Abstract
This paper is devoted to the geometry of symmetric Killing tensors and Codazzi tensors of ranks p ≥ 2 and includes, in addition to the new results obtained in this paper, a survey on this topic from earlier works.
Similar content being viewed by others
References
I. A. Aleksandrova, S. E. Stepanov, and I. I. Tsyganok, “A remark on a Laplacian operator which acts on symmetric tensors,” e-print ArXiV 1411.1928 [math.DG].
C. Barbance, “Sur les tenseurs symetriques,” C. R. Acad. Sci. Paris. Ser. A, 276, 387–389 (1973).
A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin–Heidelberg–New York (1987).
R. G. Bettiol and R. A. E. Mendes, “Sectional curvature and Weitzenböck formulae,” e-print ArXiV 1708.09033v1 [math.DG].
M. Boucetta, “Spectre des Laplacien de Lichnerowicz sur les sphères et les projectifs réels,” Publ. Mat., 43, No. 2, 451–483 (1999).
J. P. Bourguignon, “Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein,” Invent. Math., 63, 263–286 (1981).
Bourguignon J. P., “Formules de Weitzenbök en dimension 4,” in: Géométrie riemannienne en dimension 4. Semin. Arthur Besse, Paris 1978/79, Cedic / Fernand Nathan, Paris (1981), pp. 308–333.
D. M. J. Calderbank, P. Gauduchon, and M. Herzlich, “On the Kato inequality in Riemannian geometry,” in: Global Analysis and Harmonic Analysis, Soc. Math. Fr., Paris (2000), pp. 95–113.
G. Catino, C. Mantegazza, and L. Mazzieri, “A note on Codazzi tensors,” Math. Ann., 362, No. 1–2, 629–638 (2015).
B. Y. Chen, Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific, New Jersey (2017).
B. Coll, J. J. Ferrando, and J. A. Saez, “On the geometry of Killing and conformal tensors,” J. Math. Phys., 47, 062503 (2006).
N. S. Dairbekov and V. A. Sharafutdinov, “Conformal Killing symmetric tensor fields on Riemannian manifolds,” Mat. Tr., 13, No. 1, 85–145 (2010).
A. Derdzinski and C. L. Shen, “Codazzi tensor fields, curvature and Pontryagin forms,” Proc. London Math. Soc., 47, No. 3, 15-26 (1983).
L. P. Eisenhart, Riemannian Geometry, Princeton Univ. Press, Princeton (1949).
A. Gebarowski, “On nearly conformally symmetric warped product space-times,” in: Proc. Int. Conf. “Differential Geometry and Its Applications”, Bucharest (1992), pp. 61–75.
R. E. Green and H. Wu, “Integrals of subharmonic functions on manifolds of nonnegative curvature,” Invent. Mat., 27, 265–298 (1974).
K. Heil, A. Moroianu, and U. Semmelmann, “Killing tensors on tori,” J. Geom. Phys., 117, 1–6 (2017).
K. Heil, A. Moroianu, and U. Semmelmann, “Killing and conformal Killing tensors,” J. Geom. Phys., 106, 383–400 (2016).
D. L. Johnson and L. B. Whitt, “Totally geodesic foliations on 3-manifolds,” Proc. Am. Math. Soc., 76, No. 2, 355–357 (1979).
G. H. Katzin and J. Levine, “Note on the number of linearly independentmth order first integrals,” Tensor, 19, No. 1, 42–44 (1968).
Sh. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin–Heidelberg–New York (1972).
Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. 2, Interscience, New York–London–Sydney (1969).
D. Kramer, H. Stephani, M. Maccallum, E. Herl, Exact Solutions of the Einstein Field Solutions, Deutscher Verlag der Wissenschaften, Berlin (1980).
D. Kubiznak and V. P. Frolov, “The hidden symmetry of higher dimensional Kerr-NUT-AdS spacetimes,” Class. Quant. Grav., 24, No. 3, F1–F6 (2007).
G. F. Laptev and N. M. Ostianu, “Distribution of m-dimensional linear elements in spaces of projective connection, I,” Tr. Geom. Semin. VINITI, 5, 49–94 (1971).
J. Leder, A. Schwenk-Schellschmidt, U. Simon, and M. Wiehe, “Generating higher order Codazzi tensors by functions,” in: Geometry and Topology of Submanifolds. Vol. IX, World Scientific, London–Singapore (1999), pp. 174–191.
P. Li and R. Schoen, “Lp and mean value properties of subharmonic function on Riemannian manifolds,” Acta Math., 153, No. 1, 279–301 (1984).
A. Lichnerowicz, “Propagateurs et commutateurs en relativite generate,” Publ. Math. IHES, 10, No. 1, 5–56 (1961).
H. L. Liu, “Codazzi tensor and the topology of surfaces,” Ann. Glob. Anal. Geom., 16, 189–202 (1998).
H. L. Liu, U. Simon, and C. P. Wang, “Higher order Codazzi tensors on conformally flat spaces,” Beitr. Algebra Geom., 39, No. 2, 329–348 (1998).
C. A. Mantica, L. C. Molinari, Y. J. Suh, and S. Shenawy, “Perfect-fluid, generalized Robertson–Walker space-times, and Gray’s decomposition,” J. Math. Phys., 60, No. 5, 052506 (2019).
O. V. Manturov, Elements of Tensor Calculus [in Russian], Prosveshchenie, Moscow (1991).
G. Merton, “Codazzi tensors with two eigenvalue functions,” Proc. Am. Math. Soc., 141, No. 9, 3265–3273 (2013).
J. Mikeš et. al., Differential Geometry of Special Mappings, Palacky Univ. Press, Olomouc (2015).
A. G. Nikitin, “Generalized Killing tensors of arbitrary order,” Ukr. Mat. Zh., 43, No. 6, 786–795 (1991).
A. P. Norden, Spaces of Affine Connection [in Russian], Nauka, Moscow (1976).
P. Petersen, Riemannian Geometry, Springer-Verlag, Berlin (2016).
S. Pigola, M. Rigoli, and A. G. Setti, Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique, Birkhäuser, Basel (2008).
R. Ponge and H. Reckziegel, “Twisted products in pseudo-Riemannian geometry,” Geom. Dedic., 48, 15–25 (1993).
R. Rani, S. B. Edgar, and A. Barnes, “Killing tensors and conformal Killing tensors from conformal Killing vectors,” Class. Quant. Grav., 20, No. 11, 1929–1942 (2003).
B. L. Reinhart, Differential Geometry of Foliations, Springer-Verlag, Berlin–New York (1983).
W. Roter, “On a generalization of conformally symmetric metric,” Tensor, 46, 278–286 (1987).
J. H. Sampson, “On a theorem of Chern,” Trans. Am. Math. Soc., 177, 141–153 (1973).
I. G. Shandra, S. E. Stepanov, and J. Mikeˇs, “On higher-order Codazzi tensors on complete Riemannian manifolds,” Ann. Glob. Anal. Geom., 56, No. 3, 429–442 (2019).
V. A. Sharafutdinov, “Killing tensor fields on the 2-torus,” Sib. Mat. Zh., 57, No. 1, 199–221 (2016).
Yu. I. Shinkunas, “On distributions of m-dimensional planes in an n-dimensional Riemannian space,” Tr. Geom. Semin. VINITI, 5, 123–133 (1973).
P. A. Shirokov and A. P. Shirokov, Affine Differential Geometry [in Russian], Fizmatlit, Moscow (1959).
N. S. Sinyukov, Geodesic Mappings of Riemannian Spaces [in Russian], Nauka, Moscow (1979).
E. Stein and G. Weiss, “Generalization of the Cauchy–Riemann equations and representations of the rotation group,” Am. J. Math., 90, 163-196 (1968).
S. E. Stepanov, “On one class of Riemannian near product structurs,” Izv. Vyssh. Ucheb. Zaved. Mat., 7, 40–46 (1989).
S. E. Stepanov, “Geometric obstruction of the existence of a totally umbilical distribution on a compact manifold,” in: Webs and Quasigroups [in Russian], Kalinin (1990), pp. 135–137.
S. E. Stepanov, “Fields of symmetric tensors on a compact Riemannian manifold,” Mat. Zametki, 52, No. 4, 85-88 (1992).
S. E. Stepanov, “Integral formula for a compact manifold with a Riemannian near product structure,” Izv. Vyssh. Ucheb. Zaved. Mat., 7, 69-73 (1994).
S. E. Stepanov, “On an application of one theorem of P. A. Shirokov in Bochner’s technique,” Izv. Vyssh. Ucheb. Zaved. Mat., 9, 53-59 (1996).
S. E. Stepanov, “On conformal Killing 2-form of the electromagnetic field,” J. Geom. Phys., 33, No. 3, 191–209 (2000).
S. E. Stepanov and J. Mikeš, “Seven invariant classes of the Einstein equations and projective mappings,” AIP Conf. Proc., 1460, No. 1, 221–225 (2012).
S. E. Stepanov and J. Mikeš, “The spectral theory of the Yano rough Laplacian with some of its applications,” Ann. Glob. Anal. Geom., 48, 37–46 (2015).
S. E. Stepanov, J. Mikeš, and M. Jukl, “The pre-Maxwell equations,” in: Geometric Methods in Physics, Springer-Verlag, Berlin (2013), pp. 377–381.
S. E. Stepanov and V. V. Rodionov, “Addendum to one work of J.-P. Bourguignon,” Differ. Geom. Mnogoobr. Figur., No. 28, 68–72 (1997).
S. E. Stepanov and M. B. Smolnikova, “First-order fundamental differential operators on exterior and interior forms,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 11, 55–60 (2002).
S. E. Stepanov and M. B. Smolnikova, “Affine differential geometry of Killing tensors,” Izv. Vyssh. Ucheb. Zaved. Mat., 11, 82-86 (2004).
S. E. Stepanov and I. I. Tsyganok, “A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions,” Adv. Geom., 19, No. 3, 291–296 (2019).
S. E. Stepanov, I. I. Tsyganok, and M. B. Khripunova, “The Killing tensors on an n-dimensional manifold with SL(n,R)-structure,” Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 55, No. 1, 121-131 (2016).
S. E. Stepanov, I. I. Tsyganok, and J. Mikeš, “On the Sampson Laplacian,” Filomat, 33, No. 4, 342–358 (2019).
H. Stephani, “A note on Killing tensors,” Gen. Rel. Grav., 9, No. 9, 789–792 (1978).
T. Sumitomo and K. Tandai, “Killing tensor fields of degree 2 and spectrum of SO(n+1)/(SO(n−1) × SO(2)),” Osaka J. Math., 17, 649–675 (1980).
J. Tafel, “All transverse and TT tensors in flat spaces of any dimension,” Gen. Rel. Grav., 50, No. 3, 31 (2018).
M. Takeuchi, “Killing tensor fields on spaces of constant curvature,” Tsukuba J. Math., 7, No. 2, 233–256 (1963).
M. Visinescu, “Higher-order symmetries in a gauge covariant approach and quantum anomalies,” in: Proc. 6th Summer School “Modern Mathematical Physics” (Belgrade, Serbia, September 14–23, 2010), Inst. Phys., Belgrade (2011), pp. 321–332.
H. Weyl, The Classical Groups, Their Invariants and Representations, Princeton Univ. Press, Princeton, New Jersey (1997).
G. J. Weir, “Conformal Killing tensors in reducible spaces,” J. Math. Phys., 18 (1977), pp. 1782–1787.
H. Wu, “A remark on the Bochner technique in differential geometry,” Proc. Am. Math. Soc., 78, No. 3 (1980), pp. 403–407.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 179, Proceedings of the International Conference “Classical and Modern Geometry” Dedicated to the 100th Anniversary of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 1, 2020.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Stepanov, S.E., Aleksandrova, I.A. & Tsyganok, I.I. On Symmetric Killing Tensors and Codazzi Tensors of Ranks p ≥ 2. J Math Sci 276, 443–469 (2023). https://doi.org/10.1007/s10958-023-06763-w
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06763-w