Title:
|
The Killing Tensors on an $n$-dimensional Manifold with $SL(n,)$-structure (English) |
Author:
|
Stepanov, Sergey E. |
Author:
|
Tsyganok, Irina I. |
Author:
|
Khripunova, Marina B. |
Language:
|
English |
Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
ISSN:
|
0231-9721 |
Volume:
|
55 |
Issue:
|
1 |
Year:
|
2016 |
Pages:
|
121-131 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
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 $SL(n,)$-structure and discuss possible applications of obtained results in Riemannian geometry. (English) |
Keyword:
|
Differentiable manifold |
Keyword:
|
$SL(n,)$-structure |
Keyword:
|
Killing tensors |
MSC:
|
53A15 |
MSC:
|
53A45 |
idZBL:
|
Zbl 1365.53027 |
idMR:
|
MR3674606 |
. |
Date available:
|
2016-08-30T12:04:13Z |
Last updated:
|
2018-01-10 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145823 |
. |
Reference:
|
[1] Chern, S. S.: The geometry of G-structures.. Bull. Amer. Math. Soc. 72 (1966), 167–219. Zbl 0136.17804, MR 0192436, 10.1090/S0002-9904-1966-11473-8 |
Reference:
|
[2] Eisenhart, L. P.: Riemannian geometry.. Princeton Univ. Press, Princeton, NJ, 1949. Zbl 0041.29403, MR 0035081 |
Reference:
|
[3] Fulton, C. M.: Parallel vector fields.. Proc. Amer. Math. Soc. 16 (1965), 136–137. Zbl 0141.19303, MR 0172215, 10.1090/S0002-9939-1965-0172215-0 |
Reference:
|
[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. MR 0224021 |
Reference:
|
[5] Kashiwada, T.: On konformal Killing tensor.. Natural Science Report, Ochanomizu University 19, 2 (1968), 67–74. MR 0243458 |
Reference:
|
[6] Kobayashi, Sh.: Transformation Groups in Differential Geometry.. Erbeb. Math. Grenzgeb 70, Springer-Verlag, New York–Heidelberg, 1972. Zbl 0246.53031, MR 0355886 |
Reference:
|
[7] Kobayashi, Sh., Nomizu, K.: Foundations of differential geometry. Vol. 1.. Intersience, New York–London, 1963. MR 0152974 |
Reference:
|
[8] Krame, D., Stephani, H., MacCallum, M. A. H., Herit, E.: Exact solutions of Einstein’s field equations.. Cambridge Univ. Press, Cambridge, 1980. MR 0614593 |
Reference:
|
[9] Mikeš, J.: Geodesic mapping of affine-connected and Riemannian spaces.. J. Math. Sci. 78, 3 (1996), 311–333. MR 1384327, 10.1007/BF02365193 |
Reference:
|
[10] Mikeš, J., Stepanova, E., Vanžurová, A.: Differential Geometry of Special Mappings.. Palacký University, Olomouc, 2015. Zbl 1337.53001, MR 3442960 |
Reference:
|
[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, MR 0212697 |
Reference:
|
[12] Nomizu, K.: What is affine differential geometry.? In: Differential Geometry Meeting, Univ. Munster, 1982, 42–43. |
Reference:
|
[13] Nomizu, K.: On completeness in affine differential geometry.. Geometriae Dedicata 20, 1 (1986), 43–49. Zbl 0587.53010, MR 0823159, 10.1007/BF00149271 |
Reference:
|
[14] Schouten, J. A.: Ricci-calculus.. Grundlehren Math. Wiss., 10, Springer-Verlag, Berlin, 1954. Zbl 0057.37803, MR 0516659 |
Reference:
|
[15] Schirokow, P. A., Schirokow, A. P.: Affine Differentialgeometrie.. Teubner, Leipzig, 1962. Zbl 0106.14703, MR 0150666 |
Reference:
|
[16] Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the Affine Differential Geometry of Hypersurfaces.. Science University of Tokyo, Tokyo, 1991. MR 1200242 |
Reference:
|
[17] Stepanov, S. E.: The Killing-Yano tensor.. Theoretical and Mathematical Physics 134, 3 (2003), 333–338. Zbl 1178.53074, MR 2001815, 10.1023/A:1022645304580 |
Reference:
|
[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. MR 1654091 |
Reference:
|
[19] Stepanov, S. E.: On conformal Killing 2-form of the electromagnetic field.. J. Geom. Phys. 33 (2000), 191–209. Zbl 0977.53013, MR 1747041, 10.1016/S0393-0440(99)00046-7 |
Reference:
|
[20] Stepanov, S. E.: A class of closed forms and special Maxwell’s equations.. Tensor, N.S. 58 (1997), 233–242. Zbl 0954.53026, MR 1719406 |
Reference:
|
[21] Stepanov, S. E.: The vector space of conformal Killing forms on a Riemannian manifold.. J. Math. Sci. 110, 4 (2002), 2892–2906. MR 1758432, 10.1023/A:1015327018220 |
Reference:
|
[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, MR 3275956 |
Reference:
|
[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. MR 1972985 |
Reference:
|
[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. MR 1224969 |
Reference:
|
[25] Tachibana, S.-I.: On conformal Killing tensor in a Riemannian space.. Tohoku Math. Journal 21, 1 (1969), 56–64. Zbl 0182.55301, MR 0242078, 10.2748/tmj/1178243034 |
Reference:
|
[26] Yano, K., Ishihara, Sh.: Harmonic and relative affine mappings.. J. Differential Geometry 10 (1975), 501–509. MR 0390969, 10.4310/jdg/1214433157 |
Reference:
|
[27] Yano, K., Nagano, T.: Some theorems on projective and conformal transformations.. Ind. Math. 14 (1957), 451–458. Zbl 0079.15603, MR 0110993, 10.1016/S1385-7258(57)50059-0 |
. |