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ACTA UNIVERSITATIS APULENSIS

No 14/2007

ON SOME LINEAR D-CONNECTIONS ON THE TOTAL
SPACE E OF A VECTOR BUNDLE ξ = (E, π, M )

Adrian Lupu
Abstract. Using the theory introduced by R. Miron and M. Anastasiei
[2] and some results from the theory given by P. Stavre [4], [5], [6], we will
obtain in this paper the results from section 2. Based on [4]-[6], we will obtain
other results in a future paper.
2000 Mathematics Subject Classification: 53C60.
Keywords and phrases: linear d-connections on the total space of a vector
(12)

bundle, D operators, w conjugations.
1. Introduction
Let us consider ξ = (E, π, M ) a vector bundle with the base space Mn =
(M, [A], Rn ) a C ∞ differentiable, n-dimensional, paracompact manifold, with
the m-dimensional fiber type. We obtain the total space E, with the structure

En+m = (E, [A], Rn+m ) of C ∞ -differentiable, (n + m)-dimensional, paracompact manifold ([2]).
Let us consider an almost sympectic structure w, on E whose restriction
to the vertical subspace is nondegenerate. It results that (n + m) must be an
even number and that m must also be an even number. Therefore n is an even
number. In these conditions, there is a nonlinear connection, N , given by:
(w)

w(hY, vZ) = 0,

X, Z ∈ X (E).

(1)

It results the decomposition:
w = hw + vw,
119

(2)

A. Lupu - On some linear d-connections on the total space E of a vector ...

where:
(hw)(X, Z) = w(hX, hZ),

(vw)(X, Z) = w(vX, vZ),

X, Z ∈ X (E)

and h, v - the horizontal and vertical projectors associated to N . In a similar
(w)

way, if we have N we will have an horizontal distribution H : u ∈ E → Hu E
(w)

such that Tu E = Hu E ⊕ Vu E. In the followings we will use the notions and
the notations from [2].
Since En+m is C ∞ -differentiable and paracompact it results that there are
linear connections, {D}, on E. The linear connections D, on E, which have the
remarkable geometric property that regarding parallel transport, preserve the
horizontal distribution H and the vertical one, V , have an important role. This
property is important for analytical mechanics and theoretical physics. Such

connections D, are called linear d-connections (or remarkable connections). It
results:
Proposition 1. [2] A linear connection D, on E, for which is fixed a
nonlinear connection N , of h and v projectors is a linear d-connection iff:
hDX vY = 0,

vDX hY = 0.

(3)

In the followings we will consider N = N , without notice that. Into a local
(w)


δ
∂
base, adapted to N :
(i = 1, n, a = 1, m) we can write:
,
δxi ∂y a

δ
δ
= Γijk (x, y) i ,
j
δx
δxk δx

Dhδ

Dv ∂
∂y b

δ
δ
= Γijn+b i ,
j
δx
δx

∂

∂
(x, y) b
= Γn+b
n+a
k
a
∂y
δxk ∂y

(4)

∂
= Γn+c
n+a
a
∂y

(5)

Dhδ

Du∂
∂y b

n+b (x, y)

∂
,
∂y c

h
where (Γijk , Γn+b
n+a k ) = D are the local coefficients of the h-covariant derivan+c
i
tive and (Γjn+b , Γn+a n+b ) = Dv are the local coefficients of the v-covariant
derivative (i, j, k = 1, n, a, b, c = 1, m).
Now we can write the d-tensor field of torsion, which characterize the torsion T of a linear d-connection and the d-tensor field of curvature, which characterize the curvature tensor of one linear d-connection [2].

120

A. Lupu - On some linear d-connections on the total space E of a vector ...
(12)

(21)

1. Special operators D w, D w
(1)

Let us consider D, a d-linear connection on E, given in the adapter base
to N = N , by the local coefficients:
(w)

(1)

(1)

(1)

(1)

D h = ( Γ ijk , Γ n+c
n+a k ),

(1)

D v = ( Γ ij

(1)

n+c
n+b , Γ n+a

n+b )

(2)

D, a linear d-connection, given by the local coefficients:
(2)

(2)

(2)

(2)

n+c
D h = ( Γ ijk , Γ n+a
k ),

(2)

D v = ( Γ ij

(2)
n+b ,

Γ n+a
n+c n+b )

given by:

(2)

(1)

(1)

Γ n+c
n+a

(2)

Γ ij

where:

We have:

(1)

n+b

= Γ ij

k

n+b

= Γ n+c
n+a

n+c
= Γ n+c
n+a k + w

n+b

(1)

D hk wn+b

n+a

(1)

(1)

ir v
n+b + w D n+b wrj ,

(1)

(2)

Γ n+c
n+a

(1)

(2)

Γ ijk = Γ ijk + wir D hk wrj ,

n+b

+ wn+c

n+d

(2)

(1)

D vn+b wn+d

n+a




δ
∂
δ
∂
, wn+a n+b = w
,
,
wir = w
δxr δxi
∂y b ∂y a


∂
δ
= 0, wαβ wβγ = δγα .
,
wr n+b = w
δxr ∂y b


1
1
w = wik dxi ∧ dxk + wn+a
2
2

a

(3)
(4)

∧ δy b

(5)


δ
∂
where (dxi , δy a ) is a local base, dual to the local adapted base
.
,
δxr ∂y b
Let us consider:
(21)i def (2)i
τ jk = Γ jk

(1)

− Γ ijk ,

n+b δy

(2)
(21)n+c
def
τ n+a k = Γ n+c
n+a k

121

(1)

− Γ n+c
n+a

k

(6)

A. Lupu - On some linear d-connections on the total space E of a vector ...
(2)
(21)i
def
τ j n+b = Γ ij n+b

(1)

− Γ ij

(2)
(21)n+c
def
τ n+a n+b = Γ n+c
n+a n+b

n+b ,

(1)

− Γ n+c
n+a

n+b .

(7)

From (1), (2), (6), (7) it results:
(1)

(21)r
jk wrl

0 = D hk wjl − τ
(1)

0 = D hk wn+a

(8)

(21)n+c
n+a k wn+c n+b

(9)

− τ

n+b

(1)

(21)r
j n+c wrl

0 = D vn+c wjl − τ
(1)

0 = D vn+c wn+a

n+b

(10)

(21)n+d
n+a n+c wn+d n+b .

− τ

(11)

It results:
Proposition 1. Let us consider an almost symplectic structure w on E,
(1)

(1)

(1)

and D a linear d-connection with its local coefficients D h , D v . Then (1), (2)
are equivalent with (8)-(11).
Let’s give an interpretation for (8)-(11). In [6] P. Stavre has introduced
(21)

(12)

the operators: D w, D w by:
(12)

(1)

(21)

( D X w)(Y, Z) = ( D X w)(Y, Z) − w(Y, τ (X, Z))
(21)

(2)

(12)

( D X w)(Y, Z) = ( D X w)(Y, Z) − w(Y, τ (X, Z))

(12)
(13)

and has studied their general properties, where:
(21)

(2)

(1)

τ (X, Z) = D X Z − D X Z,

(12)

(1)

(2)

τ (X, Z) = D X Z − D X Z.

(14)

In the general case we have:
(12)

(12)

( D X w)(Y, Z) 6= ( D X w)(Z, Y ),
(12)

(15)

(12)

( D X w)(Y, Z) 6= −( D X w)(Z, Y )
(12)

(21)

D X w 6= D X w.
122

(16)

A. Lupu - On some linear d-connections on the total space E of a vector ...
(1)

The notion of almost simplectic conjugation of two linear connections D,
(2)

D is also introduced in [6]. It has been proved that this is equivalent with the
relation:
(12)

( D X w)(Y, Z) = −2β(X)w(Y, Z)

(17)

where β is a certain 1-form. An ample theory is obtained starting from here.
(2)

(1)

ω

In this case we write D ∼ D.
Taking into account [5] we shall use the w-conjugation theory for two linear
(1) (2)

d-connections D, D on the total space E, equipped with an almost simplectiv
structure w.
(12)

(1)

(2)

Definition 1. If D X w = 0 then we will say that D and D are (w)(2)

w

(1)

absolute conjugated. We shall write: D ∼ D.
abs

(1) (2)

Proposition 2. Let us consider D, D two linear d-connection on E, with
(2)

(1)

w

respect to N = N . Then the relation D ∼ D is characterized by:
abs

(w)

(12)

(12)

D hX hw = 0,

D hX vw = 0

(12)

(18)

(12)

D vX hw = 0,

D vX vw = 0.

(19)
(12)

(1) (2)

Proof. From (1), (2) section 1 and from the definition of D w, since D, D
(12)

are linear d-connections, it results that D w = 0 is characterized only by (18),
(19).
From Proposition 2 and from (1), (2) section 2 it results:
(2)

(2)

w

(1)

Proposition 3. A d-linear connection D such that D ∼ D is well-defined
abs

by (1), (2) and conversely.
(2)

A characterization of a d-linear connection D, defined by (1), (2), is given
in that way:
123

A. Lupu - On some linear d-connections on the total space E of a vector ...
(2)

(1)

w

Proposition 4. The relation D ∼ D is symmetric i.e.:
abs

(2)

(1)

w

(1)

w

(2)

(20)

D ∼ D ⇔ D ∼ D
abs

abs

(2)

(1)

w

Proof. If we have D ∼ D then we will have (1), (2) and conversely. By
direct calculus, it results:

abs

(2)

(12)r
ik wrj

D hk wij = τ

(2)

D hk wn+a

n+b

(21)

(12)n+c
n+a k wn+c n+b

(22)

= τ

(2)

(12)r
i n+b wrj

D vn+b wij = τ

(2)

D vn+c wn+a

(12)

n+b
(1)

(23)

(12)n+d
n+a n+c wn+d n+b

= τ

(24)

(2)

w

and therefore D w = 0. Hence D ∼ D.
abs

Like a corollary, it results:

Proposition 5. If we have (1), (2) then we will have:
(1)

(2)

(2)

Γ ijk = Γ ijk + wir D hk wrj
(2)

(1)

Γ n+c
n+a

k

n+c
= Γ n+c
n+a k + w

(1)

Γ ij

n+b

n+b

= Γ n+c
n+a

(2)

D hk wn+d

n+a

(26)

(2)

= Γ ij

ir v
n+b + w D n+b wrj

(2)

(1)

Γ n+c
n+a

(2)

n+d

(25)

n+b

+ wn+c

n+d

(1) (2) (1)

(27)

(2)

D vn+b wn+d

n+a .

(28)

(2)

Proposition 6. Let us consider D, D, D 6= D two linear d-connections
(2)

w

(1)

(1) (2)

on E, with respect to N = N . If D ∼ D then D, D won’t be w-compatible.
(w)

abs

Proof. A linear d-connection D, on E, is w-compatible (Dw = 0) iff ([2]):
h
DX
hw = 0,

124

h
DX
vw = 0

(29)

A. Lupu - On some linear d-connections on the total space E of a vector ...
v
DX
hw = 0,

v
DX
vw = 0

(1) (2)

(30)
(1)

Let us consider D, D two linear d-connections. If D is w-compatible and
(2)

w

(1)

(1)

(2)

(1)

D ∼ D then we will have (1), (2) with Dw = 0. It results D = D which is a
abs

contradiction.

(2)

(2)

w

(1)

In the same way, if D had been w-compatible and D ∼ D, taking into
abs

(2)

(1)

account of (25)-(28) it would have resulted D = D which is a contradiction.
(2)

(2)

w

(1)

Therefore, it will not exist a d-linear connection D on E so that D ∼ D
abs

(1)

if D is a linear, w-compatible connection on E.
Let us consider a linear connection D on E, with the torsion:
T (X, Y ) = DX Y − DY X − [X, Y ],

X, Y ∈ X (E).

We will denote the coefficients of T in a local base {X } (α = 1, n + m) by:
(α)

σ
T (X X ) = Tαβ
X,
(β)(α)

σ
σ
Tαβ
= −Tβα
.

(σ)

(31)

With these notations,
if D is a linear d-connection
on E and we choose the


δ
∂
adapted local base X = r ; X = α then T will be characterized by
(r)
δx (n+a) ∂y
the d-tensorial fields of the local components:
i
Tjk
= Γijk − Γikj ,

Tji n+b
i
Tn+a

n+b

=

Γij n+b ,

= 0,

n+a
n+a
= Rjk
Tjk

Tjn+a
n+b

(32)

∂Nja
n+a
=
− Γn+b
j
∂y b

n+c
n+c
Tn+a
n+b = Γn+a

n+b

− Γn+c
n+b

(33)
(34)

n+a

where ([2]):

δNka δNja
−
(35)
δxj
δxk
A problem which appears is to establish on what conditions two linear dn+a
=
Rjk

(1)

(2)

(2)

(1)

connections, w-absolute conjugated, D and D have the same torsion: T = T .
125

A. Lupu - On some linear d-connections on the total space E of a vector ...
(1) (2)

Proposition 7. Let us consider D, D two linear d-connections on E such
(2)

w

(1)

(1)

(2)

that D ∼ D. Then T = T iff:
abs

(1)

(1)

D hk wlj − D hj wlk = 0

(36)

(1)

D hk wn+d

n+a

=0

(37)

(1)

D vn+b wlj = 0

(1)

D vn+b wn+d

(38)

(1)

n+a

− D vn+a wn+d

n+b

= 0.

(1) (2)

(39)
(1) (2)

Proof. Because D, D are linear d-connections on E, their torsions T , T
in the local, adapted base have the components (32)-(35).
We already have:
(2)

(1)

n+a
n+a
T n+a
jk = T jk = Rjk

(1)

T in+a

(40)

(2)

n+b

= T in+a

n+b

=0

(41)

(2)

(1)

From the condition T ijk = T ijk and from (1) we obtain the equivalent
relation:
(1)

(1)

D hk wlj − D hj wlk = 0

(42)

From (1) it will also result the equivalent relation:
(1)

D hk wn+d

if we have the condition:

n+a

=0

(43)

(2)

(1)

n+a
T n+a
j n+b = T j n+b .

From (2) we will obtain the equivalent relation:
(1)

D vn+b wlj = 0

(1)

if we have T ij

(2)

n+b

= T ij

n+b .

126

(44)

A. Lupu - On some linear d-connections on the total space E of a vector ...
Furthermore from (2) we obtain the equivalent relation
(1)

D vn+b wn+d

(1)

n+a

− D vn+a wn+d

n+b

=0

(45)

if we have the condition:
(2)

(1)

T n+c
n+a

n+b

= T n+c
n+a

n+b .

We obtain thus (36)-(39) and conversely.
Like a corollary we get, equivalent:
(1) (2)

Proposition 8. Let us consider D, D two linear d-connections on E, so
(2)

w

(1)

(1)

(2)

that D ∼ D. Then T = T iff:
abs

(2)

(2)

D hk wij − D hj wik = 0

(46)

(2)

D hk wn+a

n+b

=0

(47)

(2)

D vn+b wij = 0

(2)

D vn+b wn+c

(48)

(2)

n+a

− D vn+a wn+d

n+b

= 0.

(49)

Definition 2. A linear d-connection on E will be called w-semicompatible
if we have:
h
vw = 0
(50)
Dkh wn+a n+b = 0, DX
v
Dn+b
wij = 0,

v
DX
hw = 0.

(51)

Definition 3. A linear d-connection D, on E, will be called w-semi
Codazzi connection if we have:
Dkh wij = Djh wik
v
Dn+b
wn+c

We have:

n+a

v
= Dn+a
wn+c
(1)

(52)
n+b

(53)

(2)

Proposition 9. Let us consider D, D two linear d-connections on E
(2)

w

(1)

(1)

(2)

(1)

such that D ∼ D. Then T = T iff D is w-semicompatible and it is w-semi
Codazzi.

abs

127

A. Lupu - On some linear d-connections on the total space E of a vector ...
(1)

(2)

Proposition 10. Let us consider D, D two linear d-connections on E
(2)

w

(1)

(1)

(2)

(2)

such that D ∼ D. Then T = T iff D is w-semi compatible and w-semi
Codazzi.

abs

If the distribution H is integrable then we will have:
n+a
= 0.
Rjk
(1) (2)

More general results and the relation between the curvature tensors R, R
will be given in a future paper.
I would like to thank Professor P. Stavre for his support during the preparation of this paper.
References
[1] A. Lupu, The w-conjugation in the vector bundle, RIGA Conference,
Bra¸sov, June 2007.
[2] R. Miron, M. Anastasiei, Fibrate vectoriale. Spat¸ii Lagrange. Aplicat¸ii
ˆın teoria relativit˘a¸tii, Ed. Academiei, 1987.
[3] A. Nastaselu, A. Lupu, On some structures F, F ∗ , National Seminar of
Finsler Lagrange Spaces, Bra¸sov, 2004.
[4] P. Stavre, Fibrate vectoriale, vol. I, II, III, Ed. Universitaria, Craiova,
2004-2006.
[5] P. Stavre, Aprofund˘
ari ˆın geometria diferent¸ial˘
a. Aplicat¸ii la spat¸ii Norden ¸si la teoria relativit˘a¸tii, vol. III, Ed. Universitaria, Craiova, 2006.
[6] P. Stavre, Introducere ˆın teoria structurilor geometrice conjugate, Ed.
Matrix, Bucure¸sti, 2007.
[7] P. Stavre, On Some Remarkable Results of R. Miron, Libertas Math.,
tom XXIV, Arlington, Texas, 2004, p. 51-58.
Author:
Adrian Lupu
Technical High School ”Decebal”,
Drobeta Turnu Severin, Romania
email: adilupudts@yahoo.ca
128