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St. Petersburg Mathematical Journal

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Harmonic diffeomorphisms of manifolds


Authors: S. E. Stepanov and I. G. Shandra
Translated by: N. Yu. Netsvetaev
Original publication: Algebra i Analiz, tom 16 (2004), nomer 2.
Journal: St. Petersburg Math. J. 16 (2005), 401-412
MSC (2000): Primary 53C43, 58E20
DOI: https://doi.org/10.1090/S1061-0022-05-00856-3
Published electronically: March 9, 2005
MathSciNet review: 2068345
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Abstract | References | Similar Articles | Additional Information

Abstract: In spite of the abundance of publications on harmonic mappings of manifolds, at present there exists neither a theory of harmonic diffeomorphisms, nor a definition of infinitesimal harmonic transformation of a Riemannian manifold, to say nothing of the theory of groups of such transformations. In the paper, this gap is partially filled, and a new subject of investigations is announced.


References [Enhancements On Off] (What's this?)

  • 1. J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. MR 0495450 (82b:58033)
  • 2. -, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385-584. MR 0956352 (89i:58027)
  • 3. I. Davidov and A. G. Sergeev, Twistor spaces and harmonic maps, Uspekhi Mat. Nauk 48 (1993), no. 3, 3-96; English transl., Russian Math. Surveys 48 (1993), no. 3, 1-91. MR 1243612 (95e:58046)
  • 4. L. P. Eisenhart, Riemannian geometry, Princeton Univ. Press, Princeton, 1949. MR 0035081 (11:687g)
  • 5. K. Yano, Differential geometry on complex and almost complex spaces, Internat. Ser. Monogr. in Pure Appl. Math., vol. 49A, Pergamon Press, New York, 1965. MR 0187181 (32:4635)
  • 6. N. S. Sinyukov, Geodesic mappings of Riemannian spaces, ``Nauka'', Moscow, 1979. (Russian) MR 0552022 (81g:53014)
  • 7. K. Yano, The theory of Lie derivatives and its applications, North-Holland, Amsterdam, 1957. MR 0088769 (19:576f)
  • 8. Sh. Kobayashi, Transformation groups in differential geometry, Ergeb. Math. Grenzgeb., vol. 70, Springer-Verlag, New York-Heidelberg, 1972. MR 0355886 (50:8360)
  • 9. H. Wu, The Bochner technique in differential geometry, Harvard Acad. Publ., London, 1987.
  • 10. S. E. Stepanov, The classification of harmonic diffeomorphisms, The 5-th International Conference on Geometry and Applications (August 24-29, 2001, Varna): Abstracts, Union of Bulgarian Mathematicians, Sofia, 2001, p. 55.
  • 11. J. Milnor, Morse theory, Ann. of Math. Stud., No. 51, Princeton Univ. Press, Princeton, NJ, 1963. MR 0163331 (29:634)
  • 12. S. E. Stepanov, On the global theory of some classes of mappings, Ann. Global Anal. Geom. 13 (1995), 239-249. MR 1344481 (96k:58062)
  • 13. R. Narasimhan, Analysis on real and complex manifolds, Adv. Stud. Pure Math., vol. 1, Masson, Paris; North-Holland, Amsterdam, 1968. MR 0251745 (40:4972)
  • 14. Sh. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. 2, Intersci. Tracts in Pure Appl. Math., No. 15, Vol. II, John Wiley and Sons, Inc., New York etc., 1969. MR 0238225 (38:6501)
  • 15. A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Math. Pura Appl. (4) 123 (1980), 35-58. MR 0581924 (81m:53045)
  • 16. L. Friedland and Ch.-Ch. Hsiung, A certain class of almost Hermitian manifolds, Tensor (N.S.) 48 (1989), 252-263. MR 1088471 (92b:53040)
  • 17. A. Gray, Nearly Kähler manifolds, J. Differential Geom. 4 (1970), 283-309. MR 0267502 (42:2404)
  • 18. S. E. Stepanov, A group-theoretic approach to the study of Einstein and Maxwell equations, Teoret. Mat. Fiz. 111 (1997), no. 1, 32-43; English transl., Theoret. and Math. Phys. 111 (1997), no. 1, 419-427. MR 1473424 (98k:83026)
  • 19. A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. (3), vol. 10, Springer, Berlin etc., 1987. MR 0867684 (88f:53087)
  • 20. D. Kramer et al., Exact solutions of Einstein's field equations, Cambridge Univ. Press, Cambridge, 1980. MR 0614593 (82h:83002)
  • 21. A. Nijenhuis, A note on first integrals of geodesics, Nederl. Akad. Wetensch. Proc. Ser. A 70 29 (1967), no. 2, 141-145. MR 0212697 (35:3563)
  • 22. S. E. Stepanov, On an application of a theorem of P. A. Shirokov in the Bochner technique, Izv. Vyssh. Uchebn. Zaved. Mat. 1996, no. 9, 53-59; English transl., Russian Math. (Iz. VUZ) 40 (1996), no. 9, 50-55 (1997). MR 1430472 (98b:53037)
  • 23. Sh. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, John Wiley and Sons, New York-London, 1963. MR 0152974 (27:2945)
  • 24. K. Yano and S. Bochner, Curvature and Betti numbers, Ann. of Math. Stud., No. 32, Princeton Univ. Press, Princeton, NJ, 1953. MR 0062505 (15:989f)

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Additional Information

S. E. Stepanov
Affiliation: Vladimir State Pedagogical University, Pr. Stroitelei 11, Vladimir 600024, Russia
Email: stepanov@vtsnnet.ru

I. G. Shandra
Affiliation: Financial Academy, Government of the Russian Federation, Leningradskii Pr. 51, Moscow 125468, Russia
Email: igor-shandra@mtu-net.ru

DOI: https://doi.org/10.1090/S1061-0022-05-00856-3
Keywords: Riemannian manifold, harmonic diffeomorphism, infinitesimal harmonic transformation
Received by editor(s): September 18, 2001
Published electronically: March 9, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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