CANONICAL ALMOST GEODESIC MAPPINGS OF TYPE

Vol. 19 (2018), No. 1, pp. 469–482 DOI: 10.18514/MMN.2018.1908

CANONICAL ALMOST GEODESIC MAPPINGS OF TYPE

.0; F /, 2 f 1; 2 g BETWEEN GENERALIZED PARABOLIC

K ¨ AHLER MANIFOLDS

MILO ˇS Z. PETROVI ´C Received 06 December, 2015

Abstract. We introduce a generalized parabolic K¨ahler manifold and consider special canon- ical almost geodesic mappings of type

2.0; F /, 2 f1; 2gbetween generalized Riemannian manifolds and between introduced generalized parabolic K¨ahler manifolds, particularly. Some invariant geometric objects with respect to these mappings are examined.

2010Mathematics Subject Classification: 53B05; 53B20; 53B35

Keywords: canonical almost geodesic mapping, generalized Riemannian manifold, generalized parabolic K¨ahler manifold, invariant geometric object

1. INTRODUCTION

The use of a non-symmetric affine connection became especially interesting after the works of A. Einstein [3] on the Unified Field Theory. In 1951, L.P. Eisenhart [4] introduced a generalized Riemannian space as a differentiable manifold equipped with a non-symmetric basic tensor. Eisenhart’s generalized Riemannian space is a particular case of a non-symmetric affine connection space. Some significant con- tributions to the study of geometry of non-symmetric affine connection spaces were made by E. Brinis, F. Graif, M. Prvanovi´c [17] and S.M. Minˇci´c [11–14].

Geodesic lines play an important role in modeling of various physical processes.

A mapping between two manifolds with linear connection, which preserves geodesics is called a geodesic mapping. Generalizing the notions of geodesic lines and geodesic mappings, Sinyukov [18] introduced the concept of almost geodesic lines and almost geodesic mappings of affine connected spaces without torsion. He indicated three types of almost geodesic mappings of manifolds without torsion,1,2and3.

The theory of geodesic and almost geodesic mappings of affine connected and Riemannian spaces is an active field of differential geometry, see for instance [1,2, 5–10,21,23].

c 2018 Miskolc University Press

Almost geodesic mappings of type2.e/; eD ˙1, from spaces with affine con- nection onto Riemannian spaces are considered in [10,23], while the paper [5] is ded- icated to canonical almost geodesic mappings of type2.eD0/between Riemannian spaces with an almost affinor structure, and between parabolic K¨ahlerian spaces, par- ticularly. Several papers are devoted to almost geodesic mappings of type

2.eD

˙1/, 2 f1; 2gand its special cases

2.eD ˙1; F /, 2 f1; 2gbetween manifolds with non-symmetric affine connection, see [15,19,21]. In the papers [16,22] some invariant geometric objects with respect to special almost geodesic mappings of type 11and

13, respectively, are examined, by considering equitorsion mappings. In [15]

we presented systems of differential equations of Cauchy type for almost geodesic mappings of the second type of manifolds with non-symmetric linear connection, also we found some invariant geometric object of almost geodesic mappings of type

2.eD 1; F /,2 f1; 2gunder some assumptions.

In the present paper, we extend and improve results from [5]. We consider ca- nonical almost geodesic mappings of type

2.0; F /, 2 f1; 2gbetween generalized Riemannian manifolds. Also, we introduce a generalized parabolic K¨ahler mani- fold and consider canonical almost geodesic mappings of type

2.0; F /, 2 f1; 2g between such manifolds. The wider class of metrics enables us to find more invariant geometric objects than in the classical (symmetric) case [5].

2. SPECIAL CANONICAL ALMOST GEODESIC MAPPINGS OF GENERALIZED

RIEMANNIAN MANIFOLDS

In the sense of Eisenhart (see [4]) a generalized Riemannian space is a differ- entiable manifoldM equipped with a metricg, which is generally non-symmetric.

Therefore, the metricgcan be described as follows g.X; Y /Dg.X; Y /Cg

_

.X; Y /;for allX; Y 2Tp.M /:

Heregdenotes the symmetric part of the metricgandg

_denotes the skew-symmetric part ofg, i.e.

g.X; Y /D1

2.g.X; Y /Cg.Y; X // and g

_

.X; Y /D1

2.g.X; Y / g.Y; X //;

whereX; Y 2Tp.M /andTp.M /is the tangent vector space ofM atp2M. The non-symmetric linear connection¹r of the generalized Riemannian manifold with the metricgis explicitly defined by

g.¹r^XY; Z/D 1

2.Xg.Y; Z/CY g.Z; X / Zg.Y; X //; X; Y; Z2Tp.M /: (2.1)

Let us denote byrthe Levi-Civita connection corresponding to the symmetric metric g. This connection is the symmetric part of non-symmetric linear connection¹r, i.e.

r^XY D1

2.¹r^XY C¹r^YX /; X; Y 2Tp.M /:

Also, it is well know that on the manifoldM with non-symmetric linear connection

1rcan be defined another non-symmetric linear connection²rin the following way

2rXY D¹rYXCŒX; Y ; X; Y 2X.M /;

where as usualX.M /denotes the set of smooth vector fields onM andŒ;denotes the Lie bracket [17].

M.S. Stankovi´c in [19] introduced two kinds of almost geodesic lines, as follows.

LetcWI !M be a curve on a manifoldM with non-symmetric linear connection

1r, satisfying the regularity conditionc⁰.t /¤0; t2I. Denote by .t /D.c.t /; c⁰.t //

the tangent vector field alongc, and let us put

1Dr;

2Dr

1; 2 f1; 2g: If the vector fields and

1are independent at any point (hence the (local) curve c is not a geodesic one) we can put DDspan.;

1/, 2 f1; 2g. The curvec is an almost geodesic line of the kind(2 f1; 2g) if and only if

22D. In [15] we gave an equivalent definition of almost geodesic lines of manifolds with non-symmetric linear connection, it is Definition1.

Definition 1([15]). LetcWI!M be a curve on a manifoldM with non-symmetric linear connection satisfying the regularity condition c⁰.t / ¤ 0 and let .t / D .c.t /; c⁰.t // be the tangent vector field along c. The curve c is called an almost geodesic of the kind .2 f1; 2g/if there exist vector fieldsX1 andX2 satisfying

rXiDa^j_iXj for some differentiable functionsa_i^j WI !Rand differentiable real functionsbⁱ.t /alongcsuch thatDb¹X1Cb²X2holds.

Definition 2([15,19]). A diffeomorphismf WM !M ofn-dimensional man- ifolds with non-symmetric linear connection is called an almost geodesic mapping of the kind .D1; 2/if any geodesic line of the manifoldM turns into an almost geodesic line of the kind of the manifoldM.

LetM andM be two generalized Riemannian manifolds of dimensionn > 2with the metricsg andg, respectively. We can consider these manifolds in thecommon coordinate system with respect to the diffeomorphismf WM !M. In this coordinate system the corresponding pointsp2M andf .p/2M have the same coordinates.

Therefore, we can supposeM M and for2 f1; 2gwe can put

P Dr r;

whereP is the tensor field of type.1; 2/, called thedeformation tensor field of linear connectionsrandr with respect to the mappingf.

In what follows we will useP

CS.;;/ to denote the cyclic sum on arguments in brackets, for instance for an arbitrary tensor fieldAwe have

X

CS.X;Y;Z/

A.X; Y; Z/DA.X; Y; Z/CA.Y; Z; X /CA.Z; X; Y /:

A diffeomorphism f WM !M is analmost geodesic mapping of the kind ; 2 f1; 2gif and only if [15]

P.X1; X2; X3/^P .X4; X5/^X6D0; Xi 2X.M /; i D1; : : : ; 6;

whereP is the deformation tensor field of connectionsr andr, with respect to the diffeomorphismf, andP

1,P

2, are tensor fields of type.1; 3/, defined by P1.X; Y; Z/D X

CS.X;Y;Z/

1r^Z1P .X; Y /C¹P .¹P .X; Y /; Z/; X; Y; Z2X.M / and

P2.X; Y; Z/D X

CS.X;Y;Z/

2rZ2P .X; Y /C²P .Z;²P .X; Y //; X; Y; Z2X.M /:

Basic equations of canonical almost geodesic mappings of type

2.eD0/,2 f1; 2g between generalized Riemannian manifolds are given by

P .X; Y /D X

CS.X;Y /

'.X /F Y C. 1/^{. 1/}K.X; Y /; (2.2) X

CS.X;Y /

rYFX . 1/K.F Y; X /

D X

CS.X;Y /

.X /F Y .FX /Y

; (2.3) whereX; Y 2X.M /,'is a1-form,Kis an anti-symmetric tensor field of type.1; 2/

defined by

K.X; Y /D1 2

1P .X; Y / ¹P .Y; X / D1

2

2P .Y; X / ²P .X; Y /

; andF is a tensor field of type.1; 1/satisfying

F²D0:

If the affinor structureF satisfies an additional condition Tr.F /DF_p^p D0;

then we denote by

2.0; F /, 2 f1; 2g.

A canonical almost geodesic mappingf WM !M of type

2.0; F /, 2 f1; 2g has theproperty of reciprocityif its inverse mappingf ¹WM !M is a canonical almost geodesic mapping of type

2.0; F /. Since the deformation tensor fields¹P and¹P of linear connections¹r and¹r with respect to the mappingsf andf ¹, respectively, satisfy the relation

1P .X; Y /D ¹P .X; Y /; X; Y 2X.M /;

without loss of generality we can suppose

'D '; F DF; KD K;

or in components

'_iD 'i; F^h_i DF_i^h; K^h_ij D K_ij^h: (2.4) Almost geodesic mappings of manifolds with non-symmetric linear connection, which satisfy the property of reciprocity are investigated in [15,19,21,22]. A necessary and sufficient condition for an almost geodesic mappingf WM !M of type

2, 2 f1; 2gto have the property of reciprocity is expressed by

F²D˛ICˇF;

whereI is the identity matrix and˛,ˇare some scalar functions.

2.1. Invariants

We use traditional tensor calculus approach “by components”. In local coordin- ates, with respect to a local chart.U; '/; 'D.x¹; : : : ; xⁿ/, we have

1ri

@

@x^j D¹r ^@

@xi

@

@x^j D ij^h

@

@x^h; ²ri

@

@x^j D²r ^@

@xi

@

@x^j D j i^h

@

@x^h; and

rⁱ @

@x^j D r ^@

@xi

@

@x^j D ij^h

@

@x^h;

whereij signifies a symmetrization with division and the functions _ij^hare general- ized Christoffel symbols.

A. Einstein [3] used two kinds of covariant differentiation of a tensora_jⁱ: a_jⁱ_j

1

mDa_j;mⁱ C pmⁱ a_j^p _{j m}^p a_pⁱ; a_jⁱ_j

2

mDa_j;mⁱ C mpⁱ a_j^p _mj^p a_pⁱ; wherea_j;mⁱ denotes the partial derivative of a tensora_jⁱ with respect tox^m.

S.M. Minˇci´c [11] has used two more kinds of covariant differentiation of tensors:

a_jⁱ_j

3

mDa_j;mⁱ C pmⁱ a_j^p _mj^p a_pⁱ; a_jⁱ_j

4

mDa_j;mⁱ C mpⁱ a_j^p _{j m}^p a_pⁱ:

Also, we can consider covariant differentiation with respect to the Levi-Civita con- nectionr, that is

rma_jⁱ a_jⁱ_ImDa_j;mⁱ C pmⁱ a_j^p _{j m}^p a_pⁱ; where _pmⁱ is the symmetric part of _pmⁱ .

Let us denote byj

andjj

,D1; : : : ; 4, the covariant derivatives with respect to the generalized Christoffel symbols _ij^hand ^h_ij, respectively.

In local coordinates the basic equation (2.2) reads

h ij h

ij D'_.iF_{j /}^hCK_ij^h; (2.5) where'i is the covariant vector corresponding to the linear form', whileF_i^hand K_ij^h are components of tensor fieldsF andK, respectively.

By using covariant differentiation of the first kind and the equation (2.5) we obtain F_i^h_jj

1

j DF_i^h_j

1

jC'pF_i^pF_j^hCK_pj^h F_i^p K_ij^pF_p^h: (2.6) After contracting the relation (2.6) over the indicesj andhand by using (2.4) we get

F^˛_ijj

1

˛C1

2K_p˛^˛ F^p_i 1

2K^p_{i ˛}F_p^˛DF_i^˛_j

1

˛C1

2K_p˛^˛ F_i^p 1

2K_{i ˛}^pF_p^˛; (2.7) i.e. the tensorA

1i defined by A1i DF_i^˛_j

1

˛C1

2K_p˛^˛ F_i^p 1

2K_{i ˛}^pF_p^˛; (2.8) is invariant with respect to the mappingf.

Analogously, by using covariant differentiation of the kind . D2; 3; 4/we can prove that the tensorsA

i,D2; 3; 4given by A2iDF_i^˛_j

2

˛C1

2K_˛p^˛ F_i^p 1

2K_˛i^pF_p^˛; A3i DF_i^˛_j

3

˛C1

2K_p˛^˛ F_i^p 1

2K_˛i^pF_p^˛; A4i DF_i^˛_j

4

˛C1

2K_˛p^˛ F_i^p 1

2K_{i ˛}^pF_p^˛;

(2.9)

are invariant with respect to the mappingf.

In the nontrivial case, whenF_i^h¤0, which is of particular importance for us, there exists a.1; 1/tensorF^h_i ¤0such thatF^˛_ˇF_˛^ˇ Dn. After contracting (2.6) withF^j_h

we find

n'pF_i^pD.F_i^˛_jj

1

ˇ F_i^˛_j

1

ˇ/F^ˇ_˛ K_pˇ^˛ F^ˇ_˛F_i^pCK_iˇ^pF^ˇ_˛F_p^˛: (2.10) From (2.4) we have

F^h_i DF^h_i; (2.11)

so we can conclude

F^h_i D

F^h_i: (2.12)

Also, the condition (2.4) ensures the relation K_ij^h D1

2.K_ij^h K^h_ij/: (2.13)

Now, from (2.6) by using (2.10)–(2.13) we obtain B1

h ij DB

1 h ij; where the tensorB

1 h

ij is defined by B1

h ij DF_i^h_j

1

j

1 n.F_i^˛_j

1

ˇC1

2K_ˇ^˛ F_i 1

2K_iˇ F^˛/F^ˇ_˛F_j^hC1

2K_j^h F_i 1

2K_ijF^h; (2.14) and the tensorB

1 h

ij is defined by B1

h

ij DF^h_ijj

1

j

1 n.F^˛_ijj

1

ˇC1

2K^˛_ˇF_i 1

2K_iˇF^˛/

F^ˇ_˛F_j^hC1

2K^h_jF_i 1

2K_ijF^h: Analogously, we can prove that the tensorsB

h

ij; D2; 3; 4, defined by B2

h ij DF_i^h_j

2

j

1 n.F_i^˛_j

2

ˇC1

2K_ˇ ^˛ F_i 1

2K_{ˇ i} F^˛/F^ˇ_˛F_j^hC1

2K_j ^h F_i 1

2K_{j i}F^h; B3

h ij DF_i^h_j

3

j

1 n.F_i^˛_j

3

ˇC1

2K_ˇ^˛ F_i 1

2K_{ˇ i} F^˛/F^ˇ_˛F_j^hC1

2K_j^h F_i 1

2K_{j i}F^h; B4

h ij DF_i^h_j

4

j

1 n.F_i^˛_j

4

ˇC1

2K_ˇ ^˛ F_i 1

2K_iˇ F^˛/F^ˇ_˛F_j^hC1

2K_j ^h F_i 1

2K_ijF^h; (2.15) are also invariant with respect to the mappingf.

The previous discussion generalize Theorem 1 from [5] to the case of generalized Riemannian manifolds. Namely, the tensors A

h

ij; D1; : : : ; 4, given by (2.8) and (2.9) are generalizations of the tensorAiDF_i^˛_I˛;while the tensorsB

h

ij; D1; : : : ; 4,

given by (2.14) and (2.15) are generalizations of the tensorB^h_ij given by B^h_ij DF_i^h_Ij 1

nF_i^˛_IˇF^ˇ_˛F_j^h; (2.16) where.I/denotes covariant differentiation with respect to the Levi-Civita connection.

When.I/denotes covariant differentiation with respect to the symmetric partrof non-symmetric linear connection¹r, it is obvious that the tensorsAiDF_i^˛_I˛andB^h_ij are invariant with respect to the mappingf of generalized Riemannian manifolds.

3. SPECIAL CANONICAL ALMOST GEODESIC MAPPINGS OF GENERALIZED PARABOLICK ¨AHLER MANIFOLDS

We use Eisenhart’s idea of generalized Riemannian spaces to generalize the notion of a parabolic K¨ahler manifold. Namely, we consider a parabolic K¨ahler manifold with a non-symmetric metric. M.S. Stankovi´c et al. [20] have already considered similar generalization for classical (elliptic) K¨ahler manifolds. They assumed that the affinorF is covariantly constant with respect to both of connections¹rand²r. We use weaker condition, by assuming that the affinorF is covariantly constant with respect to the symmetric part of non-symmetric linear connection¹r.

Definition 3. A generalized Riemannian manifoldM with a metricgis called a generalized parabolic K¨ahler manifold if there exists a .1; 1/tensor fieldF on M such that

F²D0; rF D0; g.X; Y /D!g.X; F Y /; !D ˙1; for allX; Y 2Tp.M /;

wherer denotes the Levi-Civita connection corresponding to the symmetric partg of metricg.

In what follows we consider only generalized parabolic K¨ahler manifolds for which !D1 in Definition3. Let M andM be two generalized parabolic K¨ahler manifolds of dimensionn > 2, with the metricsgandg, respectively and the affinor structureF. As in the case of usual parabolic K¨ahler manifolds, the conditions

F²D0 and Tr.F /DF_p^pD0 are satisfied.

The non-symmetric linear connection¹r, defined by (2.1), can be represented as follows

1rXY D rXY C1 2

1T .X; Y /; (3.1)

wherer denotes the symmetric part of non-symmetric connection¹r and¹T is the torsion tensor field of connection¹r.

For an anti-symmetric tensor fieldKgiven by K.X; Y /D1

2

1T .X; Y / ¹T .X; Y /

; (3.2)

according to (3.1) we have

1r^YFXCK.Y; FX /Dr^YFXC1 2

1T .Y; FX /C1 2

1T .Y; FX / 1 2

1T .Y; FX / DrYFXC1

2

1T .Y; FX /:

Analogously, we can prove the relation

2rYFX K.Y; FX /D rYFXC1 2

2T .Y; FX /:

Therefore the basic equations (2.2) and (2.3) in the case of canonical almost geodesic mappings of type

2.0; F /,2 f1; 2g(with a priori defined affinorF) between gen- eralized parabolic K¨ahler manifolds have the following form

P .X; Y /D X

CS.X;Y /

'.X /F Y C. 1/^{. 1/}K.X; Y /;

1 2

X

CS.X;Y /

T .Y; FX /D X

CS.X;Y /

.X /F Y .FX /Y

;

whereX; Y 2X.M /,'is a1-form andK is the anti-symmetric tensor field of type .1; 2/given by (3.2).

It is well known that the affinor structureF is locally integrable if and only if on a manifold exists a symmetric linear connectionrsuch thatrF D0. Therefore, the affinor structureF of a generalized parabolic K¨ahler manifold is locally integrable.

This fact enables us to consider another affinor structureF such that [5]

F_˛^hF^˛_i CF^h_˛F_i^˛Dı_i^h (3.3) holds on each local chartU of a generalized parabolic K¨ahler manifold.

In [5] it was proved that the geometric object

h ij

1

nC1F_.i^h _{j /ˇ}^˛ F^ˇ_˛ (3.4) is invariant with respect to the canonical almost geodesic mapping of type2.eD0/

between parabolic K¨ahler manifolds. In what follows we give some generalizations of the geometric object given by (3.4), to the case of a canonical almost geodesic mapping of type

2.0; F /, 2 f1; 2g between generalized parabolic K¨ahler mani- folds.

Theorem 1. Letf WM !M be a canonical almost geodesic mapping of type 2.0; F /, 2 f1; 2g between generalized parabolic K¨ahler manifolds M and M.

Then the geometric objectsC

h

ij,D1; : : : ; 4, given by C1

h ij D ij^h

1 nC1

_p

i q

F_p^qC1 n

F_p^˛_j

1

ˇ

F^ˇ_˛C1

2K_ˇ^˛ F^ˇ_˛F_p 1

2K_pˇ F^˛F^ˇ_˛ F^p_i C1

2K_{i q}^pF_p^q F_j^h

.ij /C1 2K_ij^h;

(3.5) C2

h ij D ij^h

1 nC1

_p

i q

F_p^qC1 n

F_p^˛_j

2

ˇ

F^ˇ_˛C1

2K_ˇ ^˛ F^ˇ_˛F_p 1

2K_ˇp F^˛F^ˇ_˛ F^p_i C1

2K_{i q}^pF_p^q F_j^h

.ij /

C1 2K_ij^h;

(3.6) C3

h ij D ij^h

1 nC1

_p

i q

F_p^qC1 n

F_p^˛_j

3

ˇ

F^ˇ_˛C1

2K_ˇ^˛ F^ˇ_˛F_p 1

2K_ˇp F^˛F^ˇ_˛ F^p_i C1

2K_{i q}^pF_p^q F_j^h

.ij /

C1 2K_ij^h;

(3.7) C4

h ij D ij^h

1 nC1

_p

i q

F_p^qC1 n

F_p^˛_j

4

ˇ

F^ˇ_˛C1

2K_ˇ ^˛ F^ˇ_˛F_p 1

2K_pˇ F^˛F^ˇ_˛ F^p_i C1

2K_{i q}^pF_p^q F_j^h

.ij /

C1 2K_ij^h;

(3.8) are invariant with respect to the mappingf.

Proof. Contracting the basic equation (2.5) withF^j_hwe obtain . ^p_{i q i q}^p/F_p^qD'iF_q^pF_p^qC'q

F_p^qF_i^pCK_{i q}^pF_p^q

Dn'iC'q.F_p^qF_i^pCF_p^qF^p_i F_p^qF^p_i /CK_{i q}^pF_p^q

(3.3)

D n'iC'qı_i^q 'qF_p^qF^p_i CK_{i q}^pF_p^q: Therefore,

.nC1/'i D. ^p_{i q i q}^p/F_p^qC'qF_p^qF^p_i K_{i q}^pF_p^q

(3.3)

D . ^p_{i q i q}^p/F_p^qC1 n h

.F_p^˛_jj

1

ˇ F_p^˛_j

1

ˇ/F^ˇ_˛ K_ˇ^˛ F^ˇ_˛F_p CK_pˇ F^˛F^ˇ_˛i

F^p_i K_{i q}^pF_p^q:

(3.9)

Now, after changing (3.9) into the basic equation (2.5), we get

h

ij D ij^hC 1 nC1

. ^p_{i q i q}^p/F_p^qC1 n

.F_p^˛_jj

1

ˇ F_p^˛_j

1

ˇ/F^ˇ_˛ K_ˇ^˛ F^ˇ_˛F_p CK_pˇ F^˛F^ˇ_˛

F^p_i K_{i q}^pF_p^q F_j^h

.ij /

CK_ij^h:

From the previous equation, by using (2.10)–(2.13), we obtain the following relation

h ij

1 nC1

_p

i q

F_p^qC1 n

F_p^˛_jj

1

ˇ

F^ˇ_˛C1 2K^˛_ˇ

F^ˇ_˛F_p 1

2K_pˇ F^˛

F^ˇ_˛

F^p_i C1

2K^p_{i q}

F_p^q F_j^h

.ij /

C1 2K^h_ij D ij^h

1 nC1

_p

i q

F_p^qC1 n

F_p^˛_j

1

ˇ

F^ˇ_˛C1

2K_ˇ^˛ F^ˇ_˛F_p 1

2K_pˇ F^˛F^ˇ_˛ F^p_i C1

2K_{i q}^pF_p^q F_j^h

.ij /

C1 2K_ij^h; which proves that the geometric objectC

1 h

ij defined by (3.5) is invariant with respect to the mappingf.

In a similar manner one can conclude that the geometric objectsC

h

ij,D2; 3; 4, determined by (3.6)–(3.8) are invariant with respect to the mappingf. When we consider a mapping between two affine connected manifolds with tor- sion, we can consider the so called equitorsion mapping, it is a mapping which pre- serves the torsion tensor.

Definition 4([16,22]). An almost geodesic mappingf WM !M of affine con- nected manifoldsM andM with the torsion tensorsT_ij^handT^h_ij, respectively, is an equitorsion almost geodesic mappingif the following condition holds

T_ij^hDT^h_ij:

Equation (3.2) in local coordinates readsK_ij^h D¹₂.T^h_ij T_ij^h/. Therefore the geo- metric objectsC

h

ij,D1; : : : ; 4, given by (3.5)–(3.8), with respect to an equitorsion canonical almost geodesic mapping of type

2.0; F /,2 f1; 2gbetween generalized parabolic K¨ahler manifolds take the following forms

C

h ij D ij^h

1 nC1

_p

i q

F_p^qC1 nF_p^˛_j

ˇ

F^ˇ_˛F^p_i

.ij /

; D1; : : : ; 4: (3.10)

Note that the geometric objects given by (3.5)–(3.8) and (3.10) are not tensors, since the generalized Christoffel symbols _ij^hare not tensors (see [14], p. 10).

The geometric object

C^h_ij D ij^h

1

nC1F_.i^h _{j /ˇ}^˛ F^ˇ_˛; (3.11) where _ij^h is the symmetric part of _ij^h, is evidently invariant with respect to the canonical almost geodesic mapping of type

2.0; F /,2 f1; 2gbetween generalized parabolic K¨ahler manifolds. This geometric object is a tensor as well as the geometric object given by (3.4).

Remark1. The geometric objects, given by (3.5)–(3.8), (3.10) and (3.11) are gen- eralizations of the tensor, given by (3.4).

4. CONCLUSION

Invariant geometric objects of canonical almost geodesic mappings of type

2.0; F /, 2 f1; 2g are examined. Since the available literature does not contain any results about invariants of almost geodesic mappings of type

2.e/, 2 f1; 2g foreD0, this paper somewise fills the gap in the theory of almost geodesic mappings of manifolds with non-symmetric affine connection.

A generalized parabolic K¨ahler manifold is introduced and some results concern- ing invariant geometric objects of canonical almost geodesic mappings of type2.eD 0/, between parabolic K¨ahler manifolds are extended. This fact opens up possibilities for further extension of results from usual parabolic K¨ahler manifolds to generalized parabolic K¨ahler manifolds.

ACKNOWLEDGEMENT

This work was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, Grant No. 174012.

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Author’s address

Miloˇs Z. Petrovi´c

University of Niˇs, Department of Mathematics, Viˇsegradska 33, 18000 Niˇs, Serbia E-mail address:milos.petrovic@pmf.edu.rs