Harmonic diffeomorphisms of manifolds. (English. Russian original) Zbl 1069.53051
St. Petersbg. Math. J. 16, No. 2, 401-412 (2005); translation from Algebra Anal. 16, No. 2, 154-171 (2004).
Summary: 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:
| [1] | J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1 – 68. · Zbl 0401.58003 · doi:10.1112/blms/10.1.1 |
| [2] | J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385 – 524. · Zbl 0669.58009 · doi:10.1112/blms/20.5.385 |
| [3] | Ĭ. Davidov and A. G. Sergeev, Twistor spaces and harmonic maps, Uspekhi Mat. Nauk 48 (1993), no. 3(291), 3 – 96 (Russian); English transl., Russian Math. Surveys 48 (1993), no. 3, 1 – 91. · Zbl 0851.58010 · doi:10.1070/RM1993v048n03ABEH001031 |
| [4] | Luther Pfahler Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, N. J., 1949. 2d printing. · Zbl 1141.53002 |
| [5] | Kentaro Yano, Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics, Vol. 49, A Pergamon Press Book. The Macmillan Co., New York, 1965. · Zbl 0127.12405 |
| [6] | Геодезические отображения римановых пространств, ”Наука”, Мосцощ, 1979 (Руссиан). · Zbl 0637.53020 |
| [7] | Kentaro Yano, The theory of Lie derivatives and its applications, North-Holland Publishing Co., Amsterdam; P. Noordhoff Ltd., Groningen; Interscience Publishers Inc., New York, 1957. · Zbl 0077.15802 |
| [8] | Shoshichi Kobayashi, Transformation groups in differential geometry, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70. · Zbl 0246.53031 |
| [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, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. · Zbl 0108.10401 |
| [12] | Sergey E. Stepanov, On the global theory of some classes of mappings, Ann. Global Anal. Geom. 13 (1995), no. 3, 239 – 249. · Zbl 0839.53028 · doi:10.1007/BF00773658 |
| [13] | Raghavan Narasimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968. · Zbl 0188.25803 |
| [14] | Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. · Zbl 0091.34802 |
| [15] | Alfred Gray and Luis M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35 – 58. · Zbl 0444.53032 · doi:10.1007/BF01796539 |
| [16] | Lew Friedland and Chuan-Chih Hsiung, A certain class of almost Hermitian manifolds, Tensor (N.S.) 48 (1989), no. 3, 252 – 263. · Zbl 0718.53027 |
| [17] | Alfred Gray, Nearly Kähler manifolds, J. Differential Geometry 4 (1970), 283 – 309. · Zbl 0201.54401 |
| [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 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 111 (1997), no. 1, 419 – 427. · Zbl 0964.53503 · doi:10.1007/BF02634197 |
| [19] | Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. · Zbl 0613.53001 |
| [20] | D. Kramer, H. Stephani, E. Herlt, and M. MacCallum, Exact solutions of Einstein’s field equations, Cambridge University Press, Cambridge-New York, 1980. Edited by Ernst Schmutzer; Cambridge Monographs on Mathematical Physics. · Zbl 0449.53018 |
| [21] | Albert Nijenhuis, A note on first integrals of geodesics, Nederl. Akad. Wetensch. Proc. Ser. A 70=Indag. Math. 29 (1967), 141 – 145. · Zbl 0161.18803 |
| [22] | S. E. Stepanov, On an application of a theorem of P. A. Shirokov in the Bochner technique, Izv. Vyssh. Uchebn. Zaved. Mat. 9 (1996), 53 – 59 (Russian); English transl., Russian Math. (Iz. VUZ) 40 (1996), no. 9, 50 – 55 (1997). |
| [23] | Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. · Zbl 0091.34802 |
| [24] | K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. · Zbl 0051.39402 |
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