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The Killing tensors on an \(n\)-dimensional manifold with \(\mathrm{SL}(n,\mathbb{R})\)-structure. (English) Zbl 1365.53027

Summary: In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an \(n\)-dimensional differentiable manifold \(M\) endowed with an equiaffine \(\mathrm{SL}(n,\mathbb{R})\)-structure and discuss possible applications of obtained results in Riemannian geometry.

MSC:

53C10 \(G\)-structures
53A15 Affine differential geometry
53A45 Differential geometric aspects in vector and tensor analysis
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[1] Chern, S. S.: The geometry of G-structures. Bull. Amer. Math. Soc. 72 (1966), 167-219. | | · Zbl 0136.17804 · doi:10.1090/S0002-9904-1966-11473-8
[2] Eisenhart, L. P.: Riemannian geometry. Princeton Univ. Press, Princeton, NJ, 1949. | · Zbl 0041.29403
[3] Fulton, C. M.: Parallel vector fields. Proc. Amer. Math. Soc. 16 (1965), 136-137. | | · Zbl 0141.19303 · doi:10.1090/S0002-9939-1965-0172215-0
[4] Katzin, G. H., Levine, J.: Note on the number of linearly independent \(m^{th}\)-order first integrals space of constant curvature. Tensor 19, 1 (1968), 42-44. · Zbl 0161.18801
[5] Kashiwada, T.: On konformal Killing tensor. Natural Science Report, Ochanomizu University 19, 2 (1968), 67-74. · Zbl 0179.26902
[6] Kobayashi, Sh.: Transformation Groups in Differential Geometry. Erbeb. Math. Grenzgeb 70, Springer-Verlag, New York-Heidelberg, 1972. | · Zbl 0246.53031
[7] Kobayashi, Sh., Nomizu, K.: Foundations of differential geometry. Vol. 1. Intersience, New York-London, 1963. · Zbl 0119.37502
[8] Krame, D., Stephani, H., MacCallum, M. A. H., Herit, E.: Exact solutions of Einstein’s field equations. Cambridge Univ. Press, Cambridge, 1980.
[9] Mikeš, J.: Geodesic mapping of affine-connected and Riemannian spaces. J. Math. Sci. 78, 3 (1996), 311-333. | · Zbl 0866.53028 · doi:10.1007/BF02365193
[10] Mikeš, J., Stepanova, E., Vanžurová, A.: Differential Geometry of Special Mappings. Palacký University, Olomouc, 2015. | · Zbl 1337.53001
[11] Nijenhuis, A.: A note on first integrals of geodesics. Proc. Kon. Ned. Acad. Van. Wetens., Ser. A 52 (1967), 141-145. | · Zbl 0161.18803
[12] Nomizu, K.: What is affine differential geometry. ? In: Differential Geometry Meeting, Univ. Munster, 1982, 42-43.
[13] Nomizu, K.: On completeness in affine differential geometry. Geometriae Dedicata 20, 1 (1986), 43-49. | | · Zbl 0587.53010 · doi:10.1007/BF00149271
[14] Schouten, J. A.: Ricci-calculus. Grundlehren Math. Wiss., 10, Springer-Verlag, Berlin, 1954. | · Zbl 0057.37803
[15] Schirokow, P. A., Schirokow, A. P.: Affine Differentialgeometrie. Teubner, Leipzig, 1962. | · Zbl 0106.14703
[16] Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the Affine Differential Geometry of Hypersurfaces. Science University of Tokyo, Tokyo, 1991. · Zbl 0780.53002
[17] Stepanov, S. E.: The Killing-Yano tensor. Theoretical and Mathematical Physics 134, 3 (2003), 333-338. | | · Zbl 1178.53074 · doi:10.1023/A:1022645304580
[18] Stepanov, S. E.: The Bochner technique for an \(m\)-dimensional compact manifold with \(SL(m,)\)-structure. St. Petersburg Mathematical Journal 10, 4 (1999), 703-714.
[19] Stepanov, S. E.: On conformal Killing 2-form of the electromagnetic field. J. Geom. Phys. 33 (2000), 191-209. | | · Zbl 0977.53013 · doi:10.1016/S0393-0440(99)00046-7
[20] Stepanov, S. E.: A class of closed forms and special Maxwell’s equations. Tensor, N.S. 58 (1997), 233-242. | · Zbl 0954.53026
[21] Stepanov, S. E.: The vector space of conformal Killing forms on a Riemannian manifold. J. Math. Sci. 110, 4 (2002), 2892-2906. | · Zbl 1022.53030 · doi:10.1023/A:1015327018220
[22] Stepanov, S. E., Jukl, M., Mikeš, J.: On dimensions of vector spaces of conformal Killing forms. In: Springer Proceedings in Mathematics & Statistics 85, (Algebra, geometry and mathematical physics. AGMP, Mulhouse, France, October 24-26, 2011), Springer, Berlin, 2014, 495-507. | · Zbl 1320.53041
[23] Stepanov, S. E., Smol’nikova, M. V.: Fundamental differential operators of orders one on exterior and symmetric forms. Russ. Math. J. 46, 11 (2002), 51-56.
[24] Stepanov, S. E., Tsyganok, I. I.: Vector fields in a manifolds with equiaffine connections. In: Webs and Quasigroups, Tver Univ. Press, Tver, 1993, 70-77. · Zbl 0792.53013
[25] Tachibana, S.-I.: On conformal Killing tensor in a Riemannian space. Tohoku Math. Journal 21, 1 (1969), 56-64. | | · Zbl 0182.55301 · doi:10.2748/tmj/1178243034
[26] Yano, K., Ishihara, Sh.: Harmonic and relative affine mappings. J. Differential Geometry 10 (1975), 501-509. · Zbl 0317.53044
[27] Yano, K., Nagano, T.: Some theorems on projective and conformal transformations. Ind. Math. 14 (1957), 451-458. | | · Zbl 0079.15603 · doi:10.1016/S1385-7258(57)50059-0
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