FOR EQUITORSION GEODESIC MAPPINGS

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Vol. 19 (2018), No. 1, pp. 665–675 DOI: 10.18514/MMN.2018.2219

WEYL PROJECTIVE OBJECTS W

1

; W

2

; W

3

FOR EQUITORSION GEODESIC MAPPINGS

NENAD O. VESI ´C Received 31 January, 2017

Abstract. In this paper, invariants of geodesic mappings of non-symmetric affine connection manifolds are studied. It is obtained new generalizations of the Weyl projective tensor of these manifolds. At the end of this paper, generalized invariants of a geodesic mapping between special three dimensional generalized Riemannian manifoldsGR3andGR3are obtained.

2010Mathematics Subject Classification: 53C15; 55C99; 53A55; 35R01 Keywords: invariant, affine connection, curvature tensor, Weyl projective tensor

1. INTRODUCTION AND MOTIVATION

Many research papers and monographs [1–19,21–27] are focused on development of the affine connection spaces theory, on the theory of mappings between these spaces, on the theory of invariants of these mappings and on the applications of them.

N. S. Sinyukov [21], J. Mikeˇs with his research group [9–11,22], G. Hall [6,7], G. P.

Pokhariyal [16,17], U. P. Singh [20] and many other researchers have given significant contributions to the development of the concept of symmetric affine connection spaces.

Consider anN-dimensional manifoldM_N on which a non-symmetric affine connectionr is defined. IfX.MN/ is Lie algebra of smooth vector fields andX; Y 2 X.MN/, then the mapping

r WX.M_N/X.M_N/!X.M_N/

definesthe non-symmetric connectiononM_N ifrXY ¤ rYXCŒX; Y . The following identities are satisfied:

r^Y1CY₂XD r^Y1XC r^Y2X; rf YXDfr^YX;

rY.X₁CX₂/D rYX₁C rYX₂; rY.f X /DYf XCfrYX;

Research supported by Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 174012.

c 2018 Miskolc University Press

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for X; Y; X1; X2; Y1; Y2 2X.MN/; f 2F.MN/, where F.MN/ is an algebra of smooth real functions onMN.

Definition 1 ([4,25,27]). An N-dimensional differentiable manifold MN endowed with a non-symmetric affine connectionrYX isthe non-symmetric man- ifoldGAN.

Remark1 ([4,8]). A differentiable manifold endowed with a non-symmetric metric G.X; Y / the basic tensorG.X; Y /

is the generalized Riemannian manifoldGRN. Affine connection coefficients of the manifoldGRN are Christoffel symbols of the second kind of this manifold.

Letrbe a non-symmetric affine connection. With regard to local chart x¹; :::; x^N/, in the base of @iN

iD1D @=@xⁱN

iD1, we obtain that is r@k@j D j k^˛@˛: for coefficients _{j k}ⁱ non-symmetric by indicesj andk.

In the whole paper, we shall use the capital Latin letters X; Y; Z; ::: to denote smooth vector fields on a smooth manifoldMN.

LetGAN D M_N;r

be a non-symmetric affine connection manifold. Because of the non-symmetryr^YXCŒX; Y ¤ r^XY, the symmetric and anti-symmetric part of the affine connection coefficientsr^YX are respectively defined as:

erYXD1

2 rYXCrXY ŒX; Y

; T .X; Y /D1

2 r^YX r^XYCŒX; Y :

(1.1)

It is evident

erYX erXY DŒX; Y : (1.2) The anti-symmetric part T .X; Y /of the affine connectionr isthe torsion tensor.

The coordinates of the affine connection coefficientsrYX are _{j k}ⁱ . The coordinates of the symmetric and the anti-symmetric parts of the coefficients _{j k}ⁱ are:

eSⁱ_{j k}D¹₂. _{j k}ⁱ C _kjⁱ / and T_{j k}ⁱ D¹₂. _{j k}ⁱ _kjⁱ /;

AnN-dimensional manifoldMN endowed with the symmetric affine connection er^YX is calledthe associated (symmetric affine connection) manifoldAN of the manifoldGAN. Based on the symmetry of affine connection coefficientserYX by X andY, it exists only one type of covariant differentiation with respect to the affine connection of the manifoldAN. For this reason, it exists only one curvature tensor of this manifold:

R.XIY; Z/Dre^Zre^YX re^Yer^ZXCreŒY;ZX : (1.3)

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A. Einstein [1–3] was the first who applied a non-symmetric affine connection in the research about gravity. He worked on the Unified Field Theory (Non-symmetric Gravitational Theory - NGT). In NGT, the basic tensorG.X; Y /is a non-symmetric tensor of the type.0; 2/. The symmetric part

g.X; Y /D1

2 G.X; Y /CG.Y; X /

of the tensorG.X; Y /is related to gravitation. The anti-symmetric part F .X; Y /D1

2 G.X; Y / G.Y; X /

DG.X; Y / g.X; Y /

of the basic tensorG.X; Y /is related to electromagnetism. Unlike Riemannian and generalized Riemannian manifolds, where affine connection coefficients are functions of the basic tensorsg.X; Y /andG.X; Y /, in Einstein’s works affine connection coefficients of manifolds at NGT satisfythe Einstein metricity condition:

ZG.X; Y / G rZX; Y

G X;rYZ

D0: (1.4)

The coordinate form of this condition is

@Gij

@x^k

˛

i kG˛j ˛

kjGi ˛D0; (1.5)

for affine connection coefficients _{j k}ⁱ of the manifold at NGT.

Based on the Einstein’s metricity condition, it exists two types r^C andr of covariant differentiation (see [5]). For example, for a tensorAof the type.1; 1/they are:

.rY^CA/.X /D.er^YA/.X /CT .AX; Y / A T .X; Y /

; (1.6)

.rYA/.X /D.erYA/.X / T .AX; Y /CA T .X; Y /

: (1.7)

Following the excellent Eisenhart’s book [4] and motivated by different results from the theory of symmetric affine connection manifolds, S. Minˇci´c [12–14] has obtained different significant results in the theory of non-symmetric affine connection manifolds. A geometry of these connections has been studied by F. Graiff [5], M. Prvan- ovi´c [19] and many others.

From the differences

rZ^CrY^CX rY^CrZ^CX; rZrYX rYrZX; rZrY^CX rY^CrZX;

one obtains the curvature tensors:

R1.XIY; Z/DR.XIY; Z/C reZT

.X; Y / erYT .X; Z/

CT T .X; Y /; Z

T T .X; Z/; Y

; (1.8)

R2.XIY; Z/DR.XIY; Z/ er^ZT

.X; Y /C er^YT .X; Z/

CT T .X; Y /; Z

T T .X; Z/; Y

;

(1.9)

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R3.XIY; Z/DR.XIY; Z/C reZT

.X; Y /C reYT .X; Z/

T T .X; Y /; Z

CT T .X; Z/; Y

2T T .Y; Z/; X

; (1.10) where R.XIY; Z/ is given by the formula (1.3). As one can see, these curvature tensors are introduced by using an algebraic approach to the curvature. Some of the geometric meaning of these curvature tensors has been pointed out in [18].

1.1. Geodesic mappings between affine connection manifolds

Geodesic mappings between affine connection manifolds have an important role in applications of differential geometry. Definitions and a lot of properties of mappings between symmetric affine connection manifolds are presented in books [9–11,21].

Many other authors have continued this research (see [16,17,22]).

A diffeomorphismf WGAN !GAN is called a geodesic mapping ofGAN onto GAN if any geodesic curve in GAN it maps onto a geodesic curve inGAN. The affine connectionsr androf the manifoldsGAN andGAN satisfy the equation

r^YX D r^YXC .X /YC .Y /XC .X; Y / (1.11) for a1-form and an anti-symmetric tensorof the type (0,2).

Invariants of geodesic mappings of a non-symmetric affine connection manifold GAN and some their properties are obtained in [25–27]. The main goal of this paper is to obtain some other generalizations of invariants of geodesic mappings defined on the manifoldGAN.

Having a geodesic mapping of two general affine connection manifolds, we can- not find a generalization of Weyl tensor as an invariant of geodesic mapping in general case. For this reason, one assumes that .X; Y /D0holds, that is, T .X; Y /D T .X; Y /. These mappings arethe equitorsion geodesic mappings.

2. INVARIANTS OF EQUITORSION GEODESIC MAPPINGS

Letf WGAN !GAN be an equitorsion geodesic mapping of a non-symmetric affine connection manifoldGAN. The basic equation of this mapping is (see [25–

27])

r^YXD r^YXC .X /Y C .Y /X; (2.1) for a1-form . The symmetric partsreYX andreYX of the affine connection coef- ficientsrYX andrYX satisfy the equation

erYX DreYXC 1 NC1

TrfU ! rUXg

Y C TrfU ! rUYg X

; (2.2)

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The curvature tensors R.XIY; Z/andR.XIY; Z/of the associated manifolds AN

andAN satisfy the relation

R.XIY; Z/DR.XIY; Z/C 1

NC1 Ri cc.Y; Z/ Ri cc.Z; Y / X 1

NC1 Ri cc.Y; Z/ Ri cc.Z; Y / X C 1

N² 1

NRi cc.X; Z/CRi cc.Z; X / Y

NRi cc.Y; Z/CRi cc.Z; Y / X 1

N² 1

N Ri cc.X; Z/CRi cc.Z; X / Y

N Ri cc.Y; Z/CRi cc.Z; Y / X

;

(2.3)

for the Ricci-curvature tensors Ri cc.X; Y / D Tr˚

U ! R.XIY; U / and Ri cc.X; Y /DTr˚

U !R.XIY; U / of the manifoldsAN andAN. From the equation (2.3), one obtains that the Weyl projective tensor

W .X; Y; Z/DR.XIY; Z/C 1

NC1 Ri cc.Y; Z/ Ri cc.Z; Y / X C 1

N² 1

NRi cc.X; Z/CRi cc.Z; X / Y NRi cc.X; Y /CRi cc.Y; X /

Z

(2.4)

is an invariant of the mappingf. The coordinates of the Weyl projective tensorW are

W_{j mn}ⁱ DR_{j mn}ⁱ C 1

NC1ı_jⁱR_ŒmnC N

N² 1ı_Œmⁱ R_{j n}C 1

N² 1ıⁱ_ŒmR_nj: (2.5) 2.1. Weyl projective tensorsW

1;W

2;W

3 of geodesic mappings From the equation (1.8), we obtain the following equation:

Ri cc.X; Y /DRi cc

1 .X; Y / Tr

n

U ! reUT

.X; Y /o CTr

n

U ! erYT

.X; U /o Tr

n

U !T T .X; Y /; Uo CTr

n

U !T T .X; U /; Yo

;

(2.6)

for Ricci-curvature tensorRi cc.X; Y /DTr˚

U !R.XIY; U / .

The similar equations can be derived for Ricci-curvature tensorsRi cc

2 .X; Y /and Ri cc

3 .X; Y /. In the same manner, we obtain the corresponding correlations between

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the Ricci-curvature tensorsRi cc.X; Y /andRi cc

.X; Y /; D1; 2; 3of the manifolds AN andGAN. BecauseT .X; Y /DT .X; Y /, it holds

erZT

.X; Y /D erZT

.X; Y / erT .X;Y /ZCT reZX; Y

CT X;erZY Cre_{T .X;Y /}Z T erZX; Y

T X;erZY :

(2.7)

From the expressions (2.3,2.6,2.7) involved in the correlation R1.XIY; Z/DR

1.XIY; Z/C 1

NC1 Ri cc.Y; Z/ Ri cc.Z; Y / Ri cc.Y; Z/CRi cc.Z; Y / X C 1

N² 1

NRi cc.X; Z/CRi cc.Z; X /

Y NRi cc.X; Y /CRi cc.Y; X / Z 1

N² 1

N Ri cc.X; Z/CRi cc.Z; X /

Y N Ri cc.X; Y /CRi cc.Y; X / Z C e_r_Z_T_{.X; Y /} e_rZT

.X; Y / e_r_Y_T_{.X; Z/}_C e_rYT .X; Z/;

we obtain

W1.X; Y; Z/DW

1.X; Y; Z/

for

W1.X; Y; Z/DR

1.XIY; Z/C 1

NC1 Ri cc

1 .Y; Z/ Ri cc

1 .Z; Y / X C 1

N² 1

N Ri cc

1 .X; Z/CRi cc

1 .Z; X / Y N Ri cc

1 .X; Y /CRi cc

1 .Y; X / Z

erT .X;Y /ZCT er^ZX; Y

CT X;er^ZY / CreT .X;Z/Y T erYX; Z

T X;erYZ C 2

NC1Tr˚

U !T .er^UY; Z/ T .er^UZ; Y / X C 1

NC1Tr n

U !erT .U;Z/Y T .U;Y /ZXo 1

N 1Trn

U !T er^ZX Y er^YXZ; Uo

(2.8) and the correspondingW

1.X; Y; Z/. In the same manner, we obtain W2.X; Y; Z/DW

2.X; Y; Z/ and W

3.X; Y; Z/DW

3.X; Y; Z/

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for

W2.X; Y; Z/DR

2.XIY; Z/C 1

NC1 Ri cc

2 .Y; Z/ Ri cc

2 .Z; Y / X C 1

N² 1

N Ri cc

2 .X; Z/CRi cc

2 .Z; X / Y N Ri cc

2 .X; Y /CRi cc

2 .Y; X / Z

CreT .X;Y /Z T erZX; Y

T X;erZY / erT .X;Z/Y CT er^YX; Z

CT X;er^YZ 2

NC1Tr˚

U !T .erUY; Z/ T .erUZ; Y / X 1

NC1Tr n

U !reT .U;Z/Y T .U;Y /ZXo C 1

N 1Tr n

U !T re^ZX Y er^YXZ; Uo

;

(2.9) W3.X; Y; Z/DR

3.XIY; Z/C 1

NC1 Ri cc

3 .Y; Z/ Ri cc

3 .Z; Y / X C 1

N² 1

N Ri cc

3 .X; Z/CRi cc

3 .Z; X / Y N Ri cc

3 .X; Y /CRi cc

3 .Y; X / Z erT .X;Y /ZCT erZX; Y

CT X;erZY / erT .X;Z/Y CT er^YX; Z

CT X;er^YZ C 2

NC1Tr˚

U !T .erUY; Z/ T .erUZ; Y / X C 1

NC1Tr n

U !erT .U;Z/Y T .U;Y /ZXo C 1

N 1Tr n

U !T re^ZX Y er^YXZ; Uo :

(2.10) Theorem 1. Let f WGAN !GAN be an equitorsion geodesic mapping. The geometrical objects(2.8,2.9,2.10)are invariants of the mappingf.

Corollary 1. The invariants W

1.X; Y; Z/;W

2.X; Y; Z/;W

3.X; Y; Z/and the Weyl projective tensorW .X; Y; Z/satisfy the equations:

W1.X; Y; Z/DW .X; Y; Z/CZ T .X; Y / Y T .X; Z/

CT T .X; Y /; Z

T T .X; Z/; Y

; (2.11)

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W2.X; Y; Z/DW .X; Y; Z/ Z T .X; Y /CY T .X; Z/

CT T .X; Y /; Z

T T .X; Z/; Y

; (2.12)

W3.X; Y; Z/DW .X; Y; Z/CZ T .X; Y /CY T .X; Z/

T T .X; Y /; Z

CT T .X; Z/; Y

2T T .Y; Z/; X

: (2.13) Example 1. Let a generalized Riemannian manifold GR3 be determined by the non-symmetric basic matrix

G.X; Y / D

2 6 4

1 e^x¹ e ^x² e^x¹ 1 e^x³ e ^x² e^x³ 1

3 7 5

and letf WGR3!GR3be an equitorsion geodesic mapping of the manifoldGR3. The symmetric and anti-symmetric part of the basic matrix G.X; Y /

are, respectively:

g.X; Y / D

2 4

1 0 0 0 1 0 0 0 1

3

5 and

F .X; Y / D

2 6 4

0 e^x¹ e ^x² e^x¹ 0 e^x³ e ^x² e^x³ 0

3 7 5: The symmetric part g.X; Y /

of the basic tensor G.X; Y /

is constant. For this reason, the coordinates of the curvature tensor R.XIY; Z/of the associated manifold R3 areR_{j mn}ⁱ D0. Furthermore, the coordinates of the Weyl projective tensor W .X; Y; Z/of the manifoldR3areW_{j mn}ⁱ D0. The coordinates of the torsion tensor T .X; Y /of the manifoldGR3are

T₂₃¹ D1

2e ^x²; T₃₂¹ D 1

2e ^x²; T₁₃² D 1 2e ^x²; T₃₁² D 1

2e ^x²; T₁₂³ D1

2e ^x²; T₂₁³ D 1 2e ^x² andT_{j k}ⁱ D0; i; j; kD1; 2; 3, in all other cases.

The coordinates W

1 i j mn;W

2 i j mn;W

3 i

j mn; i; j; m; n D 1; 2; 3, of the invariants W1.X; Y; Z/,W

2.X; Y; Z/,W

3.X; Y; Z/of the mappingf are:

W1 i

j mnDW_{j mn}ⁱ CT_{j m;n}ⁱ T_{j n;m}ⁱ CT_{j m}^˛ T_{˛ n}ⁱ T_{j n}^˛T_˛mⁱ ; W2

i

j mnDW_{j mn}ⁱ T_{j m;n}ⁱ CT_{j n;m}ⁱ CT_{j m}^˛ T_{˛ n}ⁱ T_{j n}^˛T_˛mⁱ ; W3

i

j mnDW_{j mn}ⁱ CT_{j m;n}ⁱ CT_{j n;m}ⁱ T_{j m}^˛ T_{˛ n}ⁱ CT_{j n}^˛T_˛mⁱ 2T_mn^˛ T_˛jⁱ

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for partial derivations denoted by commas and the above obtained coordinatesT_{j k}ⁱ of the torsion tensorT .X; Y /of the manifoldGR3. After a bit of computation, we obtain coordinates W

1 i j mn;W

2 i j mn;W

3 i

j mn of the invariants W

1.X; Y; Z/;

W2.X; Y; Z/;W

3.X; Y; Z/are:

W1 W 8 ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ :

W1 1

223DW

1 2 123DW

1 2

312DW

1 3

221D¹₂e ^x² W1

1

221DW

1 1 331DW

1 2

112DW

1 2

332DW

1 3

131DW

1 3

232D¹₄e ^2x² W1

1

232DW

1 2 132DW

1 2

321DW

1 3

212D ¹₂e ^x² W1

1

212DW

1 1 313DW

1 2

121DW

1 2

323DW

1 3

113DW

1 3

223D ¹₄e ^2x²

W2 W 8 ˆˆ ˆˆ ˆˆ

<

ˆˆ ˆˆ ˆˆ :

W2 1

232DW

2 2 132DW

2 2

321DW

2 3

212D¹₂e ^x² W2

1

221DW

2 1 331DW

2 2

112DW

2 2

332DW

2 3

131DW

2 3

232D¹₄e ^2x² W2

1

223DW

2 2 123DW

2 2

312DW

2 3

221D ¹₂e ^x² W2

1

212DW

2 1 313DW

2 2

121DW

2 2

323DW

2 3

113DW

2 3

223D ¹4e ^2x²

W3 W 8 ˆˆ ˆˆ ˆˆ ˆˆ ˆ<

ˆˆ ˆˆ ˆˆ ˆˆ ˆ:

W3 1

322De ^x²;W

3 3

122D e ^x²;W

3 1 212DW

3 2 121DW

3 3 131DW

3 3

232D³₄e ^2x² W3

2

312DW

3 2

321D¹₂e ^x²; W3

1

221DW

3 1 331DW

3 2

112DW

3 2

332D¹4e ^2x² W3

1 223DW

3 2 232DW

3 2 123DW

3 2 132DW

3 3 212DW

3 3 221DW

3 3 311DW

3 3

322D ¹2e ^x² W3

1

313DW

3 2 323DW

3 3

113DW

3 3

223D ¹4e ^2x² andW

i

j mnD0; ; i; j; m; nD1; 2; 3, in all other cases.

3. CONCLUSION

We expanded the concept of geodesic mappings defined on spaces with torsion in this paper. Invariants of equitorsion geodesic mappings are obtained above. We found an another generalization of Weyl projective tensor in this way.

The results, which are obtained in this paper, are presented in the corresponding operator forms. Moreover, some of these results are presented coordinately for different applications in physics.

In further research, we will stay focused on NGT, generalized Riemannian spaces in the sense of Eisenhart, as well as semi-symmetric and quarter-symmetric spaces.

We will also study conformal invariants of a generalized Riemannian manifoldGRN.

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4. ACKNOWLEDGEMENTS

The author thanks the anonymous reviewers for their careful reading of the manu- script and their many insightful comments and suggestions.

REFERENCES

[1] A. Einstein, “A generalization of the relativistic theory of gravitation,”Ann. of Math., vol. 45, no. 2, pp. 576–584, 1945, doi:10.2307/1969197.

[2] A. Einstein, “Bianchi identities in the generalized theory of gravitation,”Can. J. Math., no. 2, pp.

120–128, 1950, doi:10.4153/CJM-1950-011-4.

[3] A. Einstein,The meaning of relativity: Including the relativistic theory of the non-symmetric field, 5th ed. New York: Princeton, NJ: Princeton University Press, 1984.

[4] L. P. Eisenhart,Non-Riemannian Geometry. New York: American Mathematical Society (Col- loquium publications vol. VIII), 1927.

[5] F. Graiff, “Formule di comutazione e trasporto ciclico nei recenti spazi di einstein,”Ist. Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur., III. Ser., vol. 87, no. 1, pp. 105–110, 1954.

[6] G. Hall, “Projective relatedness and conformal flatness,”Cent. Eur. J. Math., vol. 10, pp. 1763–

1770, 2012, doi:10.2478/s11533-012-0087-6.

[7] G. Hall, “On the converse of weyl’s conformal and projective theorems,”Publ. Inst. Math., Nouv.

S´er., vol. 108, no. 94, pp. 55–65, 2013, doi:10.2298/PIM1308055H.

[8] S. Ivanov and M. L. Zlatanovi´c, “Connections on a non-symmetric (generalized) riemannian manifold and gravity,” Class. Quantum Grav., vol. 33, no. 7, 2016, doi: 10.1088/0264- 9381/33/7/075016.

[9] J. Mikeˇs, V. Kiosak, and A. Vanˇzurov´a,Geodesic Mappings of Manifolds with Affine Connection.

Olomouc: Olomouc: Palack´y University, Faculty of Science, 2008.

[10] J. Mikeˇs, E. Stepanova, A. Vanˇzurov´a, and et al., Differential Geometry of Special Mappings.

[11] J. Mikeˇs, A. Vanˇzurov´a, and I. Hinterleitner, Geodesic Mappings and Some Generalizations.

[12] S. M. Minˇci´c, “Ricci identities in the space of non-symmetric affine connexion,”Mat. Vesn., N.

Ser., vol. 10 (25), no. 2, pp. 161–172, 1973.

[13] S. M. Minˇci´c, “New commutation formulas in the non-symmetric affine connexion space,”Publ.

Inst. Math. (Beograd), vol. 22, no. 36, pp. 189–199, 1977.

[14] S. M. Minˇci´c and M. S. Stankovi´c, “On geodesic mappings of general affine connexion spaces and of generalized riemannian spaces,”Mat. Vesn., vol. 49, no. 2, pp. 27–33, 1997.

[15] S. M. Minˇci´c and M. S. Stankovi´c, “New special geodesic mappings of general affine connection spaces,”FILOMAT, vol. 14, no. 1, pp. 19–31, 2000.

[16] G. P. Pokhariyal, “Relativistic significance of curvature tensors,”Internat. J. Math. & Math. Sci., vol. 5, no. 1, pp. 133–139, 1982, doi:10.1155/S0161171282000131.

[17] G. P. Pokhariyal and R. S. Mishra, “Curvature tensors and their relativistic significance (ii),”Yoko- hama Math. J., vol. 19, no. 2, pp. 97–103, 1971.

[18] M. Prvanović, “Les vecteurs des courbures cycliques des courbes d’un espace riemannien et leurs queloques propriétés,”Acad. Serbe Sci., Publ. Inst. Math, vol. 13, pp. 35–46, 1959.

[19] M. Prvanovi´c, “Four curvature tensors of non-symmetric connection,”Proceedings of the Confer- ence ”150 years of Lobachevski Geometry”, pp. 199–205, 1977.

[20] U. P. Singh, “On relative curvature tensors in the subspace of a riemannian space,”Rev. Fac. Sci.

Univ. Istanbul, S´er. A, vol. 33, no. 2, pp. 69–75, 1968.

(11)

[21] N. S. Sinyukov, Geodesic Mappings of Riemannian Spaces (in Russian). Moskva: Moskva:

Nauka. 256 p. R. 1.60 (1979), 1979.

[22] V. S. Sobchuk, J. Mikeˇs, and O. Pokorna, “On almost geodesic mappings 2

between semisymmetric riemannian spaces,” Novi Sad J. Math., vol. 29, no. 3, pp.

309–312, 1999.

[23] M. S. Stanković, S. M. Minˇcić, L. S. Velimirović, and M. L. Zlatanović, “On equitorsion geodesic mappings of general affine connection spaces,”Rend. Semin. Mat. Univ. Padova, vol. 124, pp.

77–90, 2010, doi:10.4171/RSMUP/124-5.

[24] M. S. Stanković, M. L. Zlatanović, and L. S. Velimirović, “Equitorsion holomorphically projective mappings of generalized kahlerian space of the first kind,”CZECH MATH J, vol. 60, no. 3, pp.

635–653, 2010, doi:10.1007/s10587-010-0059-6.

[25] M. S. Stanković, M. L. Zlatanović, and N. O. Vesić, “Some properties of et- projective tensors obtained from weyl projective tensor,” Filomat, vol. 29, no. 3, pp.

573–584, 2015, doi:10.2298/FIL1503573S.

[26] N. O. Vesić, L. S. Velimirović, and M. S. Stanković, “Some invariants of equitorsion third type almost geodesic mappings,”Mediterr. J. Math., vol. 13, no. 6, pp. 4581–4590, 2016, doi:

10.1007/s00009-016-0763-z.

[27] M. L. Zlatanovi´c, “New projective tensors for equitorsion geodesic mappings,”Appl. Math. Lett., vol. 25, no. 5, pp. 890–897, 2012, doi:10.1016/j.aml.2011.10.045.

Author’s address

Nenad O. Vesi´c

University of Niˇs, Faculty of Science and Mathematics, Viˇsegradska 33, 18000 Niˇs, Serbia.

E-mail address:vesko1985@pmf.ni.ac.rs