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Isometric Semiparallel Immersions of Two-Dimensional Riemannian Manifolds into Pseudo-Euclidean Spaces

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New Developments in Differential Geometry, Budapest 1996

Abstract

The classification of semiparallel surfaces in Euclidean space (by Deprez 1985) is extended to those in pseudo-Euclidean space and detailed for the most interesting class of second order envelopes (SOE) of Veronese orbits. It is shown that in five-dimensional space such a SOE is a single orbit (or is part), in six-dimensional space there exist such SOEs with non-constant Gaussian curvature, into seven-dimensional space can be immersed isometrically every two-dimensional Riemannian manifold with non-zero curvature (at least locally) as a SOE of Veronese orbits. Results are important for description of the semisymmetric Riemannian manifolds, realizable as semiparallel submanifolds.

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References

  1. A.C. Asperti, F. Mercuri, Semi-parallel immersions into space forms, Boll. Unione Mat. Ital. (7) 8-B (1994), 833–895.

    MathSciNet  Google Scholar 

  2. C. Blomstrom, Symmetric immersions in pseudo-Riemannian space forms, Lect. Notes Math. 1156 (1985), 30–45.

    Article  MathSciNet  Google Scholar 

  3. C. Blomstrom, Planar geodesic immersions in pseudo-Euclidean space, Math. Ann. 274 (1986), 585–598.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Boeckx, O. Kowalski, L. Vanhecke, Riemannian Manifolds of Conullity Two, World Sc., London, 1996.

    Book  MATH  Google Scholar 

  5. B.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, P.A. Grifnths, Exterior Differential Systems, Springer, Berlin, 1991.

    MATH  Google Scholar 

  6. É. Cartan, Leçons sur la Géométrie des Espaces de Riemann, 2nd edition, Gautier-Villars, Paris, 1946.

    MATH  Google Scholar 

  7. É. Cartan, Les Systémes différentielles Exterieur et leurs Applications Géométriques, Hermann, Paris, 1946.

    Google Scholar 

  8. J. Deprez, Semi-parallel surfaces in Euclidean space, J.Geom. 25 (1985), 192–200.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Deprez, Semi-parallel hypersurfaces, Rend. Sem. Mat. Univ. Politec. Torino 44 (1986), 303–316.

    MathSciNet  MATH  Google Scholar 

  10. D. Ferus. Symmetric submanifolds of Euclidean space, Math. Ann. 247 (1980), 81–93.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ü. Lumiste, Decomposition of semi-symmetric submanifolds, Acta comment. Univ. Tartu 803 (1988), 69–78.

    MathSciNet  Google Scholar 

  12. Ü. Lumiste Classification of two-codimensional semi-symmetric submanifolds, Acta Comm. Univ. Tartu 803 (1988), 79–94.

    MathSciNet  Google Scholar 

  13. Ü. Lumiste, Semi-symmetric submanifolds with maximal first normal space, Proc. Estonian Acad. Sci., Phys. Math. 38 (1989), 453–457.

    MathSciNet  MATH  Google Scholar 

  14. Ü. Lumiste, Classification of three-dimensional semi-symmetric submanifolds in Euclidean spaces, Acta Comm. Univ. Tartu 899 (1990), 45–56.

    Google Scholar 

  15. Ü. Lumiste, Semi-symmetric submanifold as the second order envelope of symmetric submanifolds, Proc. Estonian Acad. Sci., Phys. Math. 39 (1990), 1–8.

    MathSciNet  MATH  Google Scholar 

  16. Ü. Lumiste, Semisymmetric submanifolds, Itogi nauki i tekhn. Ser. Probl. geom. 23 (1991), 3–28 (Russian).

    MathSciNet  MATH  Google Scholar 

  17. Ü. Lumiste, Second order envelopes of m-dimensional Veronese submanifolds, Acta Comm. Univ. Tartu 930 (1991), 53–66.

    Google Scholar 

  18. Ü. Lumiste, Symmetric orbits of the orthogonal Segre action and their second order envelopes, Rend. Semin. Mat. Messina; Ser, II, 1 (1991), 141–150.

    MathSciNet  MATH  Google Scholar 

  19. Ü. Lumiste, Modified Nomizu problem for semi-parallel submanifolds, Geometry and Topology of Submanifolds, VII. Differential Geometry in honour of Prof. Katsumi Nomizu, World Sc, Singapore [et al.], 1995, 176–181.

    Google Scholar 

  20. Ü. Lumiste, Symmetric orbits of orthogonal Veronese actions and their second order envelopes, Results in Math. 27 (1995), 284–301.

    MathSciNet  MATH  Google Scholar 

  21. Ü. Lumiste, Semi-parallel pseudo-Riemannian submanifolds with non-null principal normals of extremal dimension, Preprint Series, Inst. of Math., Univ. of Oslo, Nol, Febr. 1996, 1–34.

    Google Scholar 

  22. Ü. Lumiste, Symmetric orbits of orthogonal Plücker actions and triviality of their second order envelopes, Ann. Global. Anal. Geom. 14 (1996), 237–256.

    Article  MathSciNet  MATH  Google Scholar 

  23. Ü. Lumiste, Semiparallel submanifolds of cylindrical or toroidal Segre type, Proc. Estonian Acad. Sci., Phys. Math. 45 (1996), 161–177.

    MathSciNet  MATH  Google Scholar 

  24. Ü. Lumiste, Semi-parallel submanifolds as some immersed fibre bundles with flat connections, Geometry and Topology of Submanifolds, VIII, World Sc, Singapore, London, 1996, 236–244.

    Google Scholar 

  25. Ü. Lumiste, Semi-parallel time-like surfaces in Lorentzian spacetime forms, Diff. Geom. Applic. 7 (1997), 59–74.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Magid, Isometric immersions of Lorentz space with parallel second fundamental forms, Tsukuba J. Math. 8 (1984), 31–54.

    MathSciNet  MATH  Google Scholar 

  27. F. Mercuri, Parallel and semi-parallel immersions into space forms, Riv. Math. Univ. Parma (4) 17 (1991), 91–108.

    MathSciNet  Google Scholar 

  28. V.A. Mirzoyan, Semisymmetric submanifolds and their decomposition into a product, Izv. Vyssh. Uchebn. Zaved., Mat. 35 (1991), No 9, 29–38 (Russian).

    MathSciNet  Google Scholar 

  29. H. Naitoh, Pseudo-Riemannian symmetric R-spaces, Osaka J.Math. 21 (1984), 733–764.

    MathSciNet  MATH  Google Scholar 

  30. K. Nomizu, On hypersurfaces satisfying a certain condition on the curvature tensor, Tôhoku Math. J. 20 (1968), 46–59.

    Article  MathSciNet  MATH  Google Scholar 

  31. B. O’Neill, Isotropic and Kaehler immersions, Canad. J. Math. 17 (1965), 907–915.

    Article  MathSciNet  Google Scholar 

  32. K. Riives, Second order envelopes of congruent Veronese surfaces in E 6, Acta Comm. Univ. Tartu 930 (1991), 47–52

    MathSciNet  Google Scholar 

  33. K. Sekigawa, On some hypersurfaces satisfying R(X, Y) ·R = 0, Tensor 25 (1972), 105–108.

    Google Scholar 

  34. P.A. Shirokov, Selected Works on Geometry, Izd. Kazanskogo Universiteta, Kazan, 1966 (Russian).

    Google Scholar 

  35. N.S. Sinjukov, On geodesic maps of Riemannian spaces, Tr.III Vsesojuzn. mat. s’ezda (Proc. III All-Union Math. Congr), I, Izd. AN SSSR, Moskva, 1956, 167–168 (Russian).

    Google Scholar 

  36. N.S. Sinjukov, Geodesic Maps on Riemannian Spaces, Nauka, Moskva, 1979 (Russian).

    Google Scholar 

  37. Z.I. Szabó, Structure theorems on Riemannian spaces satisfying R(X, Y) ·R = 0. I. The local version. J.Differ. Geom. 17 (1982), 531–582.

    MATH  Google Scholar 

  38. Z.I. Szabó, Structure theorems on Riemannian spaces satisfying R(X, Y) ·R = 0. II. Global versions. Geom. Dedicata 19 (1985), 65–108.

    Article  MathSciNet  MATH  Google Scholar 

  39. H. Takagi. An example of Riemannian manifold satisfying R(X, Y) · R = 0 but not ▽R = 0, Tôhoku Math. J. 24 (1972), 105–108.

    Article  MATH  Google Scholar 

  40. M. Takeuchi, Parallel submanifolds of space forms, Manifolds and Lie groups: Papers of honor of Yôzo Matsushima, Basel, Birkhäuser, 1981, 429–447.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Pure Mathematics, University of Tartu, Vanemuise 46, Tartu, EE2400, Estonia

    Ülo Lumiste

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  1. Ülo Lumiste
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Editors and Affiliations

  1. Department of Geometry, Loránd Eötvös University, Budapest, Hungary

    J. Szenthe

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© 1999 Springer Science+Business Media Dordrecht

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Lumiste, Ü. (1999). Isometric Semiparallel Immersions of Two-Dimensional Riemannian Manifolds into Pseudo-Euclidean Spaces. In: Szenthe, J. (eds) New Developments in Differential Geometry, Budapest 1996. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5276-1_17

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  • DOI: https://doi.org/10.1007/978-94-011-5276-1_17

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6220-6

  • Online ISBN: 978-94-011-5276-1

  • eBook Packages: Springer Book Archive

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