| Title:
|
Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds (English) |
| Author:
|
Lumiste, Ülo |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
53 |
| Issue:
|
3 |
| Year:
|
2003 |
| Pages:
|
707-734 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A Riemannian manifold is said to be semisymmetric if $R(X,Y)\cdot R=0$. A submanifold of Euclidean space which satisfies $\bar{R}(X,Y)\cdot h=0$ is called semiparallel. It is known that semiparallel submanifolds are intrinsically semisymmetric. But can every semisymmetric manifold be immersed isometrically as a semiparallel submanifold? This problem has been solved up to now only for the dimension 2, when the answer is affirmative for the positive Gaussian curvature. Among semisymmetric manifolds a special role is played by the foliated ones, which in the dimension 3 are divided by Kowalski into four classes: elliptic, hyperbolic, parabolic and planar. It is shown now that only the planar ones can be immersed isometrically into Euclidean spaces as 3-dimensional semiparallel submanifolds. This result is obtained by a complete classification of such submanifolds. (English) |
| Keyword:
|
semisymmetric Riemannian manifolds |
| Keyword:
|
semiparallel submanifolds |
| Keyword:
|
isometric immersions |
| Keyword:
|
planar foliated manifolds |
| MSC:
|
53B25 |
| MSC:
|
53C25 |
| MSC:
|
53C42 |
| idZBL:
|
Zbl 1080.53036 |
| idMR:
|
MR2000064 |
| . |
| Date available:
|
2009-09-24T11:05:55Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127834 |
| . |
| Reference:
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[1] E. Boeckx: Foliated semi-symmetric spaces.Doctoral thesis, Katholieke Universiteit, Leuven, 1995. Zbl 0846.53031 |
| Reference:
|
[2] E. Boeckx, O. Kowalski and L. Vanhecke: Riemannian Manifolds of Conullity Two.World Sc., Singapore, 1996. MR 1462887 |
| Reference:
|
[3] É. Cartan: Leçons sur la géométrie des espaces de Riemann. 2nd editon.Gautier-Villars, Paris, 1946. MR 0020842 |
| Reference:
|
[4] J. Deprez: Semi-parallel surfaces in Euclidean space.J. Geom. 25 (1985), 192–200. Zbl 0582.53042, MR 0821680, 10.1007/BF01220480 |
| Reference:
|
[5] D. Ferus: Symmetric submanifolds of Euclidean space.Math. Ann. 247 (1980), 81–93. Zbl 0446.53041, MR 0565140, 10.1007/BF01359868 |
| Reference:
|
[6] O. Kowalski: An explicit classification of 3-dimensional Riemannian spaces satisfying $R(X,Y)\cdot R=0$.Czechoslovak Math. J. 46(121) (1996), 427–474. Zbl 0879.53014, MR 1408298 |
| Reference:
|
[7] O. Kowalski and S. Ž. Nikčević: Contact homogeneity and envelopes of Riemannian metrics.Beitr. Algebra Geom. 39 (1998), 155–167. MR 1614436 |
| Reference:
|
[8] Ü. Lumiste: Decomposition and classification theorems for semi-symmetric immersions. Eesti TA Toim. Füüs.Mat. Proc. Acad. Sci. Estonia Phys. Math. 36 (1987), 414–417. MR 0925980 |
| Reference:
|
[9] Ü. Lumiste: Semi-symmetric submanifolds with maximal first normal space. Eesti TA Toim. Füüs.Mat. Proc. Acad. Sci. Estonia Phys. Math. 38 (1989), 453–457. MR 1046557 |
| Reference:
|
[10] Ü. Lumiste: Semi-symmetric submanifold as the second order envelope of symmetric submanifolds. Eesti TA Toim. Füüs.Mat. Proc. Acad. Sci. Estonia Phys. Math. 39 (1990), 1–8. MR 1059755 |
| Reference:
|
[11] Ü. Lumiste: Classification of three-dimensional semi-symmetric submanifolds in Euclidean spaces..Tartu Ül. Toimetised 899 (1990), 29–44. Zbl 0749.53012, MR 1082921 |
| Reference:
|
[12] Ü. Lumiste: Semi-symmetric envelopes of some symmetric cylindrical submanifolds. Eesti TA Toim. Füüs.Mat. Proc. Acad. Sci. Estonia Phys. Math. 40 (1991), 245–257. MR 1163442 |
| Reference:
|
[13] Ü. Lumiste: Second order envelopes of symmetric Segre submanifolds.Tartu Ül. Toimetised. 930 (1991), 15–26. MR 1151820 |
| Reference:
|
[14] Ü. Lumiste: Isometric semiparallel immersions of two-dimensional Riemannian manifolds into pseudo-Euclidean spaces.New Developments in Differential Geometry, Budapest 1996, J. Szenthe (ed.), Kluwer Ac. Publ., Dordrecht, 1999, pp. 243–264. Zbl 0947.53032, MR 1670514 |
| Reference:
|
[15] Ü. Lumiste: Submanifolds with parallel fundamental form.In: Handbook of Differential Geometry, Vol. I, F. Dillen, L. Verstraelen (eds.), Elsevier Sc. B. V., Amsterdam, 2000, pp. 779–864. Zbl 0964.53002, MR 1736858 |
| Reference:
|
[16] Ü. Lumiste and K. Riives: Three-dimensional semi-symmetric submanifolds with axial, planar or spatial points in Euclidean spaces.Tartu Ülik. Toim. Acta et Comm. Univ. Tartuensis 899 (1990), 13–28. MR 1082920 |
| Reference:
|
[17] V. Mirzoyan: $s$-semi-parallel submanifolds in spaces of constant curvature as the envelopes of $s$-parallel submanifolds.J. Contemp. Math. Analysis (Armenian Ac. Sci., Allerton Press, Inc.) 31 (1996), 37–48. Zbl 0890.53027, MR 1693824 |
| Reference:
|
[18] V. Mirzoyan: On generalizations of Ü. Lumiste theorem on semi-parallel submanifolds.J. Contemp. Math. Analysis (Armenian Ac. Sci., Allerton Press, Inc.) 33 (1998), 48–58. MR 1714535 |
| Reference:
|
[19] K. Nomizu: On hypersurfaces satisfying a certain condition on the curvature tensor.Tôhoku Math. J. 20 (1968), 46–59. Zbl 0174.53301, MR 0226549, 10.2748/tmj/1178243217 |
| Reference:
|
[20] K. Sekigawa: On some hypersurfaces satisfying $R(X,Y)\cdot R=0$.Tensor 25 (1972), 133–136. MR 0331288 |
| Reference:
|
[21] P. A. Shirokov: Selected Works on Geometry.Izd. Kazanskogo Univ., Kazan, 1966. (Russian) MR 0221390 |
| Reference:
|
[22] N. S. Sinjukov: On geodesic maps of Riemannian spaces.Trudy III Vsesojuzn. Matem. S’ezda (Proc. III All-Union Math. Congr.), I, Izd. AN SSSR, Moskva, 1956, pp. 167–168. (Russian) |
| Reference:
|
[23] N. S. Sinjukov: Geodesic maps of Riemannian spaces.Publ. House “Nauka”, Moskva, 1979. (Russian) MR 0552022 |
| Reference:
|
[24] W. Strübing: Symmetric submanifolds of Riemannian manifolds.Math. Ann. 245 (1979), 37–44. 10.1007/BF01420428 |
| Reference:
|
[25] Z. I. Szabó: Structure theorems on Riemannian spaces satisfying $R(X,Y)\cdot R=0$, I. The local version.J. Differential Geom. 17 (1982), 531–582. MR 0683165, 10.4310/jdg/1214437486 |
| Reference:
|
[26] H. Takagi: An example of Riemannian manifolds satisfying $R(X,Y)\cdot R=0$ but not $\nabla R=0$.Tôhoku Math. J. 24 (1972), 105–108. Zbl 0237.53041, MR 0319109, 10.2748/tmj/1178241595 |
| Reference:
|
[27] M. Takeuchi: Parallel submanifolds of space forms.Manifolds and Lie Groups. Papers in Honour of Y. Matsushima, Birkhäuser, Basel, 1981, pp. 429–447. Zbl 0481.53047, MR 0642871 |
| Reference:
|
[28] J. Vilms: Submanifolds of Euclidean space with parallel second fundamental form.Proc. Amer. Math. Soc. 32 (1972), 263–267. Zbl 0229.53045, MR 0290298 |
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