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.
[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).
[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
[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).
[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).
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.