The Warped Product of Hamiltonian Spaces
In this paper, the geometric properties of warped product Hamiltonian spaces are studied. It is shown there is a close geometrical relation between a warped product Hamiltonian space and its base Hamiltonian manifolds. For example, it is proved that for nonconstant warped function f, the Sasaki lift...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1067992016-10-06T03:02:22Z The Warped Product of Hamiltonian Spaces Attarchi, H. Rezaii, M.M. In this paper, the geometric properties of warped product Hamiltonian spaces are studied. It is shown there is a close geometrical relation between a warped product Hamiltonian space and its base Hamiltonian manifolds. For example, it is proved that for nonconstant warped function f, the Sasaki lifted metric G of Hamiltonian warped product space is bundle-like for its vertical foliation if and only if based Hamiltonian spaces are pseudo-Riemannian manifolds. Изучены геометрические свойства гамильтоновых пространств в виде искривленных произведений. Показано, что между гамильтоновым пространством - искривленным произведением и его базовыми гамильтоновыми многообразиями существует тесная геометрическая связь. Например, доказано, что для непостоянной искривляющей функции f метрика Сасаки G для гамильтонова пространства - искривленного произведения является расслоенной метрикой по отношению к ее вертикальному слоению тогда и только тогда, когда базовые гамильтоновы пространства являются псевдо-римановыми многообразиями. The first author would like to thank the INSF for the partially support on the grant number 92006616. 2014 Article The Warped Product of Hamiltonian Spaces / H. Attarchi, M.M. Rezaii // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 300-308. — Бібліогр.: 11 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106799 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In this paper, the geometric properties of warped product Hamiltonian spaces are studied. It is shown there is a close geometrical relation between a warped product Hamiltonian space and its base Hamiltonian manifolds. For example, it is proved that for nonconstant warped function f, the Sasaki lifted metric G of Hamiltonian warped product space is bundle-like for its vertical foliation if and only if based Hamiltonian spaces are pseudo-Riemannian manifolds. |
| format |
Article |
| author |
Attarchi, H. Rezaii, M.M. |
| spellingShingle |
Attarchi, H. Rezaii, M.M. The Warped Product of Hamiltonian Spaces Журнал математической физики, анализа, геометрии |
| author_facet |
Attarchi, H. Rezaii, M.M. |
| author_sort |
Attarchi, H. |
| title |
The Warped Product of Hamiltonian Spaces |
| title_short |
The Warped Product of Hamiltonian Spaces |
| title_full |
The Warped Product of Hamiltonian Spaces |
| title_fullStr |
The Warped Product of Hamiltonian Spaces |
| title_full_unstemmed |
The Warped Product of Hamiltonian Spaces |
| title_sort |
warped product of hamiltonian spaces |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/106799 |
| citation_txt |
The Warped Product of Hamiltonian Spaces / H. Attarchi, M.M. Rezaii // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 3. — С. 300-308. — Бібліогр.: 11 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT attarchih thewarpedproductofhamiltonianspaces AT rezaiimm thewarpedproductofhamiltonianspaces AT attarchih warpedproductofhamiltonianspaces AT rezaiimm warpedproductofhamiltonianspaces |
| first_indexed |
2025-07-07T19:02:06Z |
| last_indexed |
2025-07-07T19:02:06Z |
| _version_ |
1837015946974199808 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2014, vol. 10, No. 3, pp. 300–308
The Warped Product of Hamiltonian Spaces
H. Attarchi and M.M. Rezaii
Department of Mathematics and Computer Science
Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
E-mail: hassan.attarchi@aut.ac.ir
mmreza@aut.ac.ir
Received November 21, 2012, revised March 31, 2014
In this paper, the geometric properties of warped product Hamiltonian
spaces are studied. It is shown there is a close geometrical relation be-
tween a warped product Hamiltonian space and its base Hamiltonian mani-
folds. For example, it is proved that for nonconstant warped function f , the
Sasaki lifted metric G of Hamiltonian warped product space is bundle-like
for its vertical foliation if and only if based Hamiltonian spaces are pseudo-
Riemannian manifolds.
Key words: warped product, Hamiltonian space, bundle-like metric.
Mathematics Subject Classification 2010: 54B10, 37J99.
1. Introduction
The notion of warped product spaces was introduced to study manifolds with
negative curvatures by Bishop and O’Neill [3]. Afterwards, the warped product
was used to model the standard space-time, especially in the neighborhood of
stars and black holes [10]. The notion of the warped product Finslerian mani-
folds was initially introduced by Kozma [5] in 2001. Recently, it was developed
by one of the present authors [1, 4, 11]. In this work, the warped product of
Hamiltonian spaces is introduced and it is shown that these spaces obtain Hamil-
tonian structure as well. Moreover, some geometric properties of warped product
Hamilton spaces such as its nonlinear connections are studied.
The Lagrange space has been certified as an excellent model for some impor-
tant problems in Relativity, Gauge Theory and Electromagnetism [6, 7]. The
geometry of Lagrange spaces gives a model for both the gravitational and elec-
tromagnetic fields. Moreover, this structure plays a fundamental role in studying
the geometry of the tangent bundle TM . The geometries of the cotangent bun-
dle T ∗M and the tangent bundle TM which follows the same outlines are related
c© H. Attarchi and M.M. Rezaii, 2014
The Warped Product of Hamiltonian Spaces
by the Legendre transformation. From this duality, the geometry of a Hamilto-
nian space can be obtained from that of certain Lagrangian space and vice versa.
Using this duality, several important results in the Hamiltonian spaces can be
obtained: the canonical nonlinear connection, the canonical metrical connection,
etc. Therefore, the theory of Hamiltonian spaces has the same symmetry and
beauty as the Lagrangian geometry. Moreover, it gives a geometrical framework
for the Hamiltonian theory of mechanics or physical fields. With respect to the
importance of these spaces in physical areas, present work is formed to develop
the concept of a warped product on Hamiltonian spaces. Aiming at our purpose,
this paper is organized in the following way:
Let (M, H) be a warped Hamiltonian space of the Hamiltonian spaces (M1,H1)
and (M2, H2). In Sec. 2, the notion of the warped product Hamiltonian spaces is
presented and some natural geometrical properties of the cotangent bundle for a
warped manifold are given. In Sec. 3, it is shown that (M,H) is a Hamiltonian
space and its canonical nonlinear connections are calculated as well. Moreover,
the Sasaki lifted metric G on T ∗M is introduced. In Sec. 4, the Levi–Civita con-
nection of pseudo-Riemannian metric G on T ∗M is calculated. Finally, in Sec. 5,
we prove some theorems that show close relation between the geometries of the
warped product Hamiltonian manifolds and their base Hamiltonian spaces.
2. Preliminaries and Notations
Here, a Hamiltonian space is a pair (M, H), where M is a real n-dimensional
manifold and H : T ∗M −→ IR is a smooth function whose Hessian with respect
to the cotangent bundle coordinate is a d-tensor field of type (2, 0) symmetric,
nondegenerate and of constant signature on T ∗M\{0}. Let Hn
1 = (M1,H1) and
Hm
2 = (M2,H2) be two Hamiltonian spaces with dim(Hn
1 ) = n and dim(Hm
2 ) = m,
respectively. The warped product of these spaces is denoted by H = (M,H),
where
M = M1 ×M2 and H = H1 + fH2 (1)
for some smooth function f : M1 −→ R+. Then a coordinate system on M is
denoted by {(U ×V, ϕ×ψ)}, where {(U,ϕ)} and {(V, ψ)} are coordinate systems
on M1 and M2, respectively, such that each x = (x, z) ∈ M has the local expres-
sion (xi, zα). It is notable that throughout the paper, the indices {i, j, k, . . .} and
{α, β, λ, . . .} are used for the ranges 1, . . . , n and 1, . . . , m, respectively. More-
over, the canonical projections of T ∗M1 on M1 and T ∗M2 on M2 are denoted by
π1 and π2, respectively. The fibre of the cotangent bundle at x = (x, z) ∈ M is
T ∗(x,z)M = T ∗xM1 ⊕ T ∗z M2, therefore T ∗M = T ∗M1 ⊕ T ∗M2.
The induced coordinate systems on T ∗M1 and T ∗M2 are (xi, pi) and (zα, qα),
respectively, whose coordinates pi and qα are called momentum variables [8]. The
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 301
H. Attarchi and M.M. Rezaii
change of these coordinates on T ∗M1 and T ∗M2 are given by
x̃i = x̃i(x1, . . . , xn),
rank
(
∂x̃i
∂xj
)
= n,
p̃i = ∂xj
∂x̃i pj .
z̃α = z̃α(z1, . . . , zm),
rank
(
∂z̃α
∂zβ
)
= m,
q̃α = ∂zβ
∂z̃α qβ.
(2)
Let (x,p) = (x, z, p, q) ∈ T ∗M = T ∗M1 ⊕ T ∗M2. The tangent space at (x,p) to
T ∗M is denoted by T(x,p)T
∗M , that is, a 2(n+m)-dimensional vector space. The
natural basis induced on T(x,p)T
∗M by the local coordinate of T ∗M1 and T ∗M2
is { ∂
∂xi ,
∂
∂zα , ∂
∂pi
, ∂
∂qα
}. These coordinates are changed with respect to transfor-
mations (2) as follows:
∂
∂xi = ∂x̃j
∂xi
∂
∂x̃j + ∂p̃j
∂xi
∂
∂p̃j
,
∂
∂zα = ∂z̃β
∂zα
∂
∂z̃β + ∂q̃β
∂zα
∂
∂q̃β
,
∂
∂pi
= ∂xi
∂x̃j
∂
∂p̃j
,
∂
∂qα
= ∂zα
∂z̃β
∂
∂q̃β
.
(3)
In the paper, the notations ∂̇i and ∂̇α are used instead of ∂
∂pi
and ∂
∂qα
, respectively,
similarly to the notations in [8]. The Jacobian matrix of transformations (3) is
Jac :=
∂x̃j
∂xi 0 0 0
0 ∂z̃β
∂zα 0 0
∂p̃j
∂xi 0 ∂xi
∂x̃j 0
0 ∂q̃β
∂zα 0 ∂zα
∂z̃β
. (4)
It follows that
det(Jac) = 1.
By means of last equation, we have the following corollary.
Corollary 2.1. The manifold T ∗M = T ∗M1 ⊕ T ∗M2 is orientable.
Let ∂̄a and ∂
∂xa be abbreviations for ∂̇iδa
i + ∂̇αδα+n
a and ∂
∂xi δ
i
a + ∂
∂zα δa
α+n,
respectively, where the indices {a, b, c, . . .} are used for the range 1, . . . , n + m.
Throughout the paper, these notations and range of the indices are established.
We know that there are some natural structures that live on the cotangent
bundle T ∗M . It would be interesting to present them on the cotangent bundle of
a warped product Hamiltonian space. First, the Liouville–Hamilton vector field
of T ∗M is given by
C∗ := pa∂̄
a = pi∂̇
i + qα∂̇α = C∗
1 + C∗
2 , (5)
302 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
The Warped Product of Hamiltonian Spaces
where C∗
1 and C∗
2 denote the Liouville-Hamilton vector fields of T ∗M1 and T ∗M2,
respectively.
Next, the Liouville 1-form θ on T ∗M is defined by
θ := padxa = pidxi + qαdzα = θ1 + θ2, (6)
where θ1 and θ2 are the Liouville 1-forms of T ∗M1 and T ∗M2, respectively.
And, the canonical symplectic structure ω on T ∗M is defined by ω = dθ and
has the local expression
ω := dpa ∧ dxa = dpi ∧ dxi + dqα ∧ dzα = ω1 + ω2, (7)
where ω1 and ω2 are canonical symplectic structures of T ∗M1 and T ∗M2, respec-
tively.
Finally, if the Poisson bracket on the cotangent bundles of T ∗M1, T ∗M2 and
T ∗M are denoted by {., .}1, {., .}2 and {., .}, respectively, then they are related
as follows:
{g, h} = ∂̄ag
∂h
∂xa
− ∂̄ah
∂g
∂xa
= {g, h}1 + {g, h}2, (8)
where g, h ∈ C∞(T ∗M).
The Hamilton vector field of the Hamiltonian function H is denoted by XH
and satisfies the equation
ιXH
ω = −dH.
Let XH1 and XH2 be Hamilton vector fields of the spacesHn
1 andHm
2 , respectively,
then the following theorem gives an expression of XH .
Theorem 2.1. Suppose that H = (M,H) is a warped product Hamiltonian
space defined in (1). Then the Hamilton vector field of H is given by
XH = XH1 + fXH2 −H2
∂f
∂xi
∂̇i.
P r o o f. By the definition of Hamilton vector fields, we have ιXH
ω = −dH.
It is a straightforward calculation to complete the prove.
3. Nonlinear Connection on Warped Product Hamiltonian
Space
For the Hamiltonian spaces Hn
1 and Hm
2 , the equations
{
gij = 1
2 ∂̇i∂̇jH1,
gαβ = 1
2 ∂̇α∂̇βH2
(9)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 303
H. Attarchi and M.M. Rezaii
define the fundamental tensors of the spaces Hn
1 and Hm
2 , respectively. The
fundamental tensor of the warped product Hamiltonian space (M,H) is given by
(gab) =
(
1
2
∂̄a∂̄bH
)
=
(
gij 0
0 fgαβ
)
. (10)
Now, it is easy to check that (M, H) is a Hamilton space as well. By the definition
of the canonical nonlinear connections of a Hamiltonian space presented in [8],
the canonical nonlinear connections of Hn
1 , Hm
2 and H, respectively, are obtained
as follows:
Nij = 1
4{gij ,H1} − 1
4
(
gik
∂2H1
∂pk∂xj + gjk
∂2H1
∂pk∂xi
)
,
Nαβ = 1
4{gαβ, H2} − 1
4
(
gαγ
∂2H2
∂qγ∂zβ + gβγ
∂2H2
∂qγ∂zα
)
,
N̄ab = 1
4{gab,H} − 1
4
(
gac
∂2H
∂pc∂xb + gbc
∂2H
∂pc∂xa
)
,
(11)
where (gij), (gαβ) and (gab) are the inverse matrices of (gij), (gαβ) and (gab),
respectively. The relation of the nonlinear connections N̄ab of the Hamiltonian
space H and those of Hn
1 and Hm
2 are given by
N̄ij = Nij + 1
4 ∂̇kgij
∂f
∂xk H2,
N̄αβ := N̄(α+n)(β+n) = Nαβ − 1
4f2 gαβ ∂̇kH1
∂f
∂xk ,
N̄iα := N̄i(α+n) = − 1
4f gαβ ∂̇βH2
∂f
∂xi .
(12)
Let π be the projection map
π := (π1, π2) : T ∗M1 ⊕ T ∗M2 −→ M1 ×M2.
Then the kernel of π∗ is known as the vertical bundle on T ∗M and denoted by
V T ∗M . The local sections of V T ∗M are given by
{ ∂
∂p1
, ...,
∂
∂pn
,
∂
∂q1
, ...,
∂
∂qm
}.
Using the nonlinear connections N̄ij , N̄iα and N̄αβ , we can define the nonholo-
morphic vector fields
{
δ∗
δ∗xi := δ∗
δ∗xi = ∂
∂xi + N̄ij ∂̇
j + N̄iα∂̇α,
δ∗
δ∗zα := δ∗
δ∗xα+n = ∂
∂zα + N̄αi∂̇
i + N̄αβ ∂̇β,
(13)
which generate the warped horizontal distribution on T ∗M denoted by HT ∗M .
The dual 1-forms of these local vector fields are given by
dxa = dxiδa
i + dzαδa
α+n,
δ∗pi := δpi = dpi − N̄ijdxj − N̄iαdzα,
δ∗qα := δpα+n = dqα − N̄αidxi − N̄αβdzβ.
(14)
304 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
The Warped Product of Hamiltonian Spaces
Moreover, the Sasaki metric G on T ∗M of the Hamiltonian structure H is defined
by
G = gijdxi ⊗ dxj +
gαβ
f
dzα ⊗ dzβ + gijδ∗pi ⊗ δ∗pj + fgαβδ∗qα ⊗ δ∗qβ. (15)
4. The Levi–Civita Connection of Metric G
The Lie brackets of the local vector fields given in previous section are pre-
sented as follows:
[ δ∗
δ∗xi ,
δ∗
δ∗xj ] = Rijk∂̇
k + Rijα∂̇α,
[ δ∗
δ∗xi ,
δ∗
δ∗zα ] = Riαj ∂̇
j + Riαβ ∂̇β,
[ δ∗
δ∗zα , δ∗
δ∗zβ ] = Rαβi∂̇
i + Rαβγ ∂̇γ ,
(16)
where
Rijk = δ∗N̄jk
δ∗xi − δ∗N̄ik
δ∗xj , Rijα = δ∗N̄jα
δ∗xi − δ∗N̄iα
δ∗xj ,
Riαk = δ∗N̄αk
δ∗xi − δ∗N̄ik
δ∗zα , Riαβ = δ∗N̄αβ
δ∗xi − δ∗N̄iβ
δ∗zα ,
Rαβk = δ∗N̄βk
δ∗zα − δ∗N̄αk
δ∗zβ , Rαβγ = δ∗N̄βγ
δ∗zα − δ∗N̄αγ
δ∗zβ .
(17)
The components Rabc are called the curvature tensors of the nonlinear connection
N̄ab and they are skew-symmetric with respect to the indices a and b. Moreover,
[∂̇i, δ∗
δ∗xj ] = ∂̇i(N̄jk)∂̇k,
[∂̇α, δ∗
δ∗xi ] = ∂̇α(N̄ik)∂̇k + ∂̇α(N̄iβ)∂̇β,
[∂̇i, δ∗
δ∗zα ] = ∂̇i(N̄αβ)∂̇β,
[∂̇α, δ∗
δ∗zβ ] = ∂̇α(N̄βk)∂̇k + ∂̇α(N̄βγ)∂̇γ .
(18)
Let ∇ be the Levi–Civita connection on (T ∗M, G) which is given by
2G(∇XY, Z) = XG(Y,Z) + Y G(X, Z)− ZG(X, Y )
−G([X, Z], Y )−G([Y,Z], X) + G([X, Y ], Z)
(19)
for any X,Y, Z ∈ Γ(TT ∗M). Then the components of ∇ are given by
∇ δ∗
δ∗xi
δ∗
δ∗xj = Γk
ij
δ∗
δ∗xk − f
2 N̄αkg
k
ijg
αβ δ∗
δ∗zβ + 1
2gijk∂̇
k + 1
2Rija∂̄
a,
∇ δ∗
δ∗xi
δ∗
δ∗zα = ∇ δ∗
δ∗zα
δ∗
δ∗xi + Riαa∂̄
a = −1
2N̄αjg
jk
i
δ∗
δ∗xk
+1
2(∂ ln f
∂xi δγ
α − N̄iβgβγ
α ) δ∗
δ∗zγ + 1
2Riαa∂̄
a,
∇ δ∗
δ∗zα
δ∗
δ∗zβ = −1
2
δ∗fgαβ
δ∗xi gij δ∗
δ∗xj + Γγ
αβ
δ∗
δ∗zγ + 1
2f2 gαβλ∂̇λ + 1
2Rαβa∂̄
a,
(20)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 305
H. Attarchi and M.M. Rezaii
∇∂̇i ∂̇α = ∇∂̇α ∂̇i = 1
8 ∂̇αH2g
ikh ∂f
∂xk
δ∗
δ∗xh
−1
2(f2∂̇i(N̄βγ)gγαgβλ + f∂̇α(N̄βk)gkigβλ) δ∗
δ∗zλ ,
∇∂̇i ∂̇j = −1
2( δ∗gij
δ∗xk + ∂̇i(N̄kt)gtj + ∂̇j(N̄kt)gti)gkh δ∗
δ∗xh
+1
8 ∂̇βH2g
ijk ∂f
∂xk
δ∗
δ∗zβ + 1
2gij
k ∂̇k,
∇∂̇α ∂̇β = −1
2( δ∗fgαβ
δ∗xk + f∂̇α(N̄kγ)gγβ + f∂̇β(N̄kγ)gγα)gkh δ∗
δ∗xh
−f2
2 ( δ∗gαβ
δ∗zγ + ∂̇α(N̄γθ)gθβ + ∂̇β(N̄γθ)gθα)gγλ δ∗
δ∗zλ + 1
2gαβ
γ ∂̇γ ,
(21)
∇ δ∗
δ∗xi
∂̇j = ∇∂̇j
δ∗
δ∗xi − ∂̇j(N̄ik)∂̇k = −1
2 ∂̇j(N̄ik)∂̇k
−1
2(gjh
i + Riksg
sjgkh) δ∗
δ∗xh − f
2Riαkg
kjgαβ δ∗
δ∗zβ
+1
2( δ∗gjk
δ∗xi + ∂̇k(N̄is)gsj)gkh∂̇h + 1
2f ∂̇α(N̄ik)gkjgαβ ∂̇β,
∇ δ∗
δ∗xi
∂̇α = ∇∂̇α
δ∗
δ∗xi − ∂̇α(N̄ia)∂̄a = −1
2 ∂̇α(N̄ia)∂̄a
f
2Rkiβgβαgkh δ∗
δ∗xh + f2
2 Rβiγgγαgβλ δ∗
δ∗zλ
+1
2( 1
f
δ∗fgαβ
δ∗xi + ∂̇β(N̄iγ)gγα)gβλ∂̇λ,
∇ δ∗
δ∗zα
∂̇i = ∇∂̇i
δ∗
δ∗zα − ∂̇i(N̄αβ)∂̇β = −1
2 ∂̇i(N̄αβ)∂̄β
1
2Rkαsg
sigkh δ∗
δ∗xh + f
2Rβαkg
kigβγ δ∗
δ∗zγ + 1
2
δ∗gik
δ∗zα gkh∂̇h
+ 1
2f ∂̇β(N̄αk)gkigβγ ∂̇γ ,
∇ δ∗
δ∗zα
∂̇β = ∇∂̇β
δ∗
δ∗zα − ∂̇β(N̄αa)∂̄a = −1
2 ∂̇β(N̄αa)∂̄a
f
2Rkαγgγβgkh δ∗
δ∗xh − 1
2(gβλ
α + f2Rαγθg
θβgγλ) δ∗
δ∗zλ
− 1
4f δβ
α
∂f
∂xj ∂̇j + 1
2( δ∗gβγ
δ∗zα gγλ + ∂̇γ(N̄αθ)gθβgγλ)∂̇λ,
(22)
where
gabc = ∂̄agbc, gabc = gcfgf
ab = gcfgbeg
ef
a = gcfgbegadg
def
and
Γk
ij =
gkh
2
(
δ∗gjh
δ∗xi
+
δ∗gih
δ∗xj
− δ∗gij
δ∗xh
)
,
Γγ
αβ =
gγλ
2
(
δ∗gβλ
δ∗zα
+
δ∗gαλ
δ∗zβ
− δ∗gαβ
δ∗zλ
)
.
5. Foliations on Warped Product Hamiltonian Spaces
In this section, we study geometric properties of the vertical distribution
V T ∗M which is bundle-like with respect to the metric G and totally geodesic.
The conditions which are equivalent to these properties show a close relation be-
tween the geometry of the warped Hamiltonian manifold and its base Hamiltonian
spaces.
Theorem 5.1. Let H = (M, H) be a warped product Hamiltonian space with
nonconstant warped function f . Then the warped Sasaki metric G is bundle-like
306 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3
The Warped Product of Hamiltonian Spaces
for the vertical foliation V T ∗M if and only if (M1, (gij)) and (M2, (gαβ)) are two
pseudo-Riemannian manifolds.
P r o o f. With respect to the bundle-like condition (see [2, 9]), G is bundle-
like for V T ∗M if and only if
G(∇XY +∇Y X, Z) = 0, ∀X, Y ∈ Γ(HT ∗M), Z ∈ Γ(V T ∗M).
It is equivalent to the following equations:
G(∇ δ∗
δ∗xi
δ∗
δ∗xj
+∇ δ∗
δ∗xj
δ∗
δ∗xi
, ∂̇k) = G(∇ δ∗
δ∗xi
δ∗
δ∗xj
+∇ δ∗
δ∗xj
δ∗
δ∗xi
, ∂̇α) = 0,
G(∇ δ∗
δ∗uα
δ∗
δ∗uβ
+∇ δ∗
δ∗uβ
δ∗
δ∗uα
, ∂̇i) = G(∇ δ∗
δ∗uα
δ∗
δ∗uβ
+∇ δ∗
δ∗uβ
δ∗
δ∗uα
, ∂̇γ) = 0,
G(∇ δ∗
δ∗xi
δ∗
δ∗uα
+∇ δ∗
δ∗uα
δ∗
δ∗xi
, ∂̇j) = G(∇ δ∗
δ∗xi
δ∗
δ∗uα
+∇ δ∗
δ∗uα
δ∗
δ∗xi
, ∂̇β) = 0.
By using (19)–(22), one can obtain that above equations are satisfied if and only
if gijk = gαβγ = 0, and this completes the proof.
Theorem 5.2. Let H = (M, H) be a warped product Hamiltonian space with
nonconstant warped function f . Then, H = (M, H) is a Landsberg–Hamilton
space if and only if the vertical foliation V T ∗M is totally geodesic.
P r o o f. With respect to the definition of the Landsberg–Hamilton space [8],
(M, H) is a Landsberg–Hamilton space if and only if
gab|∗c =
δ∗gab
δ∗xc
+ gbd∂̇a(N̄dc) + gad∂̇b(N̄dc) = 0.
By using (19)–(22), one can check that
gab|∗c = 0
is satisfied if and only if V T ∗M is totally geodesic, and this completes the proof.
Theorem 5.3. Let H = (M, H) be a warped product Hamiltonian space with
nonconstant warped function f . Then the horizontal distribution HT ∗M is a
totally geodesic one if and only if (M, H) is an Euclidean space.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 3 307
H. Attarchi and M.M. Rezaii
P r o o f. Suppose that HT ∗M is a totally geodesic distribution, then
∇ δ∗
δ∗xi
δ∗
δ∗xj
, ∇ δ∗
δ∗xi
δ∗
δ∗uα
, ∇ δ∗
δ∗uα
δ∗
δ∗xi
, ∇ δ∗
δ∗uα
δ∗
δ∗uβ
∈ Γ(HT ∗M).
From (20), the above conditions are hold if and only if
Rabc = gabc = 0.
These equations mean that (M, H) is an Euclidean space (the pseudo-Riemannian
space with zero curvature).
Combining Theorems 5.1 and 5.2, we have the following corollary.
Corollary 5.1. Let the warped product Hamiltonian space (M,H) be a pseudo-
Riemannian manifold with nonconstant warped function f , then the vertical dis-
tribution V T ∗M is totally geodesic and the metric G is bundle-like for V T ∗M .
Acknowledgement. The first author would like to thank the INSF for the
partially support on the grant number 92006616.
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