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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 36, Num. 2 (2021), 279 – 288

doi:10.17398/2605-5686.36.2.279

Available online November 3, 2021

Characterization of symplectic forms induced by some
tangent G-structures of higher order

P.M. Kouotchop Wamba 1, G.F. Wankap Nono 2

1 Department of Mathematics, Higher Teacher Training college
University of Yaoundé 1, PO.BOX 47 Yaoundé, Cameroon

2 Department of Mathematics and Computer Science, Faculty of Science
University of Ngaoundéré, PO.BOX 454 Ngaoundéré, Cameroon

wambapm@yahoo.fr , georgywan@yahoo.fr

Received September 6, 2021 Presented by J.C. Marrero
Accepted September 30, 2021

Abstract: Let (M,ω) be a symplectic manifold induced by an integrable G-structure P on M. In

this paper, we characterize the symplectic manifolds induced by the tangent lifts of higher order

r ≥ 1 of G-structure P , from M to TrM.

Key words: Complete lifts of differential forms, prolongations of functions, vector fields and G-

structures.

MSC (2020): 53C15; secondary 53C75, 53D05, 53D17.

1. Introduction

Let M be a smooth manifold of dimension n ≥ 1. The tangent bundle of
order r of M is the n(r + 1)-dimensional manifold TrM of r-jets at 0 ∈ R of
differential mapping % : R → M. We denote by πr,M : TrM → M the canon-
ical projection defined by πr,M (j

r
0%) = %(0). Let (U,x

i) be a local coordinate
system of M, we denote by (xi,xiα) the adapted local coordinate of T

rM over
the open set TrU, we have:{

xi(jr0%) = x
i(%(0)),

xiβ(j
r
0%) =

1
β!

dβ

dtβ
(xi ◦%)(t) |t=0 .

The differential geometry of the tangent bundles of higher order has been
extensively studied by many authors, for instance I. Kolar ([1]), Morimoto ([3]
and [4]). It plays an essential role in the description of a Lagrangian formalism
of higher order. On the other hand, we denote by FM the frame bundle of M,
it is GL(n)-principal bundle, where GL(n) is a linear Lie group of Rn. Let G be

ISSN: 0213-8743 (print), 2605-5686 (online)

© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.36.2.279
mailto:wambapm@yahoo.fr
mailto:georgywan@yahoo.fr
https://publicaciones.unex.es/index.php/EM
https://creativecommons.org/licenses/by-nc/3.0/


280 p.m.k. wamba, g.f. wankap nono

a Lie sub-group of linear group, we recall that, a G-structure on M is a sub-G-
principal bundle (P,M,pM ) of FM. We know that some structures of classical
differential geometry can be described by some G-structures. For instance,
the almost complex structures, the Riemannian structures, the symplectic
structures and regular foliations. That why, the study of tangent lifts of higher
order of these structures has their importance in the calculus of variations and
some problems connected as the Hamiltonization problems of higher order.

By the canonical linear action ρn : GL(n)×Rn → Rn, we define an injective
morphism of Lie groups j

〈r〉
n : T

r(GL(n)) → GL(n(r + 1)) such that, for any
manifold M, we have (see [3]) a natural principal bundle morphism j

〈r〉
M :

TrFM → FTrM over idTrM . The mapping is an embedding. Let (P,M,pM )
be a G-structure, we set

Gr = j〈r〉n (T
rG),

T rP = j〈r〉M (T
rP).

The principal bundle (T rP,TrM,Gr) is a Gr-structure on TrM. It is called
tangent G-structure of order r. In [3], it has showed that the tangent G- struc-
ture of order r is integrable if and only if the initial structure is integrable.
In particular, this prolongation generalize simultaneously the tangent lifts of
higher order of almost complex structures, Riemannian structures, symplec-
tic structures and regular foliations. In the particular case where r = 1,
A. Morimoto has shown the following result.

Theorem 1. ([2]) Let G be a Lie group generated by all elements u ∈
GL(n) invariant with respect to a bilinear form f on R2m. We denote by
(T P,TM,T pM ) the tangent lift of integrable G-structure (P,M,pM ). Let ωG
be a symplectic form induced by P and ωG1 the symplectic form induced by
(T P,TM,T pM ). We have:

ωG1 = (ωG)
(c) (1)

where (ωG)
(c) is the complete lift of ωG from M to TM.

In the case where r ≥ 2, we do not have similar result. In this paper,
we propose a generalization of this result. Thus, this paper is structured as
follows. In Section 2, we recall some prolongations of differential forms from
a manifold M to TrM. In Section 3, we prove the main result of this paper.



characterization of symplectic forms 281

2. Complete lift of differential forms

In this section we recall briefly the main result of A. Morimoto [4], about
lifts of functions, vector fields and differential forms to the tangent bundle of
higher order. These results will be used in the main section. Let M be a
smooth manifold of dimension m ≥ 1. Let f be a function of class C∞ and
α ∈{0, . . . ,r}, we define the α-lift f(α) as the function on TrM given by the
formula

f(α)(jr0%) =
1
α!

dα

dtα
(f ◦%)(t) |t=0 .

If α is negative, then we set f(α) = 0.

The family of α-lifts of functions is very important because, if X and Y
are vector fields such that X(f(α)) = Y (f(α)) for all functions f on M and
α = 0, . . . ,r, then X = Y .

Proposition 1. Let X ∈ X(M) and 0 ≤ α ≤ r. There is one and only
one vector field X(α) on TrM such that:

X(α)
(
f(β)

)
= (Xf)(β−α)

for any f ∈ C∞(M) and 0 ≤ β ≤ r.

Proof. See [4].

The vector field X(α) is called α-prolongation of X. For some properties
of X(α), see for instance [1] or [4]. Let (U,xi) be a local coordinate system of
M and (xi,xiβ) be a local coordinate system on T

rM over TrU induced by

(U,xi) such that X = ai ∂
∂xi

, with ai ∈ C∞(U) we have:

X(α) = (ai)(β−α)
∂

∂xiβ
.

Proposition 2. Let ω be a differential form of degree p. It exists on TrM
one and only one differential form of degree p denoted by ω(c) verifying:

ω(c)
(
X

(β1)
1 , . . . ,X

(βp)
p

)
=
(
ω(X1, . . . ,Xp)

)(r−(β1+···+βp))
(2)

for all X1, . . . ,Xp ∈ X(M) and β1, . . . ,βp ∈{0, . . . ,r}.

Proof. See [4].



282 p.m.k. wamba, g.f. wankap nono

Definition 1. The differential form ω(c) is called complete lift of ω from
M to TrM.

Let
{
x1, . . . ,xn

}
be a local coordinates system of M such that, the local

expression of ω is given by:

ω = ωi1···ipdx
i1 ∧·· ·∧dxip.

The local expression of ω(c) is given by:

ω(c) =
∑

β1+···+βp+β=r
(ωi1···ip)

(β)dxi1β1 ∧·· ·∧dx
ip
βp
. (3)

In the particular case where p = 2, i.e., ω = ωijdx
i∧dxj, then the differential

form ω(c) has the matrix form
 (ωij)

(r) · · · ωij
... ..

. ...
ωij · · · 0


 .

It is called matrix representation of the complete lift ω(c). In particular, if ω
is non degenerate, then ω(c) is also non degenerate.

Corollary 1. Let ω be a differential form of degree p on M. We have:

d
(
ω(c)

)
= (dω)(c). (4)

Proof. See [4].

Remark 1. This corollary shows that, if ω is closed (resp. exact) then ω(c)

is closed (resp. exact). Thus, if ω is a symplectic form on M then ω(c) is a
symplectic form on TrM. On the other hand, the method employed for the
description of the complete lifts of differential forms can be extended to a
symmetric tensor. In particular for a symmetric tensor g of type (0, 2) on M,
if locally g = gijdxi ⊗dxj, then the complete lift g(c) is given by:

g(c) = (gij)
(r−α−β)dxiα ⊗dx

j
β.



characterization of symplectic forms 283

3. The main result

Let f be a bilinear symmetric (resp. skew symmetric n = 2m) non degen-
erate form on Rn. In [3], the author shows that: if G is a Lie sub-group in
I(f), where I(f) is the Lie sub-group of all elements u ∈ GL(n) such that, for
any x,y ∈ Rn,

f
(
u(x),u(y)

)
= f(x,y),

then Gr is the Lie sub-group in the Lie group I(f〈r〉), where τr : T
r(R) → R

is defined by τr(j
r
0%) =

1
r!
dr(%)
dtr

(t) |t=0 and the bilinear form f〈r〉 = τr ◦Trf is
symmetric (resp. skew symmetric).

Remark 2. Let (P,M,pM ) be a G-structure, where G ⊂ I(f) as above,
f is skew symmetric and non degenerate (n = 2m). Let φ : M → P be a
section, for any x ∈ M the map

TxM ×TxM −→ R
(ux,vx) 7−→ f

(
φ(x)−1(ux),φ(x)

−1(vx)
)

is bilinear, skew symmetric and does not depend of φ. We denote it by ωG(x),
in particular we obtain an almost symplectic form ωG on the manifold M.
If (P,M,pM ) is integrable, then ωG is a symplectic form on M. It is called
symplectic form induced by (P,M,pM ).

Theorem 2. Let f and f〈r〉 be as above. Let G be a Lie subgroup of the
Lie group I(f). Let (P,M,pM ) be an integrable G-structure on M (dim M =
n = 2m). We denote by ωG and ωGr the symplectic forms induced by P and
T rP respectively. We have:

ωGr = (ωG)
(c). (5)

Let φ ∈ Γ(P), where Γ(P) denote the space of smooth sections of P , we
have:

φ(x)(ej) = φ
i
j(x)

( ∂
∂xi

)
x
. (6)

By the same process defined in [3], we prolong φ from P to T rP and we obtain
the section Φ on T rP . The matrix form of Φ is given by:


φij · · · 0
...

...
...

(φij)
(r) · · · φij


 .



284 p.m.k. wamba, g.f. wankap nono

Thus, we deduce that, for any i,j ∈{1, . . . ,n} and α,β ∈{0, . . . ,r}, we have:

Φi+nαj+nβ =
(
φij
)(α−β)

. (7)

Lemma. We denote by (e`+nα)`,α the canonical basis of Rn(r+1). Using
the matrix form of Φ, we deduce that:

Φ(ej+nα) =
(
φij
)(α−β) ∂

∂xiβ
(8)

for any j ∈{1, . . . ,n} and α ∈{0, . . . ,r}.

Proof. It comes from previous equation.

Proof of Theorem 2. We denote by f : Rn×Rn → R the bilinear skew sym-
metric, non degenerate form and (aij)1≤i,j≤n its matrix form (n = 2m). The
matrix form of the bilinear skew symmetric form f〈r〉 : Rn(r+1) ×Rn(r+1) → R
is given by: 

 0 · · · aij... ... ...
aij · · · 0


 .

We have, for any i,j ∈{1, . . . ,n} and α,β ∈{0, . . . ,r},

f〈r〉(ei+nα,ej+nβ) = δ
α
r−βaij (9)

where δαr−β is the Kronecker symbol. We denote by ωGr the symplectic
form induced by an integrable Gr-structure T rP . For any X ∈ TrM and
α,β ∈{0, . . . ,r},

f〈r〉(ei+nα,ej+nβ) =

= f〈r〉
(

Φ(X)−1
((
φki
)(γ−α) ∂

∂xkγ

)
, Φ(X)−1

((
φsj
)(µ−β) ∂

∂xsµ

))

= ωGr

((
φki
)(γ−α) ∂

∂xkγ
,
(
φsj
)(µ−β) ∂

∂xsµ

)

=
(
φki
)(γ−α) ·(φsj)(µ−β)ωGr( ∂∂xkγ , ∂∂xsµ

)
.



characterization of symplectic forms 285

We deduce that:

(
φki
)(γ−α) ·(φsj)(µ−β)ωGr( ∂∂xkγ , ∂∂xsµ

)
= δαr−βaij. (10)

We set

ωGr

(
∂

∂xkγ
,
∂

∂xsµ

)
= $k+nγ,s+nµ. (11)

In this case the equation 10 becomes:

(
φki
)(γ−α) ·(φsj)(µ−β)$k+nγ,s+nµ = δαr−βaij. (12)

Fact 1: For α = r. If β = r, then the equation 12 becomes

φki ·φ
s
j$k+nr,s+nr = 0. (13)

We deduce that

$k+nr,s+nr = 0. (14)

Let β 6= 0, we suppose that, for any µ > β , $k+nr,s+nµ = 0. We have:

φki ·
(
φsj
)(µ−β)

$k+nr,s+nµ =

= φki ·φ
s
j$k+nr,s+nβ +

r∑
µ=β+1

φki ·
(
φsj
)(µ−β)

$k+nr,s+nµ

= φki ·φ
s
j$k+nr,s+nβ = 0.

We deduce that:

$k+nr,s+nβ = 0 for any β 6= 0.

For β = 0, we have:

r∑
µ=0

φki ·
(
φsj
)(γ)

$k+nr,s+nγ = φ
k
i ·φ

s
j$k+nr,s +

r∑
µ=1

φki ·
(
φsj
)(γ)

$k+nr,s+nγ

= φki ·φ
s
j$k+nr,s = aij.

Now φki ·φ
s
j$k,s = aij, we deduce that:

$k+nr,s = $k,s. (15)



286 p.m.k. wamba, g.f. wankap nono

Fact 2: If α = r − 1. By a similar calculus we obtain: for any β ∈
{1, . . . ,r}, we have: {

$k+n(r−1),s+nβ = 0 if β ≥ 2,
$k+n(r−1),s+nβ = ωks if β = 1.

For β = 0, in the equation 12, we have:(
φki
)(γ−α) ·(φsj)(µ)$k+nγ,s+nµ =

=
(
φki
)(1) ·(φsj)(µ)$k+nr,s+nµ + φki ·(φsj)(µ)$k+n(r−1),s+nµ

=
(
φki
)(1) ·φsj$k+nr,s + φki ·(φsj)(1)$k+n(r−1),s+n + φki ·φsj$k+n(r−1),s

=
(
φki
)(1) ·φsj$k,s + φki ·(φsj)(1)$k,s + φki ·φsj$k+n(r−1),s

= −φki ·φ
s
j($k,s)

(1) + φki ·φ
s
j$k+n(r−1),s = 0.

We deduce that:
$k+n(r−1),s = ($k,s)

(1). (16)

Fact 3: We conjecture that $k+nα,s+nβ = ($k,s)
(r−α−β). By induction,

we fix α and we suppose that for any γ > α and λ > β, we have:

$k+nγ,s+nλ = ($k,s)
(r−γ−λ). (17)

Thus,(
φki
)(γ−α) ·(φsj)(µ−β)$k+nγ,s+nµ

= φki ·φ
s
j$k+nα,s+nβ +

∑
γ,λ>α,β

(
φki
)(γ−α) ·(φsj)(λ−β)$k+nγ,s+nλ

= φki ·φ
s
j$k+nα,s+nβ +

∑
γ,λ>α,β

(
φki
)(γ−α) ·(φsj)(λ−β)($k,s)(r−γ−λ)

= φki ·φ
s
j$k+nα,s+nβ +

(
φki ·φ

s
j$k,s

)(r−α−β) −φki ·φsj($k,s)(r−α−β)
= φki ·φ

s
j$k+nα,s+nβ + δ

α
r−βaij −φ

k
i ·φ

s
j($k,s)

(r−α−β).

Using the equation 10, we deduce that:

φki ·φ
s
j$k+nα,s+nβ = φ

k
i ·φ

s
j($k,s)

(r−α−β). (18)



characterization of symplectic forms 287

Therefore $k+nα,s+nβ = ($k,s)
(r−α−β). Thus, we deduce that

ωGr = $i+nα,j+nβdx
i
α ∧dx

j
β

= ($k,s)
(r−α−β)dxiα ∧dx

j
β.

Thus ωGr = (ωG)
(c).

Intrinsic second proof of Theorem 2. Let (P,M,pM ) be an integrable
G-structure as above. Because of the integrability of P and the naturality,
we may assume M = Rn and P = Rn × G ⊂ Rn × GL(n) = LRn. Then
TRn = Rn × Rn and ωG = idRn ×f : TRn ×Rn TRn = Rn × (Rn × Rn) → R.
More TrRn = Rn(r+1), T rP = TrRn ×Gr ⊂ TrRn ×GL(TrRn) = L(TrRn)
and T(TrRn) = TrRn × TrRn and ωGr = idTrRn × f〈r〉 : TrRn × (TrRn ×
TrRn) = T(TrRn) ×TrRn T(TrRn) → R, where f〈r〉 + τr ◦ Trf : TrRn ×
TrRn → R and τr : TrR → R is the respective functional. Further, (ωG)(c) +
τr ◦ Tr(ωG) modulo the exchange isomorphism T(TrM) = Tr(TM) and the
product preserving identification Tr(TM×M TM) = Tr(TM)×TrM Tr(TM).
Then

(ωG)
(c) = τr ◦Tr

(
idRn ×f

)
= idTrRn ×

(
τr ◦Trf

)
= idTrRn ×f〈r〉 = ωGr.

The proof is complete.

Corollary. Let G be a Lie group generated by all elements of linear
group invariant with respect to some bilinear symmetric non degenerate form
f. Let (P,M,pM ) be a G-structure on M. We denote by gG and gGr the pseudo
Riemannian metric on M and TrM induced by P and T rP respectively. We
have:

gGr = (gG)
(c). (19)

Proof. The proof is similar to the proof of Theorem 2.

Acknowledgements

The authors would like to thank the anonymous reviewers for their
valuable suggestions and remarks which improved the quality of this
paper.



288 p.m.k. wamba, g.f. wankap nono

References

[1] I. Kolar, P. Michor, J. Slovak, “Natural operations in differential geo-
metry”, Springer-Verlag, Berlin 1993.

[2] A. Morimoto, Prolongations of G-structures to tangent bundles, Nagoya
Math. J. 32 (1968), 67 – 108.

[3] A. Morimoto, Prolongations of G-structures to tangent bundles of higher
order, Nagoya Math. J. 38 (1970), 153 – 179.

[4] A. Morimoto, Liftings of some types of tensors fields and connections to
tangent bundles of pr-velocities, Nagoya Math. J. 40 (1970), 13 – 31.


	Introduction
	Complete lift of differential forms
	The main result