Invariants and Bonnet-type theorem for surfaces in ℝ4

• Autorzy: Georgi Ganchev , Velichka Milousheva
• Języki publikacji: EN

Abstrakty

EN

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.

Słowa kluczowe

EN

Surfaces in the four-dimensional Euclidean space; Weingarten map; Tangent indicatrix; Normal curvature ellipse; Fundamental theorem of Bonnet-type

Kategorie tematyczne

• 53A07: Higher-dimensional and -codimensional surfaces in Euclidean n -space
• 53A10: Minimal surfaces, surfaces with prescribed mean curvature

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 6
• Strony: 993-1008

Daty

wydano

2010-12-01

online

2010-10-30

Twórcy

autor

Georgi Ganchev

• Bulgarian Academy of Sciences

autor

Velichka Milousheva

Bibliografia

• [1] Asperti A.C., Some generic properties of Riemannian immersions, Bol. Soc. Brasil. Mat., 1980, 11(2), 191–216 http://dx.doi.org/10.1007/BF02584637
• [2] Burstall F., Ferus D., Leschke K., Pedit F., Pinkall U., Conformal Geometry of Surfaces in S 4 and Quaternions, Lecture Notes in Math., 1772, Springer, New York, 2002
• [3] Chen B.-Y., Geometry of Submanifolds, Pure and Applied Mathematics, 22, Marcel Dekker, New York, 1973
• [4] Chen B.-Y., Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures, Ann. Global Anal. Geom., 2010, 38(2), 145–160 http://dx.doi.org/10.1007/s10455-010-9205-5
• [5] Dajczer M., Tojeiro R., All superconformal surfaces in ℝ4 in terms of minimal surfaces, Math. Z., 2009, 261(4), 869–890 http://dx.doi.org/10.1007/s00209-008-0355-0
• [6] Eisenhart L.P., Minimal surfaces in Euclidean four-space, Amer. J. Math., 1912, 34(3), 215–236 http://dx.doi.org/10.2307/2370220
• [7] Ganchev G., Milousheva V., On the theory of surfaces in the four-dimensional Euclidean space, Kodai Math. J., 2008, 31(2), 183–198 http://dx.doi.org/10.2996/kmj/1214442794
• [8] Ganchev G., Milousheva V., Invariants of lines on surfaces in ℝ4, C. R. Acad. Bulgare Sci., 2010, 63(6), 835–842
• [9] Ganchev G., Milousheva V., Minimal surfaces in the four-dimensional Euclidean space preprint available at http://arxiv.org/abs/0806.3334v1
• [10] Garcia R., Sotomayor J., Lines of axial curvatures on surfaces immersed in ℝ4, Differential Geom. Appl., 2000, 12(3), 253–269 http://dx.doi.org/10.1016/S0926-2245(00)00015-2
• [11] Gheysens L., Verheyen P., Verstraelen L., Sur les surfaces A ou les surfaces de Chen, C. R. Acad. Sci. Paris, Sér. I Math., 1981, 292(19), 913–916
• [12] Gheysens L., Verheyen P., Verstraelen L., Characterization and examples of Chen submanifolds, J. Geom., 1983, 20(1), 47–62 http://dx.doi.org/10.1007/BF01917994
• [13] Guadalupe I.V., Rodriguez L., Normal curvature of surfaces in space forms, Pacific J. Math., 1983, 106(1), 95–103
• [14] Kommerell K., Riemannsche Flächen im ebenen Raum von vier Dimensionen, Math. Ann., 1905, 60(4), 546–596 http://dx.doi.org/10.1007/BF01561096
• [15] Little J.A., On singularities of submanifolds of higher dimensional Euclidean spaces, Ann. Mat. Pura Appl., 1969, 83, 261–335 http://dx.doi.org/10.1007/BF02411172
• [16] Mello L.F., Orthogonal asymptotic lines on surfaces immersed in ℝ4, Rocky Mountain J. Math., 2009, 39(5), 1597–1612 http://dx.doi.org/10.1216/RMJ-2009-39-5-1597
• [17] Petrović-Torgašev M., Verstraelen L., On Deszcz symmetries of Wintgen ideal submanifolds, Arch. Math. (Brno), 2008, 44(1), 57–67
• [18] Schouten J.A., Struik D.J., Einführung in die Neueren Methoden der Differentialgeometrie II, Noordhoff, Batavia-Groningen, 1938
• [19] Spivak M., Introduction to Comprehensive Differential Geometry, vol. I, V, 3rd ed., Publish or Perish, Berkeley, 1999
• [20] Wilson E.B., Moore C.L.E., A general theory of surfaces, Proc. Nat. Acad. Sci. U.S.A., 1916, 2(5), 273–278 http://dx.doi.org/10.1073/pnas.2.5.273
• [21] Wilson E.B., Moore C.L.E., Differential geometry of two-dimensional surfaces in hyperspaces, Proc. Am. Acad. Arts Sci., 1916, 52, 269–368
• [22] Wintgen P., Sur l’inégalité de Chen-Willmore, C. R. Acad. Sci. Paris Sér. A, 1979, 288(21), 993–995
• [23] Wong Y.-C., A new curvature theory for surfaces in a Euclidean 4-space, Comment. Math. Helv., 1952, 26(1), 152–170 http://dx.doi.org/10.1007/BF02564298

Identyfikatory

DOI

10.2478/s11533-010-0073-9