J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
84Ae95.
PROLONGATIONS OF GOLDEN STRUCTURES TO
BUNDLES OF INFINITELY NEAR POINTS
Georges Florian Wankap Nono1 .
Achille Ntyam2 , and Emmanuel Hinamari Mang-Massou1 University of NgaoundeAreA,Faculty of Science.
Department of Mathematics and Computer Science.
O BOX 454 NgaoundeAreA.
Cameroon georgywan@yahoo.
fr, hinamariemmanuel@gmail.
University of YaoundeA I.
Higher Teacher Training College.
Department of Mathematics.
O BOX 49 YaoundeA.
Cameroon ntyam achille@yahoo.
Abstract.
For a Golden-structure on a smooth manifold M and any covariant functor which assigns to M its bundle M A of infinitely near points of A-king, we define the Golden-structure A on M A and prove that is integrable if and only if so is A .
We also investigate the integrability, parallelism, half parallelism and anti-half parallelism of the Golden-structure A and their associated distributions on M A .
Key words and Phrases: Prolongations, bundles of infinitely near points, golden structures, integrability, parallelism.
INTRODUCTION
The differential geometry of Golden-structure on manifolds has been first initiated by M.
CraCsmaUreau and C.
Hretcanu in .
The concepts of a GoldenRiemannian structure and a Golden-Riemannian manifold has been introduced in .
by using a corresponding almost product structure, and some properties of Golden-Riemannian manifold have been studied.
And a few years later, some properties of the induced structure on an invariant submanifold in a Golden Reimannian manifolds were investigated by many authors such as C.
Hretcanu and M.
CraCsmaUreau in .
GoOk.
Keles and E.
Kilic in .
In .
Gezer.
Cengiz and A.
Salimov studied the problem of the integrability for Golden-Riemannian structures.
In .
Ozkan defined the prolongations of Golden-structures to tangent bundles of order r Ou 1.
The present paper is mainly focused on a study of prolongations of Goldenstructures on manifolds to bundles of infinitely near points.
Basicly, this study is inspried from the paper .
The main goal of this paper is to generalize the results of .
to the bundles of infinitely near points of kind A in the sense of A.
Weil .
The paper has three sections and is organized as follows.
In Section 2, we review the notion of bundles of infinitely near points and recall some definitions and properties of the Golden-structure.
The Section 3 is devoted to prolongations of the Golden-structure to bundles of near points of kind A and we will investigate some properties of these prolongations.
We also discuss of integrability, parallelism, 2020 Mathematics Subject Classification: 58A20, 58A32, 53C15, 53C25, 58A05 Received: 01-10-2021, accepted: 03-01-2022.
Prolongations of Golden Structures half parallelism and anti-half parallelism of a Golden-structure and the associated distributions to bundle of near points.
We end this last section by studying the prolongation to bundles of near points of Golden pseudo-Riemannian structure on a smooth manifold M .
PRELIMINARIES
Bundles of Infinitely Near Points A weil algebra or local algebra .
n sense of A.
Wei.
is a real associative, commutative and unital algebra of finite dimension over R, admitting a unique maximal ideal M such that A/M is one-dimensional over R and that Mh 1 = .
for a nonnegative integer h.
The smallest h such that Mh 6= .
and Mh 1 = .
is called the height of A.
We shall identify the field R with the subspace of A consisting of all scalar multiples of the unit.
Thus A = R Oi M.
For example,the algebra of dual numbers D = R[T ]/(T 2 ) is Weil algebra with height 1.
Let us recall this construction of bundles of A-points of M base on .
LetAos denote by M a smooth manifold.
C O (M ) the algebra of smooth functions on M and A the weil algebra with the maximal ideal M.
An infinitely near points to x on M of kind A .
r A-points of M near .
is a homomorphism of algebras i : C O (M ) Ie A such that i.
) O f .
mod(M).
O for all f OO C (M ).
We denote by MxA and M A = MxA respectively the set of all infinitely points xOOM on M of kind A and the set of all near points on M of kind A.
If M and N are two smooth manifolds and F OO C O (M.
N ) a differential map, then one defines the differential map F A : M A Ie N A , i 7Ie F A .
such that, for all g OO C O (N ).
F A .
= i.
F ).
If F is a diffeomorphism, then F A will be too.
We can identify (M y N )A with M A y N A by the following identification AM y AN : (M y N )A Ie M A y N A , i 7Ie (AM .
AN .
) where AM : (M y N )A Ie M A .
: (M y N )A Ie N A ) is a projection.
F1 : M1 Ie N1 .
F2 : M2 Ie N2 .
F1 : M1 Ie N10 and F20 : N1 Ie N are differentiable maps between manifolds, then we have the following equalities (F1 .
F10 )A (F1 y F2 ) (F1A .
F10A ) (F20 F1 )A = F20A F1A F1A y F2A .
M )A = 1M A .
If (U, .
is a local chart on M with a local coordinate system .
A A A , un ), the map uA : U A Ie An , i 7Ie .
1 ).
A A A , i.
n )) is a bijection from U A to an open subset of An and defines a local chart (U A , uA ) on M A .
Hence the set M A becomes a differentiable manifold of dimension dim(A) A dim(M ).
Now, let AA : M A Ie M.
MxA i 7Ie x.
Therefore, the manifold M A with the projection AA : M A Ie M is called the bundles of A-points of M .
r bundles of infinitely near points of M of kind A).
Wankap Nono et.
The notion of bundles of kind D = R[T ]/(T 2 ) is the same as the tangent More generally, when A = R[T1 .
A A A .
Tp ]/(T1 .
A A A .
Tp )r 1 .
M A is the space J0r (Rp .
M ) of jet of order r at 0 of differentiable map from Rp to M and the associated bundle of A-points is the bundles of pr -velocities.
Golden-Riemannian Manifolds Let M be a smooth manifold and Tqp (M ) the C O (M )-module of tensor fields of .
, .
-type on M .
An element of T11 (M ) is usually called vector 1-form .
r affino.
on M .
Let X(M ) = e(T M ) be the C O (M )-module of all vector fields on Definition 2.
) An affinor on M is called polynomial structure if it satisfies the following algebraic equation Q(X) = X n anOe1 X nOe1 A A A a1 X a0 = 0 nOe1 .
nOe2 where .
, .
A A A , .
and are linearly independent for every x OO M and is the identity transformation affinor.
The monic polynomial Q(X) is named the structure polynomial.
We recall that, a polynomial structure is integrable if the Nijenhuis tensor N vanishes identically, where N (X.
Y ) = 2 [X.
Y ] [X.
Y ] Oe [X.
Y ] Oe [X.
Y ], for all X.
Y OO X(M ).
Remark 2.
In particular, if Q(X) = X 2 Oe .
Q(X) = X 2 ), then we will have an almost product structure A .
an almost complex structure ).
When Q(X) = X 2 , we have the notion of almost tangent structure E .
Definition 2.
Let (M, .
be a Riemannian-manifold.
A Golden-structure on (M, .
is a given non-null affinor of class C O on M which verifies the following 2 Oe Oe M = 0 .
where M is the identity transformation affinor.
In this case, the pair (M, ) is called Golden-manifold.
We say that the metric g is -compatible if we have the following equality g(X.
Y ) = g(X.
Y ) .
for all vector fields X.
Y OO X(M ).
If we substitute Y into Y in .
, then Equation .
may also be written as g(X.
Y ) = g(X.
Y ) g(X.
Y ).
Definition 2.
) A Golden-Riemannian manifold is a triple (M, g, ), where (M, .
is a Riemannian-manifold, is a Golden-structure on (M, .
and g is compatible.
Definition 2.
) Let F be a smooth map from a Golden Riemannian manifold (M, g, ) to a Golden Riemannian manifold (N, h, ).
Then F is called a Golden map if the following condition is satisfied dF = dF.
In .
, we have this following proposition which show the connection between the almost product structure and the Golden-structure on M .
Prolongations of Golden Structures Proposition 2.
)Let M be a smooth manifold.
Any Golden-structure on M induces two almost product structures on M defined as follows AOe = Oe Oo .
Oe M ) and A = Oo .
Oe M ).
Conversely, any almost product structure A on M induces two Goldenstructures on M defined as follows Oo Oo .
Oe = (M Oe 5A) and = (M 5A).
Let (M, ) be a Golden-manifold.
According to .
, we define these two r = Oo ((E Oe .
M ) and s = Oo (EM Oe ) .
where the Golden ratio E = OO 1.
618 is the root of the algebraic equation t Oe t Oe 1 = 0.
We can easily have these following equalities r2 = r, s2 = s, s r = r s = 0 and s r = M .
This means that, r and s are projection operators splitting the tangent bundle T M = M D into two complementary parts, and define two globally complementary distributions R and S of M D ( see .
) as follows .
OO MxD : .
= E.
and S = .
OO MxD : .
= .
Oe E).
xOOM xOOM .
The projection operators r and s verify these following equalities:
= Er .
Oe E)s .
r = r = Er and s = s = .
Oe E)s.
MAIN RESULTS
A-Lift of Golden-structures to Bundles of Near Points Let M be a smooth manifold and M A a manifold of infinitely near points on M of kind A.
For a given affinor on a M .
Morimoto in .
gives its A-lift A and shows that A is a unique affinor on M A which verifies A (X A ) = ((X))A , .
for all OO T11 (M ) and X OO X(M ).
Hence, we can show A A = ( )A for all , .
OO T11 (M ).
When = , equation .
becomes ( 2 )A = ( A )2 .
Hence, we have these following results.
Proposition 3.
Let be an affinor on M .
The following assertions are equivalent.
is a Golden-structure on M .
A is a Golden-structure on M A .
M A Oe A is a Golden-structure on M A .
Wankap Nono et.
Proof.
It is the use of linearity of the A-lift, equation .
and the fact that (M )A = M A .
Proposition 3.
Let (M, ) a Golden-manifold.
The Golden-structure A on M A is an isomorphism on (M A )D i , for every i OO M A.
The Golden-structure A on M A is invertible and its inverse ( A )Oe1 = A Oe M A satisfies the equation ( A )Oe1 ( A )Oe1 Oe M A = 0.
Remark 3.
Let OO T11 (M ) be an almost complex .
almost produc.
structure on M .
Then A and Oe A are an almost complex .
almost produc.
structure on M A .
Morever.
A is integrable if an only if so is .
(See .
If E is an almost tangent structure on M , then E A .
-E A ) is also an almost tangent structure on M A .
The following proposition shows the connection between Golden and almost product structures on M A .
Proposition 3.
Let M be a smooth manifold.
If is the Golden-structure on M , then the Golden-structure A .
eA = M A Oe A ) on M A induces two almost product structures AA and AA defined as follows on M Oe = Oe Oo .
Oe M A ) and A = Oo .
Oe M A ).
Conversely, if A is an almost product structure on M , then the almost product AA .
AeA = OeAA ) on M A induces two Golden-structures Oe and on M defined as follows Oo Oo Oe = (M A Oe 5AA ) and = (M A 5AA ).
According to CraCsmaUreanu and Hretcanu in .
, we have the following remark.
Remark 3.
If E is an almost tangent structure on M , then its A-lift E A induces two affinor structures on M A defined as follows Oo
EeOe
= (M A Oe 5E A ) and Ee
= (M A 5E A )
and which are called tangent Golden-structures on M A .
These tangent Golden-structures satisfy the equation E A )2 Oe EeA M A = 0.
If is the complex structure on M , then its A-lift A induces two affinor
structures on M A defined as follows Oo
= (M A Oe 5 A ) and e
= (M A 5 A )
and which are called complex Golden-structures on M A .
These complex Golden-structures satisfy the equation A )2 Oe eA M A = 0.
Prolongations of Golden Structures Example 3.
rolongation to M A of triple structures on M ) Let .
A and be three affinors structures on the smooth manifold M such that = A.
According to .
, the triple (.
A, ) is called almost hyperproduct structure .
, almost biproduct complex structure .
, almost product bicomplex structure .
and almost hypercomplex structure .
on M if .
E and verify respectively the following equalities:
2 = A2 = 2 = A = M , 2 = A2 = Oe 2 = A = M .
Oe 2 = A2 = 2 = A = OeM and 2 = A2 = 2 = A = OeM .
Let Oo Oo A Oo 5A ) and eOe = (M A Oe 5 A ) eOe = (M A Oe 5 A ).
AeA Oe = (M A Oe Oo Oo A Oo .
= (M A 5 A ).
AeA
5A ) and e
= (M A 5 A ))
= (M A
be the induces structures associated to A .
AA and A respectively .
ee proposition We easily see that, those induces structures verify this following equality Oo A = 2eA AeA Oe eA Oe AeA EM A A A A A .
AeOe , eOe ) and .
Ae , e ) are and the triple .
Oe .
) on M A if and only if (.
A, ) is an .
) on M .
In this case, eA is a Golden-structure on M A .
) on M A if and only if (.
A, ) is an .
) on M .
In this case, eA is a complex Golden-structure on M A .
Integrability of Golden-structure to Bundles of Near Point The purpose of this section is to give some properties of integrability of the Golden-structure A on M A and its associated distributions.
Let (M, ) be a Golden-manifold and A a given Weil algebra.
Definition 3.
The Golden-structure A on M A is integrable if N A (X A .
Y A ) = 0, for all vector fields X.
Y in M .
Proposition 3.
is an integrable Golden-structure on M if and only if the Golden-structure A on (M A , g A ) is integrable on M A .
Proof.
It comes from the fact that N A (X A .
Y A ) = (N (X.
Y ))A by using relations .
, for all vector fields X.
Y in M .
Morimoto in .
has proved the following proposition.
Proposition 3.
) Let M be a smooth manifold.
The map X(M ) Ie X(M A ).
X 7Ie X A is a homomorphism of Lie algebras.
For all OO T11 (M ) and X OO X(M ), one has ((X))A = A (X A ).
Hence, from the above proposition, we can construct the A-lift of these two projection operators r and s on M as follows rA = Oo ((E Oe .
M A A ) and sA = Oo (EM A Oe A ).
These new operators satisfy the following equalities .
A )2 = rA , .
A )2 = sA , sA rA = rA sA = 0 and rA sA = M A r = r = Er s = s = .
Oe E)s .
Wankap Nono et.
Therefore, rA and sA are projection operators splitting the tangent bundle T M A = (M A )D into two complementary parts, and define two globally complementary distributions RA and S A of the set of D-point of M A according to .
LetAos recall this result from M.
CraCsmaUreanu and C.
Hretcanu.
Proposition 3.
)Let (M, ) be the Golden-manifold.
The distribution R .
S) is integrable if and only if .
X, rY ] OO e(R) .
X, sY ] OO e(S)) for all vector fields X.
Y in M .
We have this following definition.
Definition 3.
The distribution RA .
S A ) is integrable if the vector field .
A X A , rA Y A ] .
A X A , sA Y A ]) belongs to e(RA ) .
e(S A )) for all vector fields X and Y in M .
Hence, we have these following results.
Proposition 3.
Let (M, ) be the Golden-manifold.
The distribution RA .
S A ) is integrable if and only if R on .
S) is Proof.
It comes from the fact that sA .
A X A , rA Y A ] = .
X, rY ])A .
rA .
A X A , sA Y A ] = .
X, sY ])A ), for all vector fields X and Y in M .
Proposition 3.
Let (M, ) be the Golden-manifold.
The distribution RA .
S A ) is integrable if and only if N A .
A X A , rA Y A ) OO e(RA ) .
N A .
A X A , sA Y A ) OO e(S A )) .
for all vector fields X.
Y in M .
Proof.
For all vector fields X and Y in M , one has sA N A .
A X A , rA Y A ) = sA ( A )2 .
A X A , rA Y A ] sA [ A rA X A , z A rA Y A ] Oe sA A [ A rA X A , rA Y A ] Oe sA A .
A X A , rA Y A ] .
Oe E)2 sA .
A X A , rA Y A ] E 2 .
A X A , sA Y A ] Oe E.
Oe E)sA .
A X A , rA Y A ] Oe E.
Oe E)sA .
A X A , rA Y A ] 5sA .
A X A , rA Y A ].
With the same manner, rA N A .
A X A , sA Y A ) = 5rA .
A X A , rA Y A ].
Hence, the proof is finished.
Proposition 3.
Let be a Golden-structure on M and A its A-lift on M A .
The following assertions are equivalent:
A is integrable.
Both the distribution RA and S A are integrable.
Proof.
Let X and Y be two vector fields on M .
We have rA N A .
A X A , sA Y A ) sA N A .
A X A , rA Y A ) Hence, the proof has been finished.
5rA [(M A Oe rA )X A , (M A Oe rA )Y A ] 5(M A Oe rA ).
A X A , rA Y A ] 5.
A )2 [X A .
Y A ] 5.
A X A , rA Y A ] Oe 5rA .
A X A .
Y A ] Oe 5rA [X A , rA Y A ] 5NrA (X A .
Y A ).
Prolongations of Golden Structures Corollary 3.
Let be a Golden-structure on M and A its A-lift on M A .
The following assertions are equivalent:
A is integrable.
Both RA and S A are integrable.
Both R and S are integrable.
is integrable.
Theorem 3.
Let A be the almost product on a smooth manifold M .
The almost product AA on M A is integrable if an only if the associated GoldenA structure .
Oe ) is integrable.
Proof.
Let X.
Y OO X(M ).
AA an almost product structure on M A and Oe Oo Oo 5A ) .
= 2 (M A 5A )) The induced Golden-structure on 2 (M A Oe M A .
One has:
N A (X A .
Y A )
A A
A A A
A A A A
A 2
Y ]
[X A .
[Oe X .
Y A ] Oe Oe X .
Oe Y ] Oe Oe ) [X A .
Y A ] [Oe (Oe Oo ( A Oe 2 5AA 5(AA )2 )[X A .
Y A ] Oo
1 A A
5 A A A
5 A A A
[X .
Y ] Oe [X .
A Y ] Oe [A X .
Y ] [AA X A .
AA Y A ]
5 A A A
5 A A A A
1 A A
[X .
Y ]
A [X .
Y ]
A [A X .
Y ] Oe AA [AA X A .
Y A ] Oo
1 A A
5 A A A A
5 A A A
5 A A A
[X .
Y ]
A [X .
Y ]
[X .
A Y ] Oe A [X .
A Y ] ((AA )2 [X A .
Y A ] [AA X A .
AA Y A ] Oe AA [AA X A .
Y A ] Oe AA [X A .
AA Y A ]) N A (X A .
Y A ) 4 A With the same manner, we have N A (X A .
Y A ) = 45 NAA (X A .
Y A ).
Hence, the proof follows.
Conversely, we have this following theorem.
Theorem 3.
Let (M, ) be the Golden-manifold.
The Golden-structure A on (M A , g A ) is integrable if and only if the associated almost product AA .
AOe ) is integrable.
Parallelism.
Half Parallelism and Anti-half Parallelism of Goldenstructure on M A In this section, we discuss parallelism, half parallelism and anti half parallelism of the distributions associated with the golden structure on M A .
We recall that, a distribution D on M is called parallel with respect to the linear connection ON if the vector field ONX Y belongs to D for any vector fields Y OO e(D) and X OO X(M ) = e(T M ).
Let be a Golden-structure on M .
For all vector fields X and Y in M , letAos put 4(X.
Y ) = (ONX Y ) Oe (ONY X) Oe ONX Y ONY X.
We recall this following definition from .
Definition 3.
)Let (M, g, ) be a Golden Riemannian manifold.
The distribution R .
S) on M is called half-parallel with respect to the linear connection ON if 4(X.
Y ) OO e(R) .
e(S)), for all vector fields X OO e(R) .
e(S)) and Y OO X(M ).
Wankap Nono et.
The distribution R .
S) on M is called anti-half parallel with respect to the linear connection ON if 4(X.
Y ) OO e(S) .
e(R)), .
for all vector fields X OO e(R) .
e(S)) and Y OO X(M ).
Let ON be a linear connection on M .
Its A-lift ONA is a unique linear connection on M which satisfies this equality ONA
= (ONX Y )A ,
XA Y
where X and Y mean prolongation to M of vector fields X and Y in M .
ee Theorem 5 of .
From the above consideration, we have these following definitions.
Definition 3.
Let ON be a linear connection on a Golden-manifold (M, ) .
The distribution RA .
S A ) is parallel with respect to linear connection ONA if ONA OO e(RA ) .
e(S A )) XA Y for all vector fields X OO e(R) .
e(S)) and Y OO X(M ).
Definition 3.
Let ON be a linear connection on a Golden-manifold (M, ) .
The distribution RA .
S A ) on M A is called half-parallel with respect to the linear connection ONA if 4 A (X A .
Y A ) OO e(RA ) .
e(S A )), .
for all vector fields X OO e(R) .
e(S)) and Y OO X(M ).
The distribution RA .
S A ) on M A is called anti-half parallel with respect to the linear connection ON if 4 A (X A .
Y A ) OO e(S A ) .
e(RA )), .
for all vector fields X OO e(R) .
e(S)) and Y OO X(M ).
Let ON be a linear connection on a Golden-manifold (M, ).
According to ScA , we can associate to the pair ( A .
ONA ) two other linear connections ON and ON on M A called respectively Schouten and VraUnceanu connections, and define as ScA
A A
A A
r Y ) sA (ONX s Y) ONX Y = rA (ONX ONX A Y A = rA (ONA rA Y ) sA (ONA sA Y ) rA .
A X, rA Y ] sA .
A X, sA Y ], rA X sA X for all vector fields X and Y in M A .
Hence, we have the following results.
Theorem 3.
The Golden-structure A on M A is parallel with respect to Schouten and VraUnceanu connections.
Proof.
From the linearity of ONA and the relations .
, one has ScA
(ONX A )Y
ScA
ScA
ONX A Y Oe A ONX Y
A A
A A
rA (ONA rA A Y ) sA (ONX s A Y ) Oe A rA (ONX r Y) Oe A sA (ONA sA Y ) A A A A ErA (ONA rA Y ) .
Oe E)sA (ONX s Y ) Oe ErA (ONX r Y)Oe .
Oe E)sA (ONA sA Y ) Prolongations of Golden Structures With the same manner, (ONX A )Y = 0.
Theorem 3.
The projection operator rA .
sA ) is parallel with respect to Schouten and VraUnceanu connections.
Proof.
It comes from relations .
and the fact that ONA and the bracket of vector fields on M A are linear.
Proposition 3.
Let ON be a linear connection on a Golden-manifold (M, ).
The distribution R .
S) is parallel with respect to a fixed linear connection ON on M if and only if RA .
S A ) is parallel with respect to linear connection ONA on M A .
Proof.
Let X OO e(R) .
e(S)) and Y be a vector field in M .
From the relations .
in this order, on has sA ONA = .
ONX Y )A .
sA ONA = .
ONX Y )A ).
XA Y
XA Y
Hence, sA (ONA X A Y ) = 0 Ni s(ONX Y ) = 0 .
r (ONX A Y ) = 0 Ni r(ONX Y ) = .
Theorem 3.
The distribution RA .
S A ) is parallel with respect to the Schouten and VraUnceanu connections for every linear connection ONA on M A .
Proof.
Let ONA be a linear connection on M A .
X A OO e(RA ) and Y A OO X(M A ) be the A-lift of vector fields Y OO e(R) and X OO X(M ).
From relations .
, we easily have rY = Y and sY = 0.
Hence ONX A Y A A A A A = rA (ONA X A r Y ) s (ONX A s Y ) = rA (ONA X A .
Y ) ) s (ONX A .
Y ) ) = rA (ONA X A Y ) OO e(R ) ONX A Y
= r
ONA
Y ]A .
X)A Y OO e(RA ).
It can be proved analogously that the distribution S A is parallels with respect to the Schouten and VraUnceanu connections for a linear connection ONA .
Proposition 3.
Let be a Golden structure, parallels with respect to a linear connection ON on M .
Then A is parallels with respect to linear connection ONA on M A if and only if (ONA X A )Y for all vector fields X and Y in M.
Proposition 3.
Let ON be a linear connection on a Golden manifold (M, ) and ONA its A-lift on (M A .
A ).
The distribution R .
S) on M is half parallels with respect to ON if and only if the distribution RA .
S A ) on M A is also half parallel with respect to ONA .
Proof.
It comes from Equation .
and Equality .
Proposition 3.
Let ON be a linear connection on a Golden manifold (M, ) and ONA its A-lift on (M A .
A ).
The distribution RA .
S A ) on M A is anti-half parallel with respect to ONA .
Wankap Nono et.
Proof.
Let X A OO e(RA ) and Y A OO X(M A ) be the A-lift of vector fields X OO e(R) and Y OO X(M ).
From relations .
, we have rA A = ErA and X = EX.
Hence r (ONX A Y ) Oe (ONY A X ) Oe ON(X)A Y ONY A (X) since ONA is linear.
Therefore, 4 A (X A .
Y A ) OO e(S A ) and RA is anti-half parallel with respect to ONA .
S A is anti-half parallel with respect to ONA by using the same method.
Proposition 3.
Let ON be a fixed linear connection on Golden-manifold (M, ).
The the distribution R .
S) is half parallels with respect to Schouten and VraUnceanu connections if and only if so is RA .
S A ).
Prolongation to M A of Golden Pseudo-Riemannian Structure on M Let g be a pseudo-Riemannian metric on M .
Its A-lift is a unique pseudoRiemannian metric on M A which satisfies g A (X A .
Y A ) = .
(X.
Y ))A , .
where X A and Y A mean prolongation to M A of vector fields X and Y in M .
ee proposition 12 of .
Hence, the pair (M A , g A ) becomes a pseudo-Riemannian Then, we easily have the following results.
Proposition 3.
If the triple (M, g, ) is a Golden pseudo-Riemannian manifold, then so is the triple (M A , g A .
A ).
Corollary 3.
Let (M, g, ) be a pseudo-Riemannian manifold.
For all vector fields in M , we have .
g A .
A X A .
Y A ) = g A (X A , rA Y A ) .
g A .
A X A .
Y A ) = g A (X A , sA Y A )):
This means that the projection operators rA and sA are g A -symmetric .
g A .
A X A , sA Y A ) = 0: This means that the distribution RA and S A are g A -orthogonal.
N A ( A X A .
Y A ) = N A (X A .
A Y A ).
This means that the Golden structure A is N A -symmetric.
Remark 3.
If .
A) is a pseudo-Riemannian almost product on M .
hat is.
A is a g-symmetric almost product structure on pseudo-Riemannian manifold (M, .
, then the pair .
A .
AA ) is also a pseudo-Riemannian almost product on M A and the triple (M A , g A .
A ) is a Golden pseudo-Riemannian structure on M A where A is the Golden-structure on M A induced by AA .
ee Proposition 3.
Proposition 3.
If F : M Ie N is a Golden map between Golden pseudoRiemannian manifolds (M, g, ) and (N, h, ), then F A : M A Ie N A is a Golden map between Golden pseudo-Riemannian manifolds (M A , g A .
A ) and (N A , hA .
A ).
Proof.
Since F is a Golden map, then we have:
dF = dF.
Taking the A-lift on the both sides of the above equality and from the relation .
, we obtain dF A A = A dF A .
Prolongations of Golden Structures Acknowledgement.
The authors would like to thank the anonymous reviewers for their valuable suggestions and remarks which improved the quality of this paper.
References