On symmetric Killing tensors and Codazzi tensors of ranks $p\geq 2$

S. E. Stepanov ; I. A. Aleksandrova ; I. I. Tsyganok

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 1, Tome 179 (2020), pp. 94-120.

Résumé

This paper is devoted to the geometry of symmetric Killing tensors and Codazzi tensors of ranks $p\ge 2$ and includes, in addition to the new results obtained in this paper, a survey on this topic from earlier works.

Détail

Keywords: Killing tensor, Codazzi tensor, geodesic, Riemannian manifold, curvature tensor.

Bibliographie

[13] Besse A., Mnogoobraziya Einshteina, Mir, M., 1990

[14] Burginon Zh.-P., “Formuly Veitsenbeka v razmernosti $4$”, Chetyrekhmernaya rimanova geometriya. Seminar Artura Besse 1978/1979, Mir, M., 1985, 260–279

[15] Veil G., Klassicheskie gruppy. Ikh invarianty i predstavleniya, IL, M., 1947

[16] Dairbekov N. S., Sharafutdinov V. A., “Konformno killingovy simmetrichnye tenzornye polya na rimanovykh mnogoobraziyakh”, Mat. tr., 13:1 (2010), 85–145 | Zbl

[17] Kobayasi Sh., Gruppy preobrazovanii v differentsialnoi geometrii, Fizmatlit, M., 1986

[18] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii. T. 2, Nauka, M., 1981 | MR

[19] Kramer D., Shtefani Kh., Kherlt E., Mak-Kallum M., Tochnye resheniya uravnenii Einshteina, Energoizdat, M., 1982 | MR

[20] Laptev G. F., Ostianu N. M., “Raspredelenie $m$-mernykh lineinykh elementov v prostranstve proektivnoi svyaznosti, I”, Tr. geom. semin. VINITI., 5 (1971), 49–94

[21] Manturov O. V., Elementy tenzornogo ischisleniya, Prosveschenie, M., 1991

[22] Nikitin A. G., “Obobschennye tenzory Killinga proizvolnogo poryadka i poryadka”, Ukr. mat. zh., 43:6 (1991), 786–795 | MR | Zbl

[23] Norden A. P., Prostranstva affinnoi svyaznosti, Nauka, M., 1976 | MR

[24] Sinyukov N. S., Geodezicheskie otobrazheniya rimanovykh prostranstv, Nauka, M., 1979 | MR

[25] Stepanov S. E., “Ob odnom klasse rimanovykh struktur pochti proizvedeniya”, Izv. vuzov. Mat., 7 (1989), 40–46

[26] Stepanov S. E., “Geometricheskoe prepyatstvie k suschestvovaniyu vpolne ombilicheskogo raspredeleniya na kompaktnom mnogoobrazii”, Tkani i kvazigruppy, Kalinin, 1990, 135–137 | Zbl

[27] Stepanov S. E., “Polya simmetricheskikh tenzorov na kompaktnom rimanovom mnogoobrazii”, Mat. zametki., 52:4 (1992), 85-–88 | Zbl

[28] Stepanov S. E., “Integralnaya formula dlya kompaktnogo mnogoobraziya s rimanovoi strukturoi pochti proizvedeniya”, Izv. vuzov. Mat., 7 (1994), 69-–73 | Zbl

[29] Stepanov S. E., “O primenenii odnoi teoremy P. A. Shirokova v tekhnike Bokhnera”, Izv. vuzov. Mat., 9 (1996), 53-–59 | Zbl

[30] Stepanov S. E., Rodionov V. V., “Dopolnenie k odnoi rabote Zh. P. Burginona”, Differ. geom. mnogoobr. figur., 1997, no. 28, 68–72 | Zbl

[31] Stepanov S. E., Smolnikova M. B., “Fundamentalnye differentsialnye operatory pervogo poryadka na vneshnikh i simmetricheskikh formakh”, Izv. vuzov. Mat., 2002, no. 11, 55–60 | Zbl

[32] Stepanov C. E., Smolnikova M. B., “Affinnaya differentsialnaya geometriya tenzorov Killinga”, Izv. vuzov. Mat., 11 (2004), 82-–86 | Zbl

[33] Sharafutdinov V. A., “Killingovy tenzornye polya na $2$-tore”, Sib. mat. zh., 57:1 (2016), 199–221 | MR | Zbl

[34] Shinkunas Yu. I., “O raspredelenii $m$-mernykh ploskostei v $n$-mernom rimanovom prostranstve”, Tr. geom. semin. VINITI., 5 (1973), 123–133

[35] Shirokov P. A., Shirokov A. P., Affinnaya differentsialnaya geometriya, Fizmatlit, M., 1959

[36] Eizenkhart L. P., Rimanova geometriya, IL, M., 1948

[37] Aleksandrova I. A., Stepanov S. E., Tsyganok I. I., A remark on a Laplacian operator which acts on symmetric tensors, arXiv: 1411.1928 [math.DG]

[38] Barbance C., “Sur les tenseurs symetriques”, C. R. Acad. Sci. Paris. Ser. A., 276 (1973), 387–389 | MR | Zbl

[39] Bettiol R. G., Mendes R. A. E., Sectional curvature and Weitzenböck formulae, arXiv: 1708.09033v1 [math.DG]

[40] Boucetta M., “Spectre des Laplacien de Lichnerowicz sur les sphères et les projectifs réels”, Publ. Mat., 43:2 (1999), 451–483 | DOI | MR | Zbl

[41] Bourguignon J. P., “Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d'Einstein”, Invent. Math., 63 (1981), 263–286 | DOI | MR | Zbl

[42] Calderbank D. M. J., Gauduchon P., Herzlich M., “On the Kato inequality in Riemannian geometry”, Global Analysis and Harmonic Analysis, Soc. Math. Fr., Paris, 2000, 95–113 | MR | Zbl

[43] Catino G., Mantegazza C., Mazzieri L., “A note on Codazzi tensors”, Math. Ann., 362:1–2 (2015), 629–638 | DOI | MR | Zbl

[44] Coll B., Ferrando J. J., Saez J. A., “On the geometry of Killing and conformal tensors”, J. Math. Phys., 47 (2006), 062503 | DOI | MR | Zbl

[45] Derdzinski A., Shen C. L., “Codazzi tensor fields, curvature and Pontryagin forms”, Proc. London Math. Soc., 47:3 (1983), 15–-26 | DOI | MR | Zbl

[46] Gebarowski A., “On nearly conformally symmetric warped product space-times”, Proc. Int. Conf. “Differential Geometry and Its Applications”, Bucharest, 1992, 61–75 | MR

[47] Chen B. Y., Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific, New Jersey, 2017 | MR | Zbl

[48] Green R. E., Wu H., “Integrals of subharmonic functions on manifolds of nonnegative curvature”, Invent. Mat., 27 (1974), 265–298 | DOI | MR | Zbl

[49] Heil K., Moroianu A., Semmelmann U., “Killing tensors on tori”, J. Geom. Phys., 117 (2017), 1–6 | DOI | MR | Zbl

[50] Heil K., Moroianu A., Semmelmann U., “Killing and conformal Killing tensors”, J. Geom. Phys., 106 (2016), 383–400 | DOI | MR | Zbl

[51] Johnson D. L., Whitt L. B., “Totally geodesic foliations on $3$-manifolds”, Proc. Am. Math. Soc., 76:2 (1979), 355–357 | MR | Zbl

[52] Katzin G. H., Levine J., “Note on the number of linearly independent mth-order first in-tegrals space of constant curvature”, Tensor., 19:1 (1968), 42–44 | MR | Zbl

[53] Kubiznak D., Frolov V. P., “The hidden symmetry of higher dimensional Kerr-NUT-AdS spacetimes”, Class. Quant. Grav., 24:3 (2007), F1–F6 | DOI | MR | Zbl

[54] Leder J., Schwenk-Schellschmidt A., Simon U., Wiehe M., “Generating higher order Codazzi tensors by functions”, Geometry and Topology of Submanifolds, v. IX, World Scientific, London–Singapore, 1999, 174–191 | DOI | MR

[55] Li P., Schoen R., “$L_p$ and mean value properties of subharmonic function on Riemannian manifolds”, Acta Math., 153:1 (1984), 279–301 | MR | Zbl

[56] Lichnerowicz A., “Propagateurs et commutateurs en relativite generate”, Publ. Math. IHES., 10:1 (1961), 5–56 | DOI | MR

[57] Liu H. L., “Codazzi tensor and the topology of surfaces”, Ann. Glob. Anal. Geom., 16 (1998), 189–202 | DOI | MR | Zbl

[58] Liu H. L., Simon U., Wang C. P., “Higher order Codazzi tensors on conformally flat spaces”, Beitr. Algebra Geom., 39:2 (1998), 329–348 | MR | Zbl

[59] Mantica C. A., Molinari L. C., Suh Y. J., Shenawy S., “Perfect-fluid, generalized Robertson–Walker space-times, and Gray’s decomposition”, J. Math. Phys., 60:5 (2019), 052506 | DOI | MR | Zbl

[60] Merton G., “Codazzi tensors with two eigenvalue functions”, Proc. Am. Math. Soc., 141:9 (2013), 3265–3273 | DOI | MR | Zbl

[61] Mikeš J. et. al., Differential Geometry of Special Mappings, Palacky Univ. Press, Olomouc, 2015 | MR | Zbl

[62] Petersen P., Riemannian geometry, Springer-Verlag, 2016 | MR | Zbl

[63] Pigola S., Rigoli M., Setti A. G., Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique, Birkhäuser, Basel, 2008 | MR | Zbl

[64] Ponge R., Reckziegel H., “Twisted products in pseudo-Riemannian geometry”, Geom. Dedic., 48 (1993), 15–25 | DOI | MR | Zbl

[65] Rani R., Edgar S. B., Barnes A., “Killing tensors and conformal Killing tensors from conformal Killing vectors”, Class. Quant. Grav., 20:11 (2003), 1929–1942 | DOI | MR | Zbl

[66] Reinhart B. L., Differential Geometry of Foliations, Springer-Verlag, Berlin–New York, 1983 | MR | Zbl

[67] Roter W., “On a generalization of conformally symmetric metric”, Tensor., 46 (1987), 278–286 | Zbl

[68] Sampson J. H., “On a theorem of Chern”, Trans. Am. Math. Soc., 177 (1973), 141–153 | DOI | MR | Zbl

[69] Shandra I. G., Stepanov S. E., Mikeš J., “On higher-order Codazzi tensors on complete Riemannian manifolds”, Ann. Glob. Anal. Geom., 56:3 (2019), 429–442 | DOI | MR | Zbl

[70] Stein E., Weiss G., “Generalization of the Cauchy–Riemann equations and representations of the rotation group”, Am. J. Math., 90 (1968), 163-–196 | DOI | MR | Zbl

[71] Stepanov S. E., “On conformal Killing 2-form of the electromagnetic field”, J. Geom. Phys., 33:3 (2000), 191–209 | DOI | MR | Zbl

[72] Stepanov S. E., Mikeš J., “Seven invariant classes of the Einstein equations and projective mappings”, AIP Conf. Proc., 1460:1 (2012), 221–225 | DOI

[73] Stepanov S. E., Mikeš J., “The spectral theory of the Yano rough Laplacian with some of its applications”, Ann. Glob. Anal. Geom., 48 (2015), 37–46 | DOI | MR | Zbl

[74] Stepanov S., Mikeš J., Jukl M., “The pre-Maxwell equations”, Geometric Methods in Physics, Springer-Verlag, 2013, 377–381 | MR

[75] Stepanov S. E., Tsyganok I. I., “A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions”, Adv. Geom., 19:3 (2019), 291–296 | DOI | MR | Zbl

[76] Stepanov S. E., Tsyganok I. I., Khripunova M. B., “The Killing tensors on an $n$-dimensional manifold with $SL(n,R)$-structure”, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 55:1 (2016), 121-–131 | MR | Zbl

[77] Stepanov S. E., Tsyganok I. I., Mikeš J., “On the Sampson Laplacian”, Filomat., 33:4 (2019), 342–358 | DOI | MR

[78] Stephani H., “A note on Killing tensors”, Gen. Rel. Grav., 9:9 (1978), 789–792 | DOI | MR | Zbl

[79] Sumitomo T., Tandai K., “Killing tensor fields of degree 2 and spectrum of $SO(n+1)/(SO(n-1)\times SO(2))$”, Osaka J. Math., 17 (1980), 649–675 | MR | Zbl

[80] Tafel J., “All transverse and TT tensors in flat spaces of any dimension”, Gen. Rel. Grav., 50:3 (2018), 31 | DOI | MR | Zbl

[81] Takeuchi M., “Killing tensor fields on spaces of constant curvature”, Tsukuba J. Math., 7:2 (1963), 233–256 | DOI | MR

[82] Visinescu M., “Higher-order symmetries in a gauge covariant approach and quantum anomalies”, Proc. 6th Summer School “Modern Mathematical Physics” (Belgrade, Serbia, September 14–23, 2010), Inst. Phys., Belgrade, 2011, 321–332

[83] Weir G. J., “Conformal Killing tensors in reducible spaces”, J. Math. Phys., 18 (1977), 1782–1787 | DOI | Zbl

[84] Wu H., “A remark on the Bochner technique in differential geometry”, Proc. Am. Math. Soc., 78:3 (1980), 403–407 | DOI | MR