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Статья
2017

Harmonic and conformally Killing forms on complete Riemannian manifold


S. E. StepanovS. E. Stepanov, I. I. TsyganokI. I. Tsyganok, T. V. DmitrievaT. V. Dmitrieva
Русская математика
https://doi.org/10.3103/S1066369X17030057
Abstract / Full Text

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.

Author information
  • Financial University at the Government of the Russian Federation, Leningradskii pr. 49–55, Moscow, 125993, RussiaS. E. Stepanov & I. I. Tsyganok
  • Russian State Social University, ul. Vil’gelma Pika 4, Bld. 1, Moscow, 129226, RussiaT. V. Dmitrieva
References
  1. Hodge, W. V.D. The Theory and Applications of Harmonic Integrals (Cambridge Univ. Press, Cambridge, 1989).
  2. de Rham, G. Variétés Différentiables (Hermann & Cie, Paris, 1955; In. Lit.,Moscow, 1956).
  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).
  4. Petersen, P. Riemannian Geometry (Springer, New York, 2006).
  5. Tachibana, S. On Conformal Killing Tensor in a Riemannian Space, Tohoku Math. J. 21, No. 1, 56–64 (1969).
  6. Kashiwada, T. On Conformal Killing Tensor, Natural. Sci. Rep. Ochanomizu Univ. 19, No. 2, 67–74 (1968).
  7. Kobayashi S., Nomizu K. Foundations of Differential Geometry. I (Interscience Publishers, 1969; Nauka, Moscow, 1981).
  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).
  9. Moroianu, A., Semmelmann, U. Twistor Forms on Kähler Manifolds, Ann. Sc. Norm. Super Pisa Cl. Sci. (5) 2, No. 4, 823–845 (2003).
  10. Semmelmann, U. Conformal Killing Forms on Riemannian Manifolds, Math. Z. 245, No. 3, 503–527 (2003).
  11. Belgun, F., Moroianu, A., Semmelmann, U. Killing Forms on Symmetric Spaces, Diff. Geom. Appl. 24, No. 3, 215–222 (2006).
  12. Stepanov, S. E. On Conformal Killing 2-Formof the Electromagnetic Field, J. Geom.Phys. 33, No. 3–4, 191–209 (2000).
  13. Stepanov, S. E. A Class of Closed Forms and Special Maxwell’s Equations, Tensor (N. S.) 58, No. 3, 233–242 (1997).
  14. Stepanov, S. E., Mikes, J. Betti and Tachibana Numbers, MiskolcMath. Notes 14, No. 3, 265–276 (2013).
  15. Stepanov, S. E., Mikes, J. Betti and Tachibana Numbers of Compact RiemannianManifolds, Diff.Geom. Appl. 31, No. 4, 486–495 (2013).
  16. Stepanov, S. E. Curvature and Tachibana Numbers, Sb.Math. 202, No. 7, 1059–1069 (2011).
  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.
  18. Besse, A. L. Einstein Manifolds (Springer, 1987;Mir,Moscow, 1991).
  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).
  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).
  21. Stepanov, S. E. A New Strong Laplacian on Differential Forms, Math.Notes 76, No. 3, 420–425 (2004).
  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].
  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).
  24. Scott, C. L p Theory of Differential Forms on Manifolds, Trans. Amer.Math. Soc. 247, No. 6, 2075–2096 (1995).