Skip to main content
SpringerLink
Log in

Harmonic and conformally Killing forms on complete Riemannian manifold

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We present a classification of complete locally irreducible Riemannian manifolds with nonnegative curvature operator, which admit a nonzero and nondecomposable harmonic form with its square-integrable norm. We prove a vanishing theorem for harmonic forms on complete generic Riemannian manifolds with nonnegative curvature operator. We obtain similar results for closed and co-closed conformal Killing forms.

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. Hodge, W. V.D. The Theory and Applications of Harmonic Integrals (Cambridge Univ. Press, Cambridge, 1989).

    MATH  Google Scholar 

  2. de Rham, G. Variétés Différentiables (Hermann & Cie, Paris, 1955; In. Lit.,Moscow, 1956).

    MATH  Google Scholar 

  3. Yau, S.-T. Some Function-Theoretic Properties of Complete Riemannian Manifold and Their Applications to Geometry, Indiana Univ.Math. J. 25, No. 7, 659–670 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  4. Petersen, P. Riemannian Geometry (Springer, New York, 2006).

    MATH  Google Scholar 

  5. Tachibana, S. On Conformal Killing Tensor in a Riemannian Space, Tohoku Math. J. 21, No. 1, 56–64 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  6. Kashiwada, T. On Conformal Killing Tensor, Natural. Sci. Rep. Ochanomizu Univ. 19, No. 2, 67–74 (1968).

    MathSciNet  MATH  Google Scholar 

  7. Kobayashi S., Nomizu K. Foundations of Differential Geometry. I (Interscience Publishers, 1969; Nauka, Moscow, 1981).

    MATH  Google Scholar 

  8. Tachibana, S., Yamaguchi, S. The First Proper Space of for p-Forms in Compact Riemannian Manifolds of Positive Curvature Operator, J. Diff. Geom. 15, No. 1, 51–60 (1980).

    MathSciNet  MATH  Google Scholar 

  9. Moroianu, A., Semmelmann, U. Twistor Forms on Kähler Manifolds, Ann. Sc. Norm. Super Pisa Cl. Sci. (5) 2, No. 4, 823–845 (2003).

    MathSciNet  MATH  Google Scholar 

  10. Semmelmann, U. Conformal Killing Forms on Riemannian Manifolds, Math. Z. 245, No. 3, 503–527 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  11. Belgun, F., Moroianu, A., Semmelmann, U. Killing Forms on Symmetric Spaces, Diff. Geom. Appl. 24, No. 3, 215–222 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  12. Stepanov, S. E. On Conformal Killing 2-Formof the Electromagnetic Field, J. Geom.Phys. 33, No. 3–4, 191–209 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  13. Stepanov, S. E. A Class of Closed Forms and Special Maxwell’s Equations, Tensor (N. S.) 58, No. 3, 233–242 (1997).

    MathSciNet  MATH  Google Scholar 

  14. Stepanov, S. E., Mikes, J. Betti and Tachibana Numbers, MiskolcMath. Notes 14, No. 3, 265–276 (2013).

    MathSciNet  MATH  Google Scholar 

  15. Stepanov, S. E., Mikes, J. Betti and Tachibana Numbers of Compact RiemannianManifolds, Diff.Geom. Appl. 31, No. 4, 486–495 (2013).

    Article  MATH  Google Scholar 

  16. Stepanov, S. E. Curvature and Tachibana Numbers, Sb.Math. 202, No. 7, 1059–1069 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  17. Bourguignon, J. P. Formules de Weitzenbok en Dimension 4, Séminare A. Besse sur la géométrie Riemannienne dimension 4 (Cedic. Ferman, Paris, 1981), pp. 308–331.

    Google Scholar 

  18. Besse, A. L. Einstein Manifolds (Springer, 1987;Mir,Moscow, 1991).

    MATH  Google Scholar 

  19. Bérard, P. H. From Vanishing Theorems to Estimating Theorems: The Bochner Technique Revisited, Bull. Amer.Math. Soc. (N. S.) 19, No. 2, 371–406 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  20. Kora, M. On Conformal Killing Forms and the Proper Space D for p-Forms, Math. J. Okayama Univ. 22, No. 2, 195–204 (1980).

    MathSciNet  MATH  Google Scholar 

  21. Stepanov, S. E. A New Strong Laplacian on Differential Forms, Math.Notes 76, No. 3, 420–425 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  22. Stepanov, S. E., Tsyganok, I. I. Comparative Analysis of the Spectral Properties of the Hodge–de Rham and Tachibana Operators, The Results on Science and Tekhn. VINITI. Ser. Contemp. Math. and its Appslication. Thematic reviews 127, 151–182 (2014) [in Russian].

    MATH  Google Scholar 

  23. Stepanov, S. E., Mikes, J. The Hodge–de Rham Laplacian and Tachibana Operator on a Compact Riemannian Manifold with Curvature Operator of Definite Sign, Izv. Math. 79, No. 2, 375–387 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  24. Scott, C. L p Theory of Differential Forms on Manifolds, Trans. Amer.Math. Soc. 247, No. 6, 2075–2096 (1995).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Financial University at the Government of the Russian Federation, Leningradskii pr. 49–55, Moscow, 125993, Russia

    S. E. Stepanov & I. I. Tsyganok

  2. Russian State Social University, ul. Vil’gelma Pika 4, Bld. 1, Moscow, 129226, Russia

    T. V. Dmitrieva

Authors
  1. S. E. Stepanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. I. Tsyganok
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. V. Dmitrieva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. E. Stepanov.

Additional information

Original Russian Text © S.E. Stepanov, I.I. Tsyganok, T.V. Dmitrieva, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 3, pp. 51–57.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stepanov, S.E., Tsyganok, I.I. & Dmitrieva, T.V. Harmonic and conformally Killing forms on complete Riemannian manifold. Russ Math. 61, 44–48 (2017). https://doi.org/10.3103/S1066369X17030057

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X17030057

Keywords

Search

Navigation