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The Sampson Laplacian on Negatively Pinched Riemannian Manifolds

Year 2021, Volume: 14 Issue: 1, 91 - 99, 15.04.2021

Vladimir Rovenski , Sergey Stepanov Irina Tsyganok

https://doi.org/10.36890/iejg.780995

Abstract

We prove vanishing theorems for the kernel of the Sampson Laplacian, acting on symmetric tensors on a Riemannian manifold and estimate its first eigenvalue on negatively pinched Riemannian manifolds. Some applications of these results to conformal Killing tensors are presented.

Keywords

Riemannian manifold Sampson Laplacian spectral and vanishing theorems conformal Killing tensor

References

  • [1] Becce A.: Einstein manifolds, Springer-Verlag, Berlin, (1987).
  • [2] Benn M. and Charlton P.: Dirac symmetry operators from conformal Killing-Yano tensors. Class. Quantum Grav. 14, 1037-1042 (1997).
  • [3] Berger M. and Ebine D.: Some decomposition of the space of symmetric tensors of a Riemannian manifold. Journal of Differential Geometry. 3,379–392 (1969).
  • [4] Bochner S. and Yano K.: Curvature and Betti numbers, Princeton Univ. Press, Princeton, (1953).
  • [5] Bouguignon J.-P.: Formules deWeitzenböck en dimension 4, Geometrie riemannienne en dimension 4. Semin. Arthur Besse, Paris 1978/79, Textes Math., Cedic, Paris. 3, 308–333( 1981).
  • [6] Burns K. and Katok A.: Manifolds with non-positive curvature. Ergodic Theory of Dynamical Systems. 5:2, 307–317 (1985).
  • [7] Chow B., Lu P. and Ni L.: Hamilton’s Ricci flow, Providence, AMS, (2006).
  • [8] Craioveanu M., Puta M. and Rassias T.M.: Old and new aspects in spectral geometry, Kluwer Academic Publishers, London, (2001).
  • [9] Dairbekov N.S. and Sharafutdinov V.A.: Conformal Killing symmetric tensors on Riemannian manifolds. Mat. Tr. 13:1, 85–145 (2010).
  • [10] Eastwood M.: Higher symmetries of the Laplacian. Annals of Mathematics. 161, 1645–1665 (2005).
  • [11] Gibbons G.W. and Perry M.J.:Quantizing gravitational instantons. Nuclear Physics B. 146,I,90–108 (1978).
  • [12] Gilkey P.R.: Invariant theory, the heat equation, and the Atiyah-Singer index theorem. CRC Press, Washington, (1995).
  • [13] Goldberg S.I.: Curvature and homology. Dover Publications, New-York, (1998).
  • [14] Gromov M. and Thurston W.: Pinching constants for hyperbolic manifolds. Invent. Math.89, 1–12 (1987).
  • [15] Hamenstädt U.: Compact manifolds with 1=4-pinched negative curvature. Lectures Notes in Math., 1481. Global Differential Geometry and Global Analysis, Springer-Verlag, Berlin-Heidelberg, 73–78 (1991).
  • [16] Heil K., Moroianu A. and Semmelmann U.: Killing and conformal Killing tensors. J. Geom. Phys., 106,383–400 (2016).
  • [17] Hitchin, N.: A note on vanishing theorems, In: Geometry and Analysis on Manifolds. Progr. Math. 308, 373–382 (2015).
  • [18] Kashiwada T.: On conformal Killing tensor. Natural. Sci. Rep. Ochanomizu Univ. 19:2, 67–74 (1968).
  • [19] Kashiwada T.: On the curvature operator of the second kind. Natural Science Report, Ochanomizu University. 44:2, 69-73 (1993).
  • [20] Lichnerowicz A.: Propagateurs et commutateurs en relativité générate. Publ. Mathématiques de l’IHÉS.10:1, 293–344(1961).
  • [21] Michel R.: Problème d’analyse géomètrique lié à la conjecture de Blaschke. Bull. Soc. Math. France, 101,17–69 (1973).
  • [22] Mikeš J. and Stepanov S.E.: Betti and Tachibana numbers of compact Riemannian manifolds. Differential Geometry and its Applications. 31:4, 486–495 (2013).
  • [23] Mikeš J., Sandra I.G. and Stepanov S.E.:On higher order Codazzi tensors on complete Riemannian manifolds. Annals of Global Analysis and Geometry. 56, 429–442 (2019).
  • [24] Mikeš J., Rovenski V. and Stepanov S.E.: An example of Lichnerowicz-type Laplacian. Annals of Global Analysis and Geometry.58:1, 19–34 (2020).
  • [25] Petersen P.: Riemannian Geometry. Springer Science, New-York, (2016).
  • [26] Pilch K. and Schellekens N.: Formulas of the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces. J. Math. Phys.25:12, 3455–3459 (1984).
  • [27] Rovenski V., Stepanov S.E. and Tsyganok I.I.: On the Betti and Tachibana numbers of compact Einstein manifolds. Mathematics. 7:12,1210 (6 pp.) (2019).
  • [28] Sampson, J.H.: On a theorem of Chern. Trans. AMS. 177, 141–153 (1973).
  • [29] Stepanov S.E.: Curvature and Tachibana numbers. Sb. Math.202:7, 1059–1069 (2011).
  • [30] Stepanov S.E. and Mikeš J.: On the Sampson Laplacian. Filomat. 33:4, 1059-1070 (2019).
  • [31] Stepanov S.E. and Mikeš J.: The spectral theory of the Yano rough Laplacian with some of its applications. Ann. Global Anal. Geom.48:137–46 (2015).
  • [32] Stepanov S.E. and Shandra I.G.: Geometry of infinitesimal harmonic transformations. Ann. Global Anal. Geom. 24:3, 291–299 (2003).
  • [33] Stepanov S.E., Tsyganok I.I. and Mikesh J.: On a Laplacian which acts on symmetric tensors. arXiv: 1406.2829 [math.DG].1, 14pp. (2014).
  • [34] Stepanov S.E.: Fields of symmetric tensors on a compact Riemannian manifold. Mathematical Notes.52:4, 1048–1050 (1992).
  • [35] Stepanov S.E. and Tsyganok I.I.:Theorems of existence and of vanishing of conformally killing forms. Russian Mathematics. 58:10, 46–51 (2014).
  • [36] Stepanov S.E. and Rodionov V.V.: Addition to a work of J.-P. Bourguignon, Differ. Geom. Mnogoobr. Figur, 28 ,68–72 (1997).
  • [37] Stepanov S.E.:On conformal Killing 2-form of the electromagnetic field. Journal of Geometry and Physics.33, no. 3-4, 191–209 (2000).
  • [38] Stepanov S.E. and Tsyganok I.I.: Conformal Killing L2-forms on complete Riemannian manifolds with nonpositive curvature operator. J. of Math. Analysis and Applications. 458:1, 1–8 (2018).
  • [39] Stephani H., Kramer D., Mac Callum M., Hoenselaers C. and Herlt E.: Exact solutions of Einstein’s field equations. Cambridge University Press, (2003).
  • [40] Tachibana S.: On conformal Killing tensor in a Riemannian space. Tohoku Math. Journal. 21, 56–64 (1969).
  • [41] Tachibana S. and Ogiue K.: Les variétés riemanniennes dont l’opérateur de coubure restreint est positif sont des sphères d’homologie réelle. C. R. Acad. Sc. Paris.289, 29–30 (1979).
  • [42] Tandai K. and Sumitomo T.: Killing tensor fields of degree 2 and spectrum of SO(n + 1)=SO(n - 1) x SO(2). Osaka J. Math. 17, 649–675 (1980).
  • [43] Tsagas G.: A relation between Killing tensor fields and negative pinched Riemannian manifolds. Proceedings of the AMS, 22:2, 476–478 (1969).
  • [44] Vasy A. and Wunsch J.: Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature. Mathematical Research Letters.12:5, 673–684 (2005).
  • [45] Warner N.P.: The spectra of operators on CPn. Proc. R. Soc. Lond. A. 383, 217–230 (1982).

Year 2021, Volume: 14 Issue: 1, 91 - 99, 15.04.2021

Vladimir Rovenski , Sergey Stepanov Irina Tsyganok

https://doi.org/10.36890/iejg.780995

Abstract

References

  • [1] Becce A.: Einstein manifolds, Springer-Verlag, Berlin, (1987).
  • [2] Benn M. and Charlton P.: Dirac symmetry operators from conformal Killing-Yano tensors. Class. Quantum Grav. 14, 1037-1042 (1997).
  • [3] Berger M. and Ebine D.: Some decomposition of the space of symmetric tensors of a Riemannian manifold. Journal of Differential Geometry. 3,379–392 (1969).
  • [4] Bochner S. and Yano K.: Curvature and Betti numbers, Princeton Univ. Press, Princeton, (1953).
  • [5] Bouguignon J.-P.: Formules deWeitzenböck en dimension 4, Geometrie riemannienne en dimension 4. Semin. Arthur Besse, Paris 1978/79, Textes Math., Cedic, Paris. 3, 308–333( 1981).
  • [6] Burns K. and Katok A.: Manifolds with non-positive curvature. Ergodic Theory of Dynamical Systems. 5:2, 307–317 (1985).
  • [7] Chow B., Lu P. and Ni L.: Hamilton’s Ricci flow, Providence, AMS, (2006).
  • [8] Craioveanu M., Puta M. and Rassias T.M.: Old and new aspects in spectral geometry, Kluwer Academic Publishers, London, (2001).
  • [9] Dairbekov N.S. and Sharafutdinov V.A.: Conformal Killing symmetric tensors on Riemannian manifolds. Mat. Tr. 13:1, 85–145 (2010).
  • [10] Eastwood M.: Higher symmetries of the Laplacian. Annals of Mathematics. 161, 1645–1665 (2005).
  • [11] Gibbons G.W. and Perry M.J.:Quantizing gravitational instantons. Nuclear Physics B. 146,I,90–108 (1978).
  • [12] Gilkey P.R.: Invariant theory, the heat equation, and the Atiyah-Singer index theorem. CRC Press, Washington, (1995).
  • [13] Goldberg S.I.: Curvature and homology. Dover Publications, New-York, (1998).
  • [14] Gromov M. and Thurston W.: Pinching constants for hyperbolic manifolds. Invent. Math.89, 1–12 (1987).
  • [15] Hamenstädt U.: Compact manifolds with 1=4-pinched negative curvature. Lectures Notes in Math., 1481. Global Differential Geometry and Global Analysis, Springer-Verlag, Berlin-Heidelberg, 73–78 (1991).
  • [16] Heil K., Moroianu A. and Semmelmann U.: Killing and conformal Killing tensors. J. Geom. Phys., 106,383–400 (2016).
  • [17] Hitchin, N.: A note on vanishing theorems, In: Geometry and Analysis on Manifolds. Progr. Math. 308, 373–382 (2015).
  • [18] Kashiwada T.: On conformal Killing tensor. Natural. Sci. Rep. Ochanomizu Univ. 19:2, 67–74 (1968).
  • [19] Kashiwada T.: On the curvature operator of the second kind. Natural Science Report, Ochanomizu University. 44:2, 69-73 (1993).
  • [20] Lichnerowicz A.: Propagateurs et commutateurs en relativité générate. Publ. Mathématiques de l’IHÉS.10:1, 293–344(1961).
  • [21] Michel R.: Problème d’analyse géomètrique lié à la conjecture de Blaschke. Bull. Soc. Math. France, 101,17–69 (1973).
  • [22] Mikeš J. and Stepanov S.E.: Betti and Tachibana numbers of compact Riemannian manifolds. Differential Geometry and its Applications. 31:4, 486–495 (2013).
  • [23] Mikeš J., Sandra I.G. and Stepanov S.E.:On higher order Codazzi tensors on complete Riemannian manifolds. Annals of Global Analysis and Geometry. 56, 429–442 (2019).
  • [24] Mikeš J., Rovenski V. and Stepanov S.E.: An example of Lichnerowicz-type Laplacian. Annals of Global Analysis and Geometry.58:1, 19–34 (2020).
  • [25] Petersen P.: Riemannian Geometry. Springer Science, New-York, (2016).
  • [26] Pilch K. and Schellekens N.: Formulas of the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces. J. Math. Phys.25:12, 3455–3459 (1984).
  • [27] Rovenski V., Stepanov S.E. and Tsyganok I.I.: On the Betti and Tachibana numbers of compact Einstein manifolds. Mathematics. 7:12,1210 (6 pp.) (2019).
  • [28] Sampson, J.H.: On a theorem of Chern. Trans. AMS. 177, 141–153 (1973).
  • [29] Stepanov S.E.: Curvature and Tachibana numbers. Sb. Math.202:7, 1059–1069 (2011).
  • [30] Stepanov S.E. and Mikeš J.: On the Sampson Laplacian. Filomat. 33:4, 1059-1070 (2019).
  • [31] Stepanov S.E. and Mikeš J.: The spectral theory of the Yano rough Laplacian with some of its applications. Ann. Global Anal. Geom.48:137–46 (2015).
  • [32] Stepanov S.E. and Shandra I.G.: Geometry of infinitesimal harmonic transformations. Ann. Global Anal. Geom. 24:3, 291–299 (2003).
  • [33] Stepanov S.E., Tsyganok I.I. and Mikesh J.: On a Laplacian which acts on symmetric tensors. arXiv: 1406.2829 [math.DG].1, 14pp. (2014).
  • [34] Stepanov S.E.: Fields of symmetric tensors on a compact Riemannian manifold. Mathematical Notes.52:4, 1048–1050 (1992).
  • [35] Stepanov S.E. and Tsyganok I.I.:Theorems of existence and of vanishing of conformally killing forms. Russian Mathematics. 58:10, 46–51 (2014).
  • [36] Stepanov S.E. and Rodionov V.V.: Addition to a work of J.-P. Bourguignon, Differ. Geom. Mnogoobr. Figur, 28 ,68–72 (1997).
  • [37] Stepanov S.E.:On conformal Killing 2-form of the electromagnetic field. Journal of Geometry and Physics.33, no. 3-4, 191–209 (2000).
  • [38] Stepanov S.E. and Tsyganok I.I.: Conformal Killing L2-forms on complete Riemannian manifolds with nonpositive curvature operator. J. of Math. Analysis and Applications. 458:1, 1–8 (2018).
  • [39] Stephani H., Kramer D., Mac Callum M., Hoenselaers C. and Herlt E.: Exact solutions of Einstein’s field equations. Cambridge University Press, (2003).
  • [40] Tachibana S.: On conformal Killing tensor in a Riemannian space. Tohoku Math. Journal. 21, 56–64 (1969).
  • [41] Tachibana S. and Ogiue K.: Les variétés riemanniennes dont l’opérateur de coubure restreint est positif sont des sphères d’homologie réelle. C. R. Acad. Sc. Paris.289, 29–30 (1979).
  • [42] Tandai K. and Sumitomo T.: Killing tensor fields of degree 2 and spectrum of SO(n + 1)=SO(n - 1) x SO(2). Osaka J. Math. 17, 649–675 (1980).
  • [43] Tsagas G.: A relation between Killing tensor fields and negative pinched Riemannian manifolds. Proceedings of the AMS, 22:2, 476–478 (1969).
  • [44] Vasy A. and Wunsch J.: Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature. Mathematical Research Letters.12:5, 673–684 (2005).
  • [45] Warner N.P.: The spectra of operators on CPn. Proc. R. Soc. Lond. A. 383, 217–230 (1982).
There are 45 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Vladimir Rovenski 0000-0003-0591-8307

Sergey Stepanov This is me 0000-0003-1734-8874

Irina Tsyganok This is me 0000-0001-9186-3992

Publication Date April 15, 2021
Acceptance Date November 23, 2020
Published in Issue Year 2021 Volume: 14 Issue: 1

Cite

APA Rovenski, V., Stepanov, S., & Tsyganok, I. (2021). The Sampson Laplacian on Negatively Pinched Riemannian Manifolds. International Electronic Journal of Geometry, 14(1), 91-99. https://doi.org/10.36890/iejg.780995
AMA Rovenski V, Stepanov S, Tsyganok I. The Sampson Laplacian on Negatively Pinched Riemannian Manifolds. Int. Electron. J. Geom. April 2021;14(1):91-99. doi:10.36890/iejg.780995
Chicago Rovenski, Vladimir, Sergey Stepanov, and Irina Tsyganok. “The Sampson Laplacian on Negatively Pinched Riemannian Manifolds”. International Electronic Journal of Geometry 14, no. 1 (April 2021): 91-99. https://doi.org/10.36890/iejg.780995.
EndNote Rovenski V, Stepanov S, Tsyganok I (April 1, 2021) The Sampson Laplacian on Negatively Pinched Riemannian Manifolds. International Electronic Journal of Geometry 14 1 91–99.
IEEE V. Rovenski, S. Stepanov, and I. Tsyganok, “The Sampson Laplacian on Negatively Pinched Riemannian Manifolds”, Int. Electron. J. Geom., vol. 14, no. 1, pp. 91–99, 2021, doi: 10.36890/iejg.780995.
ISNAD Rovenski, Vladimir et al. “The Sampson Laplacian on Negatively Pinched Riemannian Manifolds”. International Electronic Journal of Geometry 14/1 (April 2021), 91-99. https://doi.org/10.36890/iejg.780995.
JAMA Rovenski V, Stepanov S, Tsyganok I. The Sampson Laplacian on Negatively Pinched Riemannian Manifolds. Int. Electron. J. Geom. 2021;14:91–99.
MLA Rovenski, Vladimir et al. “The Sampson Laplacian on Negatively Pinched Riemannian Manifolds”. International Electronic Journal of Geometry, vol. 14, no. 1, 2021, pp. 91-99, doi:10.36890/iejg.780995.
Vancouver Rovenski V, Stepanov S, Tsyganok I. The Sampson Laplacian on Negatively Pinched Riemannian Manifolds. Int. Electron. J. Geom. 2021;14(1):91-9.
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