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The Sampson Laplacian on Negatively Pinched Riemannian Manifolds

Yıl 2021, Cilt: 14 Sayı: 1, 91 - 99, 15.04.2021

Vladimir Rovenski Sergey Stepanov Irina Tsyganok

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

Öz

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.

Anahtar Kelimeler

Riemannian manifold Sampson Laplacian spectral and vanishing theorems conformal Killing tensor

Kaynakça

  • [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).

Yıl 2021, Cilt: 14 Sayı: 1, 91 - 99, 15.04.2021

Vladimir Rovenski Sergey Stepanov Irina Tsyganok

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

Öz

Kaynakça

  • [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).
Toplam 45 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Vladimir Rovenski 0000-0003-0591-8307

Sergey Stepanov Bu kişi benim 0000-0003-1734-8874

Irina Tsyganok Bu kişi benim 0000-0001-9186-3992

Yayımlanma Tarihi 15 Nisan 2021
Kabul Tarihi 23 Kasım 2020
Yayımlandığı Sayı Yıl 2021 Cilt: 14 Sayı: 1

Kaynak Göster

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. Nisan 2021;14(1):91-99. doi:10.36890/iejg.780995
Chicago Rovenski, Vladimir, Sergey Stepanov, ve Irina Tsyganok. “The Sampson Laplacian on Negatively Pinched Riemannian Manifolds”. International Electronic Journal of Geometry 14, sy. 1 (Nisan 2021): 91-99. https://doi.org/10.36890/iejg.780995.
EndNote Rovenski V, Stepanov S, Tsyganok I (01 Nisan 2021) The Sampson Laplacian on Negatively Pinched Riemannian Manifolds. International Electronic Journal of Geometry 14 1 91–99.
IEEE V. Rovenski, S. Stepanov, ve I. Tsyganok, “The Sampson Laplacian on Negatively Pinched Riemannian Manifolds”, Int. Electron. J. Geom., c. 14, sy. 1, ss. 91–99, 2021, doi: 10.36890/iejg.780995.
ISNAD Rovenski, Vladimir vd. “The Sampson Laplacian on Negatively Pinched Riemannian Manifolds”. International Electronic Journal of Geometry 14/1 (Nisan 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 vd. “The Sampson Laplacian on Negatively Pinched Riemannian Manifolds”. International Electronic Journal of Geometry, c. 14, sy. 1, 2021, ss. 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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