On F−flat structures in vector bundles over foliated manifolds
C. APREUTESEI
Department of Mathematics
“Alexandru Ioan Cuza” University of Iaşi
Iaşi - 700506, Carol Blv. 11
ROMANIA
Abstract: We give the definition of the families of F−flat structures and F−flat connections in vector bundles
over F−foliated manifolds. Essential: existence of a F−flat structure is equivalent to the existence of a F−flat
connection. Let {λ
ξ}be a family of subbundles of a vector bundle ξ. There exists a family of F−flat structure
{λ
Λ}in ξ, relative at
λ
ξ, if and only if exists a family of F−flat connections {λ
∇} in ξ(Theorem III.5). F−flat
structures (Theorem III.1), and integrable F−flat structures (Theorem III.5), are considered. Finally, integrable
Γ−structure and F−flat structures on total space of a vector bundle are presented (Theorem IV.1).
Key-Words: F−flat structure, F−flat connection, integrable F−flat structure.
Received: December 21, 2022. Revised: June 6, 2023. Accepted: June 25, 2023. Published: July 25, 2023.
1 Introduction
The notion of foliation of manifold is of great interest
for geometers. It is the basis of some results regarding
the decomposition of tangent bundle of the foliated
manifold into the tangent bundle to the leaves and the
transverse bundle. Many authors have dealt with this
topic from different point of view. The tangent bundle
can be structured in various ways, [1], [2], [6].
In,[5] the foliations studied are induced by geo-
metric structures. In our paper, on the contrary, we
use the F−flat structures to obtain (affine) geomet-
ric structures on the leaves. In other work, [6], the
leaves have remarkable structures, that is piecewise-
linear, differentiable or analytic structure.
The origin of the present work can be found in,
[4]. The notion of F−flat structures was introduced
by us, [3]. In present paper we obtain some interest-
ing characteristic results regarding the links between
F−flat structures and F−flat connections for para-
compact manifolds. We also prove an existence theo-
rem of F−flat structures and formula for connection
which define a F−flat structure.
Here the word “foliation” means a foliated atlas
and a decomposition of a manifold Minto connected
submanifolds of dimension p. We suppose that the
manifolds, foliations, maps are C∞−differentiable
(C∞−diff.) on the morphisms of vector bundles are
of constant rank. We use terms “fiber bundle with
structure group”, or “vector bundle”. The F−flat
structures and F−flat connections are defined in vec-
tor bundles over foliated manifolds, for which the
transition functions are constant along the leaves of
foliation F. Suppose that the leaf topology admits a
countable base.
Convention: i, i′, j, j′, k, k′, ... = 1,2, ..., p;
bi,bi′,b
j, b
j′,b
k, b
k′, ... =p+ 1, p + 2, ...;a, b, c, ... =
1,2, ..., m (or m+n). m=dimM.
We use the classical summation convention for in-
dices.
2 Families of F−flat structures
(F.f.s.) and F−flat connections
(F.f.c.) on vector bundles
The principal tool of this section is to present some
relations between F.f.s.
λ
Λand F.f.c.
λ
∇defined
in vector bundles over foliated manifolds. Let M
be a C∞−diff., paracompact, F−foliated manifold,
where F={(Ud,Ψα)}α∈I={(Ud, xk, x
b
k)}α∈I.
Let ξ= (E, π, M)be a vector bundle over M;Eis
total space of ξ,n=its projection, and Ru=fiber of
ξ.
Denote: T M is tangent bundle of M, and TFis
the tangent bundle of F. Here (xk)are coordinates
in a leaf of F,bx= (x
b
k) =secondary coordinates.
Consider a family of subgroups {λ
G}of GL(n, R),
λ= 1,2, ... and r=rangξ. The set of all sections of a
vector bundle (; ) is denoted Γ(; ). Let
λ
ξ= ( λ
E,
λ
Π, M)
be the subbundle of ξwith structure group
λ
G;
λ
Eis to-
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tal space of
λ
ξ,
λ
Π =projection, and Ru=fiber of
λ
ξ.
Denote
λ
Aα,β :Uα∩Uβ→λ
Gthe transition functions
of ξwhich define
λ
ξ,α, β ∈I. We use sections of ξ
relative to ξ, i.e. sd:Ud→λ
E/Uα. Consider the open
covering {Uα}α∈Iof Mand sα, sβtwo local frames
fields of
λ
ξon Uα, Uβ, respectively.
Definition 2.1. A set
λ
Λ = {(sα,
λ
Aαβ)/sα=
λ
Aαβsβ}defines an F−flat structure in vector bundle
ξrelative to
λ
ξif the transition functions
λ
Aαβ are con-
stant along the leaves of F, i.e.
λ
Aαβ(x) = λ
Aαβ(x
b
k),
x= Ψα(xk, x
b
k) = Ψ−1
β(xk′, x
b
k′). The vector bundle
ξendowed with an F−flat structure
λ
Λis called an
F−flat vector bundle relative to
λ
ξ.
Let
λ
∇be a connection of
λ
ξ.
Definition 2.2. The frame fields σα, σβof
λ
ξare
parallel at the connection
λ
∇along leaves of F−flat
(p. a. l. F) if
λ
∇Xσα=λ
∇Xσβ= 0,∀X∈Γ(TF).
Lemma 2.3. Let
λ
∇be a connection on
λ
ξ,h=
(Uα;xk, x
b
k)∈F, and σα, σβframe fields of
λ
ξ
(p.a.l.F), and σα=Bαβ σβon Uα∩Uβ.
If
λ
∇Xσα=λ
∇Xσβ= 0, then:
Bαβ are constant along the leaves of F.
Proof. Let ω, ω′be the connection forms of
λ
∇with respect to σα, σβ, respectively. Then
ω(X) = B−1
αβ ω′(X)Bαβ +B−1
αβ dBαβ (X), and
dBαβ (X) = 0, for X=∂
∂xk∈Γ(TF). Hence
∂Bαβ
∂xk= 0.
Using this lemma, we are able to study some
properties of F-f.s. of ξrelative to
λ
ξ.
Definition 2.4. The connection
λ
∇of
λ
ξis F-flat
along the leaves of Fif its curvature Ω( λ
∇)satisfies
the condition Ω( λ
∇(X, Y )) = 0,∀X, Y ∈Γ(TF).
The following result justifies the denomination of
“F-flat structure”.
Theorem 2.5. Consider a vector bundle
ξ= (E, π, M)over a paracompact, F-foliated
manifold M. Let
λ
ξbe a subbundle of ξ. There exists
an F-flat structure
λ
Λin
λ
ξif and only if exists an
F-flat connection
λ
∇in
λ
ξ.
Proof. Consider, for λarbitrary fixed, an F-flat
connection
λ
∇in
λ
ξ, and esα= (esb
a)a frame field of
E/Uα,a, b = 1,2, ..., n. We determine a frame field
sd=λ
Adβ ·esβ,sα= (sb
a),
λ
Aαβ = (Ab
a)such that
λ
∇Xsα= 0,X∈Γ(TF)where
λ
Aαβ is an unknown
matrix. Denote λ
ω= (λ
ωb
a)the connection form of
λ
∇
relative to esα= (esb
a), where λ
ωb
a=λ
Γb
akdxk+λ
Γb
a
b
kdx
b
k.
On Uα∩Uβ=∅, we have:
λ
∇∂
∂qksα=λ
∇∂
∂k
q
(λ
Ab
aesb) =
∂
λ
Ab
a
∂xk+λ
Ac
a
λ
Γb
ckesb= 0. Therefore, (λ
Ab
a)satisfies
the equations
∂
λ
Ab
a
∂xk+λ
Ac
a
λ
Γb
ck = 0.(2.1)
In this system, x= (xk)are independent variables
and ex= (x
e
k)are parameters. We transform this sys-
tem in the Pfaff system d
λ
Ab
c+λ
Ac
aΓb
ckdxk= 0, where d
is the exterior differentiation operator. Using Frobe-
nius theorem, [7], and det(λ
Ab
a)= 0, we obtain the
following compatibility conditions:
∂
λ
Γb
ai
∂xk−∂
λ
Γb
ak
∂xi+λ
Γc
ai
λ
Γb
ck −λ
Γc
ak
λ
Γb
ci = 0.(2.2)
On the other hand, Ω( λ
∇)∂
∂xk,∂
∂xi= 0, where
Ω( λ
∇)denotes the curvature of
λ
∇. These relations co-
incide with the relation (2.2). Hence exists:
λ
Aαβ=
(λ
Ab
a)and sα= (sa), where
λ
∇Xsα=λ
∇Xsβ= 0,
X∈Γ(TF). Then, using the lemma 2.3, the set
λ
Λ = {(sα,
λ
Aαβ)}αβ∈Iis an Fp.s. in
λ
ξ.
Conversely: let
λ
Λ = {(sα,
λ
Aαβ)}αβ∈Ibe an
F.f.s. in
λ
ξ, where
λ
Aαβ(x) = λ
Aαβ(x
λ
k),x∈Uα∩Uβ.
Over Uαdefine an operator
λ
∇on E/Uα, hence:
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λ
∇α
Xsα= 0,X∈Γ(TF),sα∈λ
Λ. Extends
λ
∇α
X
to s=λasa:
λ
∇α
Xs=X(λa)sa,λa:Ud→R
are functions. Now, let {aα}be a partition of
unity subordinate to {Uα}. Then, we define:
(λ
∇α
Xs)(x) = Pαaα(x)( λ
∇α
Xs)(x),∀x∈M. The
operator
λ
∇αsatisfies the condition from the defini-
tion of a connection, for X∈Γ(TF). Let T⊥F
be a subbundle of T M complementary of TF,
T M =TF⊕T⊥F. Now, define a connection
in
λ
ξ,D: Γ(T M)×Γ( λ
E)→Γ( λ
E)by the relation
DZs=λ
∇α
Xs+∇
s
X⊥, where ∇is a fixed connection
in
λ
ξ,Z=X+X⊥∈Γ(T M),X∈Γ(TF),
X⊥∈Γ(T⊥F). The connection Dis F-flat.
Indeed, DXs=
λ
∇α
Xs, and DXsα= Θα(X)sα= 0,
where Θα= (Θb
a)is the matrix of connection D
relative to sαand Ω(D)∂
∂xi,∂
∂xk= 0;Ω(D)
denote the curvature of D. Hence, Dcorresponds to
λ
Λ. The theorem is proved.
Use precedent notations.
Proposition 2.6. Let Rbe the principal frame
bundle of ξ. Then exists a Ff.s.
λ
Λin ξif and only if
there exists a subbundle
λ
Rof R.
Proof. The total space of
λ
Ris
λ
E=[
x∈M
{sα(x)∈λ
Λ(x)}x∈I,
where
λ
Λ(x) = (sα(x), Aα,β (x))α,β∈I.
The projection λ
πof
λ
Ris λ
π:λ
E→M,λ
π(sα(x)) =
x,x∈M.
The reciprocal results using the method to demon-
strate the precedent theorem.
3 Remarkable F-flat structures
The aim of this Section is to highlight the link between
the integrability of Γ-structures and F−f.s. This in-
tergrability can be achieved using a special atlas of
differentiable structure of manifold.
These remarkable F−f.s. show interest in the to-
tal space of a vector bundle.
3.1 Families of F-flat structures and tensor
fields
Let T1
2(M)be the set of C∞−diff tensor fields of
type (1
2)defined on M. Consider a family of connec-
tions
α
∇=D+αt, where Dis a given connection
on M,t∈T1
2(M)and α∈R. Connection Dand
tare symmetric along the leaves of Tif the torsion
of D,T(D)(X, Y ) = 0 and t(X, Y ) = t(Y, X),
∀X, Y ∈Γ(TT). Denote Ω( α
D)b
kℓa,Ω(D)b
kℓatb
ka the
components of curvatures Ω( α
D),Ω(D)and t, along
the leaves of F, relative to chart h= (U;xk, z
b
k).
Using precedent notations, we have
Theorem 3.1. Let Mbe a C∞−diff., paracom-
pact, and F−foliated manifold.
1. If Dand tare symmetric along the leaves of F,
then
α
∇is a symmetric connection along the leaves of
F.
2. If Ddefines an F-flat structure λon M, then
α
∇defines an F-flat structure
α
∇on Mif and only if
∂tb
ℓa
∂xk−∂tb
ka
∂xℓ+tc
ℓaΓb
kc −tc
kaΓb
ℓc = 0, k, ℓ = 1,2, ..., p.
Proof. 1. This affirmation is clear from the re-
lation T(α
∇)(X, Y ) = T(D)(X, Y )+[t(X, Y )−
t(Y, X)],X, Y ∈Γ(TF).
2. Using the definition of curvatures Ω( α
∇),Ω(D)
along the leaves of F, we obtain the relations
Ω( α
∇)∂
∂kk,∂
∂xℓ∂
∂xa= Γ( α
∇)b
kℓa
∂
∂xb+
t∂tb
ℓa
∂xk−∂tb
kc
∂xℓ+tc
ℓaΓb
kc −∂tb
kc
∂yℓtc
kaΓb
ℓc∂
∂xb,
∂
∂xb∈Γ(T M /U),∂
∂xk,∂
∂xℓ∈Γ(TF/U),
k, ℓ = 1,2, ..., p;a, b, c = 1,2, ..., m.
Remark (*). The Theorem 2.5 gives, for ξ=
T M =λ
ξ, the result: there exists an F.f.s in T M
if and only if there exists an F.f.c. in T M .
Now, the affirmation 2 results from the precedent
remark.
3.2 Integrable F-flat structures
The following is a consequence of Lemma 2.3 for
ξ=T M =λ
ξ.
Lemma 3.2. Let σα, σβ,δα=Bαβ δβbe two
arbitrary frame fields of T M/Uα∩Uβ,σα=Bαβ σβ
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on Uα∩Uβ, and ∇a connection of T M . If
∇∂
∂xkσα=∇∂
∂xkσβ= 0, then Bαβ are independent
from (xk).
Definition 3.3. We say that an F-flat structure
Λon T M is integrable if Λis defined by the family
∂
∂xaof natural frames and Jacobian matrices Jαβ =
n∂xa′
∂xao,α, β ∈I.
Let {Γa
bc}be the coefficients of a connection ∇
relative to the chart h= (U;xk, x
b
k).
Definition 3.4. An arbitrary vector field t=ta∂
∂xa
on Mis parallel relative to ∇, along the leaves of F
(p.a.l.F) if covariant derivative of tin connection ∇,
along the leaves of F, is null: ta
|k=∂ta
∂xk+Γa
kbtb= 0,
j, k = 1,2, ..., p;a, b = 1,2, ..., m.
Theorem 3.5. Let Mbe a C∞-diff., paracom-
pact, F-foliated manifold and ∇a connection on
M. Consider an arbitrary C∞-diff. vector field t
on M,t(x)= 0,∀x∈M. Then, there exists an
F-flat structure Λon T M if tis parallel relative to
∇, along the leaves of F. Moreover, in precedent
conditions, ∇is F-flat.
Proof. Step I. Consider t=ta∂
∂xaand ta
|k= 0 (ta
|k
denotes covariant derivative). Then: ∂ta
∂xk=−Γa
kbtb,
and
∂2ta
∂xk∂xi=∂2ta
∂xi∂xk↔
∂Γa
jb
∂xk−∂Γa
kb
∂xj+ Γa
kcΓc
jb −Γa
jcΓc
kbtb= 0.
Because tis arbitrary, t(x)= 0,x∈M, precedent
relations give
∂Γa
jb
∂xk−∂Γa
kb
∂xj+ Γa
kcΓc
jb −Γa
jcΓc
kb = 0.(3.1)
Step II. Let ∂
∂zbbe an arbitrary frame field of
T M/U. We prove that there exists a frame field
∂
∂yaparallel in connection ∇along the leaves
of F. Indeed, let ∂
∂ya=Ab
a∂
∂zkbe a frame
field (p.a.l.F), where A= (Ab
a)is a unknown
matrix, that satisfies the conditions ∇∂
∂xk
∂
∂ya=
0. Therefore, ∇∂
∂xk
∂
∂ya=∇∂
∂xkAb
a∂
∂zb=
∂Ab
a
∂xk+Ac
aΓb
kc∂
∂zb= 0. Hence, Aαβ = (Ab
a)satis-
fies the equations
∂Ab
a
∂xk+Ac
aΓb
kc = 0.(3.2)
To study this system we use Fröbenius theorem,
[7]. The result coincides with the relations (3.1).
Therefore, the system (3.2) is compatible. Hence,
there exists: Aαβ (x) = (Ab
a(x
b
k)) and ∇∂
∂xk
∂
∂ya= 0.
Now, we use Lemma 3.2. Consequently,
there exists on T M, the F-flat structure Λ =
n ∂
∂ya, Aαβ (x
b
k)oαβ∈I.
The second statement follows from the definition
of curvature Ω(∇)along the leaves of F. Indeed, let
esα= (esb)be a frame field of T M/U. Then:
Ω(∇)∂
∂xk,∂
∂xjesb=
=∇∂
∂xk(Γb
jaesb)− ∇ ∂
∂xk(Γb
kaesb) =
= ∂Γb
ja
∂xk−∂Γb
ka
∂xj+ Γc
jaΓb
kc −Γc
kaΓb
jc!esb= 0.
This proves the theorem.
3.3 F-flat structures and vector fields
Lemma 3.6. Let Mbe a C∞diff, paracompact,
F-foliated manifold, taC∞-diff. vector field on M,
t(x)= 0,∀x∈M. A symmetric connection ∇on M
is F-flat if and only if the second covariant deriva-
tions of t, in connection ∇, and along the leaves
of F, coincide; i.e. ta
|ij =ta
ji,i, j = 1,2, ..., p;
a, b = 1,2, ..., m.
Proof. Expression of tin the chart h=
U;xi, x
b
iis t=ta∂
∂xa. Now, we use the defini-
tions of curvature and torsion of a connection (along
the leaves of F). Partial covariant derivations of t
in connection ∇, along the leaves of F, satisfies the
relations:
ta
|i=∂ta
∂xi+ Γa
i btb,and
ta
|ij −ta
|ji = Ω(∇)a
ijbtb−T(∇)b
ij ta
|b
(3.3)
Since ∇is symmetric, Ta
bc(∇)=0, and hence
T(∇)a
ij = 0. Therefore, the relations (3.3) implies:
ta
|ij =ta
|ji = Ω(∇)a
ijbtb.
Then, precedent relations prove Lemma 3.6.
Using precedent results and Remark (*) one obtain
Theorem 3.7. Let Mbe a C∞-diff, paracompact,
F-foliated manifold. Let tbe a C∞-diff. vector field,
t(x)= 0,∀x∈M. If ∇is an F−symmetric connec-
tion on M, then the following affirmations are equiv-
alent:
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1) ∇is an F-flat connection;
2) ∇determines an F-flat structure on M;
3) Mixed covariant derivations of tin connection
∇, along the leaves of F, coincide.
Proof. It is clear, from the Lemma 3.6, that 1)↔2).
From the Remark (*) and Lemma 3.6 follows that
2)↔3). This proves the theorem.
4 Integrable F-flat structures on the
total space of a vector bundle over
an F-foliated manifold
Define a differentiable structure on the total space E
of ξ= (E, π, M)in the following way. Consider
a trivializing atlas A1={(Uα, φα, Rm)}α∈Iof E
and F={(Uα, ψα)}α∈I=nUα;xk, x
b
koα∈I.
Then, the atlas of Eis A=π−1Uα, hαα∈I=
nUα;xk, x
b
k, yaox∈Iwhere hα:π−1Uα→
Rm×Rn,hα(u) = ψα(π(u)), φα,π(u)(u),u∈
π−1Uα⊂E. The coordinates change for Ais: xk′=
xk′xk, x
b
k,x
b
k′=xˆ
k′xˆ
k,ya′=Ma′
a(x)ya,x=
ψ−1
αxk, x
b
k=ψ−1
βxk′xˆ
k′, where (Ma′
a(x)) is
a field-matrices that describes the precedent coordi-
nates change in π−1(x),(Ma′
a(x)) ∈GL(n, R).
An F-flat structure Λon Eis integrable if Λis
defined by the family of frames ∂
∂xk,∂
∂x
b
k,∂
∂ya,a=
1,2, ..., n.
Denote Γ = ( α β 0
0γ0
0δ ε !) where α, β, γ
are p×p,(n−p)×(n−p),n×nreal matrices,
respectively. We remark that Γis a subgroup of
GL(m+n, R).
Theorem 4.1. Let Mbe a C∞-diff., paracom-
pact, F-foliated manifold and ξ= (E, π, M)a vec-
tor bundle over M. Suppose that Ehas a differen-
tiable structure defined by the atlas A. Then:
1) The atlas F={(Uα, xk, x
b
k)}α∈Idefines an
integrable F-flat structure Λon T M if and only if
Finduces a locally affine structure on the leaves of
F. Moreover, in this case,
Λ =  ∂
∂xk,∂
∂x
b
k;1
Jαβ(x)α,β∈I
, x ∈Uα∩Uβ,
where
1
Jαβ(x) = Ai′
i
∂bi′(x
b
k)
∂x
b
i
0∂x
b
i′
∂x
b
i!(x),Ai′
iis a
constant p×pmatrix, det Ai′
i= 0 and bi′x
b
rare
arbitrary real functions on Uα∩Uβ.
2) The atlas Fdefines an integrable F-flat struc-
ture Λ = ∂
∂xk,∂
∂x
b
k,∂
∂ya;2
Jαβ(x)α,β∈I
on T M
if and only if the atlas Adefines an integrable Γ-
structure on the manifold E, where
2
Jαβ(u)∈Γ,
2
Jαβ(u) =
Ai′
i
∂bi′
∂x
b
i0
0∂x
b
k′
∂x
b
k0
0∂e
ga′
a(x
b
k)
∂x
b
ke
ga′
a(x
b
k)
(u),
u∈π−1(Uα∩Uβ),ega′
a(x
b
k)are some real functions,
det(ega′
a(x
b
k)) = 0.
Proof. 1) We determine
1
Jαβ(x). To obtain xi′
consider the system P.D.E. ∂xi′
∂xi=Ai′
i(bx),∂xi′
∂x
b
i=
Bi′
b
i(bx),bx=x
b
i, where Ai′
i(bx),Bi′
b
i(bx)are arbitrary
real functions. The solution of the first equations is
xi′=Ai′
i(bx) + bi′(bx), where bi′(bx)denote arbitrary
real functions. Now, we require that the functions
(xi′)to verify the last equations:
∂Ai′
i
∂x
b
i+∂bi′
∂x
b
i=Bi′
b
i(bx),
i, i′, j, j′= 1,2, ..., p;bi,bi′... =p+1, p+2, ..., m. The
integrability conditions for this system of P.D.E. are
∂2xi′
∂xi∂x
b
j=∂Ai′
i
∂x
b
j∂xi=∂2xi′
∂x
b
j∂xi=∂Bi′
b
i
∂xi= 0.
Therefore, Ai′
i(bx) =constant. Using precedent rela-
tions, we have ∂bi′
∂xi=Bi′
i(bx), and hence bi′depends
only on bx=x
b
i. The coordinate transformations
are: xi′=Ai′
ixi+bi′(bx),x
b
i′=x
b
i′(x
b
i). Hence (xi′)
defines an affine structure on a leaf of F. Therefore
Λ =  ∂
∂xi,∂
∂x
b
i;1
Jαβ(x)α,β∈I
,
where
1
Jαβ(x) = Ai′
i
∂bi′(b
x)
∂x
b
i
0∂x
b
i′
∂x
b
i!(x).
Λis an integrable F-flat structure on T M.
Conversely: Let Fbe an arbitrary leaf of Fde-
fined by the equations x
b
i=constant. Using precedent
WSEAS TRANSACTIONS on MATHEMATICS
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Volume 22, 2023
notations, the locally affine structure on Fis given by
xi′=Ai′
ixi+bi′. Consequently:
Λ =  ∂
∂xi,∂
∂x
b
i;1
Jαβ(x),
1
Jαβ(x) = Ai′
i
∂bi′
∂x
b
i
0∂x
b
i′
∂x
b
i!(x).
2) Remark: If Fdefines an F.f.s. on T M given
by
1
Jαβ(x), then
1
Jαβ(x)is a submatrix of
2
Jαβ(u),
π(u) = x∈Uα∩Uβ. Hence, the problem is to de-
termine all the elements of
2
Jαβ(u)in the hypothesis
that
2
Jαβ(u)is independent from (xi). To obtain the
last line of
2
Jαβ(u), we use the coordinate transforma-
tions:
xk′=Ak′
kxk+bk′(x
b
k), x
b
k′=x
b
k′(x
b
k),
ya′=Ma′
a(x)ya
x=ψ−1
α(xk, x
b
k) = ψ−1
β(xk′, x
b
k′).
(4.1)
By an abuse of notation, we write Ma′
a(xk, x
b
k), for
(Ma′
a◦Ψ−1
α)(xk, x
b
k).
We obtain the system of P.D.E.:
∂ya′
∂xk=∂(Ma′
a(xk, x
b
k))
∂xkya,
∂ya′
∂x
b
k=∂Ma′
a(xk, x
b
k)
∂x
b
kya,
∂ya′
∂ya=Ma′
a(xk, x
b
k).
(4.2)
The solution of the system (4.2) do not depend on
(xk)if and only if there exist some functions fa′
ai(x
b
k),
ga′
a
b
i(x
b
k)so that
∂Ma′
a
∂xi=fa′
ai(x
b
k),∂Ma′
a
∂x
b
k=ga′
ab
i(x
b
k),(4.3)
i, j, k = 1,2, ..., p;bi, b
j, b
k=p+1, p+2, ..., n,a, a′=
1,2, ..., m.
Solve the system (4.3). Obtain Ma′
a=
fa′
ai (x
b
k)xi+ega′
a(x
b
k), where ega′
a(x
b
k)are arbitrary real
functions, det(ega′
a(x
b
k)) = 0. Now we want that Ma′
a
to verify the last equations (4.3):
∂Ma′
a
∂x
b
i=∂fa′
ai(x
b
k)
∂x
b
i+∂ega′
a(x
b
k)
∂x
b
i=ga′
a
b
j(x
b
k).(4.4)
For the integrability conditions we use (4.3) and
(4.4):
∂2Ma′
a
∂xi∂x
b
j=∂fa′
ai
∂x
b
j=∂2Ma′
a
∂x
b
j∂xi=∂ga′
ab
i
∂xi= 0.
Hence fa′
ai=constant, and therefore ∂ega′
a
∂x
b
i=ga′
ab
i(x
b
k),
i.e. ega′
ado not depend on (xi).
Coordinate change of Aare:
xk′=Ak′
ixi+bk′(x
b
k), x
b
k′=x
b
k′(x
b
k),
ya′= (fa′
aixi+ega′
a(x
b
k))ya.(4.5)
The last elements of
2
Jαβ(u)are:
∂ya′
∂xk=∂
∂xk(fa′
akxk+ega′
a)ya=fa′
akya,
∂ya′
∂x
b
k=∂(ega′
a(x
b
k))
∂x
b
kya,
∂ya′
∂ya=fa′
akxk+ega′
a(x
b
k).
The matrix
2
Jαβ(u)defines an F.f.s. if
2
Jαβ(x) =
2
Jαβ(x
b
k). Therefore, fa′
ak = 0.
The last element of
2
Jαβ(u)are:
0,∂e
ga′
a(x
b
k)
∂x
b
k,ega′
a(x
b
k)and hence
2
Jαβ(u)∈Γ.
Therefore, Adefines an integrable Γ-structure on E.
Conversely: If there exists the matrix
2
Jαβ(u)from
theorem, then the existence of
2
Jαβ(u)determines the
existence of
1
Jαβ(u),x=π(u)∈Uα∩Uβ. The
affirmation follows.
Conclusions: The F-flat structures are based on
families of local frames linked to each other by con-
stant matrices along the leaves of foliation.
These structures are usefully in the study of vector
field on Riemannian foliated manifolds. The remark-
able structures included in the work are of interest for
the total space of a vector bundle.
It would be interesting to develop the study of
these structures on analytic complex manifold.
References:
[1] Apreutesei C., Automorphismes d’une
GT−structure, C.R. Acad. Sc., Paris, Vol.
271, 1970, pp. 481-484.
[2] Apreutesei C., Quelques classes caracteristiques
et GT-structures, C.R. Acad. Sc., Paris, 280, 1975,
pp. 41-44.
[3] Apreutesei C., Sur les structures F-plates dans
les fibres vectoriels, An. Şt. Univ. Al. I. Cuza Iaşi,
1, 2004, pp. 105-110.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.63
C. Apreutesei
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Volume 22, 2023
[4] Apreutesei C., Partial trivial structures in real vec-
tor bundles, An. Şt. Univ. Al. I. Cuza Iaşi, LXIII,
2017, pp. 429-440.
[5] Bejancu A., Farran H. R., Foliations and Geomet-
ric Structures, Springer 2006.
[6] Fuks D.B., ”Foliations”, J. Soviet Math. 18 No.
2(1982) , 255-291.
[7] Papuc I. Dan, Geometrie diferenţială, Editura Di-
dactică şi Pedagogică, Bucureşti, 1982.
1/. Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting Pol-
icy)
The authors equally contributed in the present re-
search, at all stages from the formulation of the prob-
lem to the final findings and solution.
2/. Sources of Funding for Research Presented
in a Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
3/. Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.63
C. Apreutesei
E-ISSN: 2224-2880
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Volume 22, 2023