannian met-ricGon the tangent sphere bundleTrM of constant radiusr has the expression
where theM-tensor fields involved as coefficients have the expressions:
Ah
We mention that A and C have these quite simple expressions, since they are the coefficients of the tangential part of the Levi-Civita connection. All the terms containingyh∂T
h = 0 have been cancelled.
The invariant form of the Levi-Civita connection fromTrM is
WhenX andY are both of them tangential vector fields, or horizontal vector fields, respectively, (3.15) becomes
After some quite long computations we get
1
and it follows easily that these relations are identically fulfilled.
which, after the computations, become
[a1(Ah
ij−Bijh) + 2αλg0jδih]∂hT = 0, [a2(
1
r2g0jC
h
i0−Cijh)−a1Bjih]∂hT = 0,
and they are satisfied, since the final values,
− 1 2r2
£ a1gij−
b1λ+ (a1+r2b1)µ
λ g0ig0j ¤
yh∂hT, −
1
2r2(a1gij−
a1
r2g0ig0j)y
h∂T h,
are zero due to the relationyh∂hT = 0, fulfilled by the generators∂hT.
4
η
−
Einstein Sasakian tangent sphere bundles of
natural diagonal lifted type
In this section we shall find the condition under which the normal contact metric manifold (TrM, ϕ, ξ, η, G), withϕ, ξ, η, and Ggiven respectively by (3.6), (3.7), and
(3.8), isη−Einstein, i.e. the corresponding Ricci tensor may be written as
(4.1) Ric=ρG+ση⊗η,
whereρand σare smooth real functions.
In the case whereA, B, C are given by (3.16), it can be shown (see e.g. [7]) that their covariant derivatives are:
(4.2)
(
˙
∇kAhij = 0, ∇˙kCijh =−12∇˙kR
h lijyl,
˙
∇kBhij=−c 2 2c1
˙
∇kRjilh yl− c2d1 2c1(c1+r2d1)
˙
∇kRiljrylyryh,
where ˙∇kRhkij, ∇˙kRhkij are the usual local coordinate expressions of the covariant
derivatives of the curvature tensor field and the Riemann-Christoffel tensor field of the Levi Civita connection ˙∇from the base manifoldM. If the base manifold (M, g) is locally symmetric then, obviously, ˙∇kAhij = 0, ∇˙kBhij= 0, ∇˙kCijh = 0.
In the paper [7] we obtained by a standard straightforward computation the hor-izontal and tangential components of the curvature tensor field K, denoted by se-quences of H and T, to indicate horizontal, or tangential argument on a certain position. For example, we have
K(δi, δj)∂kT =HHT Hkijh δh+HHT Tkijh ∂hT.
The expressions of the non-zeroM-tensor fields which appear as coefficients may be found in [7].
Let us remark some facts concerning the obtaining of the Ricci tensor field for the tangent bundleT M. We have the well known formula
Ric(Y, Z) =trace(X →K(X, Y)Z),
g
RicHHjk=Ricg(δj, δk) =HHHHkhjh +V HHVkhjh ,
g
RicV Vjk=Ricg(∂j, ∂k) =V V V Vkhjh −V HV Hkjhh ,
where the components V V V Vh
kij, V HV Hkijh , V HHVkijh , are obtained from the
cur-vature tensor field onT M in a similar way as the componentsT T T Tkijh , T HT Hkijh ,
T HHTh
kij are obtained from the curvature tensor field on TrM. In the expression
of RicHVg jk = Ricg(δj, ∂k) = RicV Hg jk = Ricg(∂j, δk) there are involved the
covari-ant derivatives of the curvature tensor fieldR. If (M, g) is locally symmetric then
RicHVjk =RicV Hjk = 0. In particular we have RicHVjk = RicV Hjk = 0 in the
case where (M, g) has constant sectional curvature.
Equivalently, we may use an orthonormal frame (E1, . . . , E2n) onT M and we may
use the formula
g
Ric(Y, Z) = 2n
X
i=1
G(Ei, K(Ei, Y)Z).
We may choose the orthonormal frame (E1, . . . , E2n) such that the first n vectors
E1, . . . , En are the vectors of a (orthonormal) frame inHT M and the lastn vectors
En+1, . . . E2n are the vectors of a (orthonormal) frame inV T M. Moreover, we may
assume that the last vectorE2n is the unitary vector of the normal vectorN =yi∂i
toTrM.
The components of the Ricci tensor field of TrM can be obtained in a similar
way by using the above traces. However the vector fields ∂T
1, . . . , ∂Tn are not
in-dependent. On the open set from TrM, where yn 6= 0 we can consider the basis
{δ1, . . . , δn, ∂1T, . . . , ∂nT−1}forT TrM. The last vector∂nT is expressed as
(4.3) ∂T
n =−
1
yn n−1
X
i=1
yi∂T i .
Remark that the basis{δ1, . . . , δn, ∂T1, . . . , ∂nT−1} can be completed with the normal vectorN =yV =yh∂
h.
Using the relation (4.3), we obtained in [7] the components of the Ricci tensor on
TrM:
RicHHjk=Ric(δj, δk) =HHHHkhjh +T HHTkhjh −
d1(4c1+r2d1) 4c1c2r2
g0jg0k.
RicT Tjk=Ric(∂jT, ∂kT) =HT T Hkhjh +T T T Tkhjh −
1
r2gjk+ 1
r4g0jg0k.
Taking into account the above relations and the η−Einstein condition (4.1), we obtain that
RicT Tjk−ρG
(2)
jk =− c2
2 4c2
1
Ri
hj0Rhik0−
c2 2d1 2c2
1(c1+r2d1)Rh0j0R h
0k0
+2c 2
1(n−2)−2c1[ρc1c2−d1(n−2)]r 2
−d1(2ρc1c2+d1)r 4 2c1r2(c1+r2d1) (gjk−
1
RicHHjk−ρG
In the sequel we shall prove the main theorem of this paper.
Theorem 4.1. The Sasakian manifold (TrM, ϕ, ξ, η, G), with ϕ, ξ, η, and G given
r2 is the constant sectional curvature of the base manifold M. Replacingc by the the mentioned value, the cases become
Case I)c1=a21c2, ρ=−
curvaturec, so the relation (4.4) becomes
RicT Tjk−ρG(2)jk =
The above quantity vanishes if and only if the functionρhas the expression
ρ= 2c1(n−2)(c1+r
2d1) +r4(c2c2 2−d21) 2c1c2r2(c1+r2d1)
.
Replacing this value into (4.5), and taking into account that the base manifold has constant sectional curvaturec, we obtain
From the relation (4.6), we have that
RicHHjk=ρG(1)jk +ση(δj)η(δk)
if and only if the coefficients ofgjk andg0jg0k vanish.
From the vanishing condition of the first coefficient in (4.6), we obtain two cases:
I)c1=cc2r2, andII)d1= cc2r 2
−c1(n−2) (n−1)r2 .
In the first case the numerator of the second coefficient in (4.6) becomes
−(cc2+d1)2r6{λ2[cc2(n−2)−d1n] + 2σcc22(cc2+d1)r2},
and it vanishes if and only ifI.1)d1=−cc2, orI.2)σ=λ22dcc1n2−cc2(n−2)
2(cc2+d1)r2. The subcase
d1=−cc2 reduces toc1+r2d1= 0,which should be excluded, since the constantρ, some components ofK, as well as the Levi Civita connection are not defined. Thus, case I reduces to the subcase I.2, and the values ofc1, ρ, σ are those presented in the theorem.
In the second case the coefficient ofg0jg0k in (4.6) has the numerator equal to
−(c1+cc2r2) n
2σc1c2(c1+cc2r2)2+λ2hn2(n
−2)(c21+r4c2c22)−2cc1c2 £
n[6+(n−4)n]−2
¤
r2io.
Thus the subcases are
II.1) c1=−cc2r2,
II.2)σ=λ22r
2cc
1c2{n[n(n−4) + 6]−2} −n2(n−2)(c21+c2c22r4) 2c1c2(c1+cc2r2)2
.
The first subcase reduces again toc1+r2d1= 0,which should be excluded. Hence the caseIIpresented in the theorem is practically the subcase II.2.
Thus the theorem is proved.
Acknowledgement. Supported by the Program POSDRU/89/1.5/S/49944, ”AL. I. Cuza” University of Ia¸si, Romania.
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Authors’ addresses:
Simona-Luiza Drut¸˘a-Romaniuc
Science Department, ”AL. I. Cuza” University of Ia¸si, 54 Lascar Catargi Street, RO-700107 Ia¸si, Romania. E-mail: simona.druta@uaic.ro
Vasile Oproiu
Faculty of Mathematics, ”AL. I. Cuza” University of Ia¸si, Bd. Carol I, No. 11, RO-700506 Ia¸si, Romania;
Institute of Mathematics ”O. Mayer”, Romanian Academy, Ia¸si Branch, Romania.
