Skip to main content
SpringerLink
Log in

An example of Lichnerowicz-type Laplacian

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We consider the Sampson Laplacian acting on covariant symmetric tensors on a Riemannian manifold. This operator is an example of the Lichnerowicz-type Laplacian. It is of fundamental importance in mathematical physics and appears in many problems in Riemannian geometry including the theories of infinitesimal Einstein deformations, the stability of Einstein manifolds and the Ricci flow. We study the Sampson Laplacian using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting Weitzenböck decomposition and further of estimating its lowest eigenvalue. In addition, we also survey a series of results that we obtained earlier.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agricola, L., Friedrich, T.: Global Analysis. Differential Forms in Analysis, Geometry and Physics, AMS (2002)

  2. Arnold, V.I., Kresin, B.A.: Topological Methods in Hydrodynamics. Springer, Berlin (1998)

    Book  Google Scholar 

  3. Bérard, P.H.: From vanishing theorems to estimating theorems: the Bochner technique revisited. Bull. AMS 19(2), 371–406 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on Riemannian manifolds. J. Diff. Geom. 3, 379–392 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  5. Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  6. Géometrie riemannienne en dimension 4: Séminaire Arthur Besse 1978/1979, Cedic/Fernand Nathan, Paris (1981)

  7. Bochner, S., Yano, K.: Curvature and Betti Numbers. Princeton University Press, Princeton (1953)

    MATH  Google Scholar 

  8. Boucetta, M.: Spectre des Laplaciens de Lichnerowicz sur les sphères et les projectifs réels. Publ. Mat. 43, 451–483 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bouguignon, J.-P., Karcher, H.: Curvature operators: pinching estimates and geometric examples. Ann. Sc. Éc. Norm. Sup. 11, 71–92 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Calabi, E.: An extension of E. Hopf’s maximum principle with an application to Riemannian geometry. Duke Math. J., 25, 45–56 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chow B., Lu P., Ni L.: Hamilton’s Ricci flow, GSM 77, AMS (2006)

  12. Collinson, C.D., Howarth, L.: Generalized Killing tensors. Gen. Relativity Gravitation 32(9), 1767–1776 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Craioveanu, M., Puta, M., Rassias, T.M.: Old and New Aspects in Spectral Geometry. Kluwer Acad Publ, Cambridge (2001)

    Book  MATH  Google Scholar 

  14. Dairbekov, N.S., Sharafutdinov, V.A.: Conformal Killing symmetric tensors on Riemannian manifolds. Mat. Tr. 13(1), 85–145 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Dolan, P., Kladouchou, A., Card, C.: On the significance of Killing tensors. Gen. Relativity Gravitation 21(4), 427–437 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eastwood, M.: Higher symmetries of the Laplacian. Ann. Math. 161(3), 1645–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86(1), 109–160 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gibbons, G.W., Perry, M.J.: Quantizing gravitational instantons. Nuclear Phys. B 146(1), 90–108 (1978)

    Article  MathSciNet  Google Scholar 

  19. Gilkey, P.R.: Invariant Theory, the Heat Equation, and the Atiyah–Singer Index Theorem. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  20. Heil, K., Moroianu, A., Semmelmann, U.: Killing and conformal Killing tensors. J. Geom. Phys. 106, 383–400 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hitchin, N.: A note on vanishing theorems, In: Geometry and Analysis on Manifolds, Progr. Math., 308, 373–382 (2015)

  22. Kashiwada, T.: On the curvature operator of the second kind, Natural Science Report. Ochanomizu Univ. 44(2), 69–73 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Kobayashi, Sh: Transformation Groups in Differential Geometry. Springer, Berlin (1995)

    MATH  Google Scholar 

  24. Kobayashi, Sh., Nomizu, K.: Foundations of Differential Geometry, vol. II. Int. Publishers, New York-London-Sydney (1969)

  25. Li, P.: Geometric Analysis. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  26. Li, P., Schoen, R.: \(L^{p}\) and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153(1), 279–301 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lichnerowicz, A.: Propagateurs et commutateurs en relativité générate. Publ. Math. de l’IHÉS 10(1), 293–344 (1961)

    MATH  Google Scholar 

  28. Lichnerowicz, A.: Spineurs harmoniques. C. R. Acad. Sci. Paris Ser. A-B 257, 7–9 (1963)

    MathSciNet  MATH  Google Scholar 

  29. Lichnerowicz, A.: Opérateurs différentials invariants sure space homogène, Ann. Sci. École Norm. Sup., Ser. 3, 81 : 4, 341–385 (1964)

  30. Michel, J.-P.: Higher symmetries of the Laplacian via quantization. Ann. Inst. Fourier 64(4), 1581–1609 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nishikawa, S.: On deformation of Riemannian metrics and manifolds with positive curvature operator. LNM 1201, 202–211 (1986)

    MathSciNet  MATH  Google Scholar 

  32. Palais, R.: Seminar on the Atiyah-Singer Index Theorem. Princeton University Press, Princeton (1965)

    Book  MATH  Google Scholar 

  33. Petersen, P.: Riemannian Geometry. Springer, Berlin (2016)

    Book  MATH  Google Scholar 

  34. Pigola, S., Rigoli, M., Setti, A.G.: Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Bochner Technique. Birkhauser, Basel (2008)

    MATH  Google Scholar 

  35. Pilch, K., Schellekens, N.: Formulas of the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces. J. Math. Phys. 25(12), 3455–3459 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rovenski V., Stepanov S.E., Tsyganok I., Geometry in the large of the kernel of Lichnerowicz Laplacians and its applications, preprint, arXiv:1903.10230, 19 pp. (2019)

  37. Sampson, J.H.: On a theorem of Chern. Trans. Amer. Math. Soc. 177, 141–153 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  38. Stepanov, S.E.: Fields of symmetric tensors on a compact Riemannian manifold. Math. Notes 52(4), 1048–1050 (1992)

    Article  MathSciNet  Google Scholar 

  39. Stepanov S.E., Tsyganok I.I.: Theorems of existence and non-existence of conformal Killing forms, Russian Math. (Iz. VUZ), 58 : 10, 46–51 (2014)

  40. Stepanov, S.E., Tsyganok, I.I., Mikeš, J.: On the Sampson Laplacian. Filomat 33(4), 1059–1070 (2019)

    Article  MathSciNet  Google Scholar 

  41. Stepanov, S.E., Mikeš, J.: The spectral theory of the Yano rough Laplacian with some of its applications. Ann. Glob. Anal. Geom. 48, 37–46 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Stepanov, S.E.: Vanishing theorems in affine, Riemannian and Lorentz geometries. J. Math. Sci. (New York) 141(1), 929–964 (2007)

    Article  MathSciNet  Google Scholar 

  43. Stepanov S.E., Tsyganok I.I., Aleksandrova I.A., A remark on the Laplacian operator which acts on symmetric tensors, arXiv: 1411.1928 [math.DG], 4, pp. 8 (2014)

  44. Stepanov S.E., Tsyganok I.I., Mikesh J., On a Laplacian which acts on symmetric tensors, arXiv: 1406.2829 [math.DG], 14 pp. (2014)

  45. Stepanov, S.E., Shandra, I.G.: Geometry of infinitesimal harmonic transformations. Ann. Glob. Anal. Geom. 24, 291–299 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Stepanov, S.E., Tsyganok, I.I.: Conformal Killing \(L^2\)-forms on complete Riemannian manifolds with nonpositive curvature operator. J. Math. Anal. Appl. 458(1), 1–8 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  47. Stepanov, S.E., Tsyganok, I.I.: The theory of infinitesimal harmonic transformations and its applications to the global geometry of Riemann solitons. Balkan J. Geom. Appl. 24(1), 113–121 (2019)

    MathSciNet  Google Scholar 

  48. Sumitomo, T.: On the commutator of differential operators. Hokkaido Math. J. 1, 30–42 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  49. Sumitomo, T., Tandai, K.: Killing tensor fields on the standard sphere and spectra of \(SO(n+1)/\) \((SO(n-1)\times SO(n))\) and \(O(n+1)/(O(n-1) \times O(n))\). Osaka J. Math. 20(1), 51–78 (1983)

    MathSciNet  MATH  Google Scholar 

  50. Warner, N.P.: The spectra of operators on \(CP^{n}\). Proc. R. Soc. Lond. A 383, 217–230 (1982)

    Article  MATH  Google Scholar 

  51. Wu, H.: A remark on the Bochner technique in differential geometry. Proc. AMS 78(3), 403–408 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  52. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970)

    MATH  Google Scholar 

  53. Yau, S.-T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25(7), 659–670 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of the first author was supported by the Internal Grant Agency of the Faculty of Science of Palacky University, Olomouc (grant No. 2020014 “Mathematical Structures”). The authors would like to express their thanks to Professor Ivor Hall (BA Hons) for her editorial suggestions that contributed to the improvement in this paper as well as to the referee for his careful reading of the previous version of the manuscript and suggestions about this paper which led to various improvements.

Author information

Authors and Affiliations

  1. Department of Algebra and Geometry, Palacky University, 12, 17. listopadu, 77146, Olomouc, Czech Republic

    Josef Mikeš

  2. Department of Mathematics, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Vladimir Rovenski

  3. Dept. of Data Analysis and Financial Technology, Finance University, 49-55, Leningradsky Prospect, Moscow, Russia, 125468

    Sergey E. Stepanov

Authors
  1. Josef Mikeš
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Rovenski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sergey E. Stepanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Josef Mikeš.

Additional information

Dedicated to the memory of the professor Thomas Friedrich.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mikeš, J., Rovenski, V. & Stepanov, S.E. An example of Lichnerowicz-type Laplacian. Ann Glob Anal Geom 58, 19–34 (2020). https://doi.org/10.1007/s10455-020-09714-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-020-09714-9

Keywords

Mathematics Subject Classification

Search

Navigation