SOME NEW IDENTITIES FOR THE SECOND COVARIANT DERIVATIVE OF THE CURVATURE TENSOR

Miroslav D. Maksimovic, Mića S. Stanković

DOI Number
https://doi.org/10.22190/FUMI200930038M
First page
519
Last page
528

Abstract


In this paper we study the second covariant derivative of Riemannian curvature tensor. Some new identities for the second covariant derivative are given. Namely, identities obtained by cyclic sum with respect to three indices are given. In the first case, two curvature tensor indices and one covariant derivative index participate in the cyclic sum, while in the second case one curvature tensor index and two covariant derivative indices participate in the cyclic sum.

Keywords

covariant derivative, curvature tensor, Riemannian manifold, second order identity

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References


bibitem{calmet2018}

{sc X. Calmet}: textit{Vanishing of quantum gravitational corrections to vacuum solutions of general relativity at second order in curvature}. Physics Letters B. {bf 787} (2018), 36--38.

bibitem{chongshan1986}

{sc L. Chongshan}: textit{On concircular transformations in Riemannian spaces}. J. Austral. Math. Soc. {bf 40} (1986), 218--225.

bibitem{cui2019}

{sc J. Cui, J. C. Yong, H. T. Yun {rm and} P. Zhao}: textit{On a projective conformal semi-symmetric connection}. Filomat. {bf 33}, No.12 (2019), 3901--3912.

bibitem{eisenhart1967} {sc L.P. Eisenhart}: textit{Riemannian spaces}, Princeton Univ. Press, 1967.

bibitem{hinter2009} {sc I. Hinterleitner {rm and} J. Mikev s}: textit{Geodesic mappings onto Weyl manifolds}, J. Appl. Math. {bf 2}, (2009), 125--133.

bibitem{mantica2011}

{sc C. A. Mantica {rm and} L. G. Molinari}: textit{A second order identity for the Riemann tensor and applications}. Colloq. Math. {bf 122} (2011), 69--82.

bibitem{mikes2015} {sc J. Mikev s}: textit{Differential geometry of special mappings}, Palacky Univ. Press, Olomouc, 2015.

bibitem{nieuw1977}

{sc P. van Nieuwenhuizen {rm and} C. C. Wu}: textit{On integral relations for invariants constructed from three Riemann tensors and their applications in quantum gravity}. J. Math. Phys. {bf 18}, 182 (1977).

bibitem{petrovic2019}

{sc M.Z. Petrovi'c}: textit{Generalized para-Kahler spaces in Eisenhart's sense admitting a holomorphically projective mapping}. Filomat {bf 33}, No.13 (2019), 4001--4012.

bibitem{prasad2017}

{sc R. Prasad {rm and} A. Haseeb}: textit{Conformal curvature tensor on $K$-contact manifolds with respect to the quarter-symmetric metric connection}. Facta Universitatis, Ser. Math. Inform. {bf 32}, No.4 (2017), 503--514.

%%%%%%%%%%%%%%%%%%%%

bibitem{micapi1}

{sc M. S. Stankovic}: textit{First type almost geodesic mappings

of general affine connection spaces}. Novi Sad J. Math. {bf 29}, No. 3

(1999), 313--323.

bibitem{micapi2}

{sc M. S. Stankovi'c}: textit{On a canonic almost geodesic

mappings of the second type of affine spaces}. Filomat {bf 13}, (1999),

--114.

bibitem{micapi3}

{sc M. S. Stankovi'c}: textit{On a special almost geodesic

mappings of third type of affine spaces}. Novi Sad J. Math. Vol.

{bf 31}, No. 2, 2001, 125--135.

bibitem{micasg} {sc M. S. Stankovi'c}: textit{Special equitorsion almost

geodesic mappings of the third type of non-symmetric affine

connection spaces}. Applied Mathematics and Computation, {bf 244},

(2014), 695--701.

bibitem{micamincic}

{sc M. S. Stankovi'c {rm and} S. M. Minv ci'c}: textit{New special

geodesic mappings of generalized Riemannian space}. Publ. Inst.

Math. (Beograd) (N. S) {bf 67}(81) (2000), 92--102.

bibitem{micamincicljubica2}

{sc M. S. Stankovi'c, S. M. Minv ci'c {rm and} Lj. S. Velimirovi'c}:

textit{On holomorphically projective mappings of generalized

Kahlerian spaces}. Matematicki vesnik {bf 54} (2002), 195--202.

bibitem{micamincicljubica1}

{sc M. S. Stankovi'c, S. M. Minv ci'c {rm and} Lj. S. Velimirovi'c}:

textit{On equitorsion holomorphically projective mappings of

generalised Kahlerian spaces}. Czechosl. Math. J.,

{bf 54} (129) (2004), No. 3, 701-715.

bibitem{micamilanljubica2}

{sc M. S. Stankovi'c, M. Lj. Zlatanovi'c {rm and} Lj. S.

Velimirovi'c}: textit{Equitorsion holomorphically projective

mappings of generalized Kahlerian space of the second kind},

International Electronic Journal of Geometry, Vol. {bf 3}, No. 2

(2010), 26--39.

bibitem{micazlatja} {sc M. S. Stankovi'c, M. Lj.

Zlatanovi'c {rm and} N. O. Vesi'c}: textit{Basic equations of

$G$-almost geodesic mappings of the second type, which have the

property of reciprocity}, Czechosl. Math. J.,

(2015) Vol. {bf 65}, No. 3, pp. 787--799.

%%%%%%%%%%%%%%%%%%%%

bibitem{tani1967}

{sc M. Tani}: textit{On a conformally flat Riemannian space with positive Ricci curvature}. Tohoku Math. Journ. {bf 19}, 2 (1967).

bibitem{veblen1922}

{sc O. Veblen}: textit{Normal coordinates for the geometry of paths}. Proceedings of Nat. Acad. of Sciences {bf 8} (1922), 192--197.

bibitem{walker1950}

{sc A. G. Walker}: textit{On Ruse's spaces of recurrent curvature}. In: Proceedings of the London Mathematical Society. {bf 52} (1950), 36--64.

bibitem{zakhary2001}

{sc E. Zakhary {rm and} J. Carminati}: textit{On the problem of algebraic completeness for the invariants of the Riemann tensor I}. J. Math. Phys. {bf 42} (2001).

bibitem{zlatanovic2014}

{sc M.Lj. Zlatanovi'c, I. Hinterleitner {rm and} M.S. Najdanovi'c}: textit{On equitorsion concircular tensors of generalized Riemannian spaces}. Filomat. {bf 28}, No.3 (2014), 463--471.

bibitem{zlatanovic2015}

{sc M.Lj. Zlatanovi'c, S.M. Minv{c}i'c {rm and} M.Z. Petrovi'c}: textit{Curvature tensors and pseudotensors in a generalized Finsler space}. Facta Universitatis, Ser. Math. Inform. {bf 30}, No.5 (2015), 741--752.

bibitem{ZlatanovicStankovicJMAA2017}

{sc M.Lj. Zlatanovi'c {rm and} V.M. Stankovi'c}: textit{Some invariants

of holomorphically projective mappings of generalized Kahlerian

spaces}. J. Math. Anal. Appl., Vol. {bf 450}, (2017), 480--489.

bibitem{ZlatanovicStankovicJMAA2018}

{sc M.Lj. Zlatanovi'c {rm and} V.M. Stankovi'c}: textit{Geodesic

mapping onto K"ahlerian space of the third kind}. J. Math. Anal.

Appl., Vol. {bf 458}, (2018), 601--610.




DOI: https://doi.org/10.22190/FUMI200930038M

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