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BIHARMONIC CURVES INTO QUADRICS

Published online by Cambridge University Press:  26 August 2014

S. MONTALDO and
A. RATTO
S. MONTALDO
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di CagliariVia Ospedale 7209124 Cagliari, Italia e-mail: montaldo@unica.it; rattoa@unica.it
A. RATTO
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di CagliariVia Ospedale 7209124 Cagliari, Italia e-mail: montaldo@unica.it; rattoa@unica.it
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Abstract

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We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves in surfaces defined by a polynomial equation: In particular, we use it to give a complete classification of biharmonic curves in real quadrics of the three-dimensional Euclidean space.

Keywords

58E20

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 1 , January 2015, pp. 131 - 141
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Caddeo, R., Montaldo, S. and Oniciuc, C., Biharmonic submanifolds of ${\mathbb S}^3$, Int. J. Math. 12 (2001), 867–876.Google Scholar
2.Caddeo, R., Montaldo, S., Oniciuc, C. and Piu, P., The Euler-Lagrange method for biharmonic curves, Mediterr. J. Math. 3 (2006), 449–465.CrossRefGoogle Scholar
3.Caddeo, R., Montaldo, S. and Piu, P., Biharmonic curves on a surface, Rend. Mat. Appl. 21 (2001), 143–157.Google Scholar
4.Caddeo, R., Oniciuc, C. and Piu, P., Explicit formulas for non-geodesic biharmonic curves of the Heisenberg group, Rend. Sem. Mat. Univ. Politec. Torino 62 (2004), 265–277.Google Scholar
5.Cho, J. T., J-I. Inoguchi and J-E. Lee, Biharmonic curves in 3-dimensional Sasakian space forms, Ann. Mat. Pura Appl. 186 (2007), 685–701.CrossRefGoogle Scholar
6.Dimitric, I., Submanifolds of ${\mathbb E}^m$ with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica 20 (1992), 53–65.Google Scholar
7.Eells, J. and Lemaire, L., Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, vol. 50 (American Mathematical Society, Providence, RI 1983).CrossRefGoogle Scholar
8.Fetcu, D., Biharmonic-Legendre curves in Sasakian space forms, J. Korean Math. Soc. 45 (2008), 393–404.CrossRefGoogle Scholar
9.Fetcu, D., A note on biharmonic curves in Sasakian space forms, Ann. Mat. Pura Appl. 189 (2010), 591–603.CrossRefGoogle Scholar
10.Goldman, R., Curvature formulas for implicit curves and surfaces, Comput. Aided Geom. Des. 22 (2005), 632–658.Google Scholar
11.Inoguchi, J-I. and Lee, J-E., Affine biharmonic curves in 3-dimensional homogeneous geometries, Mediterr. J. Math. 10 (2013), 571–592Google Scholar
12.Jiang, G. Y., 2-harmonic maps and their first and second variation formulas, Chin. Ann. Math. A 7 (1986), 389–402.Google Scholar
13.Laugwitz, D., Differential and Riemannian geometry (Academic Press, Waltham, MA, 1965).Google Scholar
14.Montaldo, S. and Oniciuc, C., A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47 (2006), 1–22.Google Scholar
15.Montaldo, S. and Onnis, I. I., Biharmonic curves on an invariant surface, J. Geom. Phys. 59 (2009), 391–399.CrossRefGoogle Scholar
16.Monterde, J., Surfaces with a family of non-geodesic biharmonic curves, Rend. Mat. Appl. 28 (2008), 123–131.Google Scholar