Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds

Lumiste, Ülo

Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 707-734

Abstract (VO)

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.

Détail

Classification : 53B25, 53C25, 53C42
Keywords: semisymmetric Riemannian manifolds; semiparallel submanifolds; isometric immersions; planar foliated manifolds

Bibliographie

[1] E.  Boeckx: Foliated semi-symmetric spaces. Doctoral thesis, Katholieke Universiteit, Leuven, 1995. | Zbl

[2] E.  Boeckx, O.  Kowalski and L.  Vanhecke: Riemannian Manifolds of Conullity Two. World Sc., Singapore, 1996. | MR

[3] É.  Cartan: Leçons sur la géométrie des espaces de Riemann. 2nd editon. Gautier-Villars, Paris, 1946. | MR

[4] J.  Deprez: Semi-parallel surfaces in Euclidean space. J.  Geom. 25 (1985), 192–200. | DOI | MR | Zbl

[5] D.  Ferus: Symmetric submanifolds of Euclidean space. Math. Ann. 247 (1980), 81–93. | DOI | MR | Zbl

[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. | MR | Zbl

[7] O.  Kowalski and S. Ž.  Nikčević: Contact homogeneity and envelopes of Riemannian metrics. Beitr. Algebra Geom. 39 (1998), 155–167. | MR

[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

[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

[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

[11] Ü.  Lumiste: Classification of three-dimensional semi-symmetric submanifolds in Euclidean spaces. Tartu Ül. Toimetised 899 (1990), 29–44. | MR | Zbl

[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

[13] Ü.  Lumiste: Second order envelopes of symmetric Segre submanifolds. Tartu Ül. Toimetised. 930 (1991), 15–26. | MR

[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. | MR | Zbl

[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. | MR | Zbl

[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

[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. | MR | Zbl

[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

[19] K.  Nomizu: On hypersurfaces satisfying a certain condition on the curvature tensor. Tôhoku Math. J.  20 (1968), 46–59. | DOI | MR | Zbl

[20] K.  Sekigawa: On some hypersurfaces satisfying $R(X,Y)\cdot R=0$. Tensor 25 (1972), 133–136. | MR

[21] P. A.  Shirokov: Selected Works on Geometry. Izd. Kazanskogo Univ., Kazan, 1966. (Russian) | MR

[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)

[23] N. S.  Sinjukov: Geodesic maps of Riemannian spaces. Publ. House “Nauka”, Moskva, 1979. (Russian) | MR

[24] W.  Strübing: Symmetric submanifolds of Riemannian manifolds. Math. Ann. 245 (1979), 37–44. | DOI

[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. | DOI | MR

[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. | DOI | MR | Zbl

[27] M.  Takeuchi: Parallel submanifolds of space forms. Manifolds and Lie Groups. Papers in Honour of Y.  Matsushima, Birkhäuser, Basel, 1981, pp. 429–447. | MR | Zbl

[28] J.  Vilms: Submanifolds of Euclidean space with parallel second fundamental form. Proc. Amer. Math. Soc. 32 (1972), 263–267. | MR | Zbl