Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T23:31:50.606Z Has data issue: false hasContentIssue false
  • English
  • Français

SLANT CURVES AND PARTICLES IN THREE-DIMENSIONAL WARPED PRODUCTS AND THEIR LANCRET INVARIANTS

Part of: Local differential geometry Global differential geometry

Published online by Cambridge University Press:  19 October 2012

CONSTANTIN CĂLIN  and
MIRCEA CRASMAREANU
CONSTANTIN CĂLIN*
Affiliation:
Department of Mathematics, Technical University ‘Gh. Asachi’, 700049 Iaşi, Romania (email: [email protected])
MIRCEA CRASMAREANU
Affiliation:
Faculty of Mathematics, University ‘Al. I. Cuza’, 700506 Iaşi, Romania (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Slant curves are introduced in three-dimensional warped products with Euclidean factors. These curves are characterised by the scalar product between the normal at the curve and the vertical vector field, and an important feature is that the case of constant Frenet curvatures implies a proper mean curvature vector field. A Lancret invariant is obtained and the Legendre curves are analysed as a particular case. An example of a slant curve is given for the exponential warping function; our example illustrates a proper (that is, not reducible to the two-dimensional) case of the Lancret theorem of three-dimensional hyperbolic geometry. We point out an eventuality relationship with the geometry of relativistic models.

Keywords

warped productslant curveLegendre curveLancret invariantFrenet frame

MSC classification

Secondary: 53B25: Local submanifolds 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C40: Global submanifolds
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 1 , August 2013 , pp. 128 - 142
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

References

[1]Baikoussis, Ch. & Blair, D. E., ‘On Legendre curves in contact $3$-manifolds’, Geom. Dedicata 49(2) (1994), 135–142.CrossRefGoogle Scholar
[2]Barros, M., ‘General helices and a theorem of Lancret’, Proc. Amer. Math. Soc. 125(5) (1997), 1503–1509.CrossRefGoogle Scholar
[3]Belkhelfa, M., Hirica, I.-E., Rosca, R. & Verstraelen, L., ‘On Legendre curves in Riemannian and Lorentzian Sasaki spaces’, Soochow J. Math. 28(1) (2002), 81–91.Google Scholar
[4]Bishop, R. L. & O’Neill, B., ‘Manifolds of negative curvature’, Trans. Amer. Math. Soc. 145 (1969), 1–49.CrossRefGoogle Scholar
[5]Blair, D. E., Riemannian Geometry of Contact and Symplectic Manifolds, 2nd edn. Progress in Mathematics, 203 (Birkhäuser, Boston, MA, 2010).CrossRefGoogle Scholar
[6]Blair, D. E., Dillen, F., Verstraelen, L. & Vrancken, L., ‘Deformations of Legendre curves’, Note Mat. 15(1) (1995), 99–110 (1997).Google Scholar
[7]Călin, C. & Crasmareanu, M., ‘Slant curves in three-dimensional normal almost contact geometry’, Mediterr. J. Math., to appear; doi:10.1007/s00009-012-0217-1.CrossRefGoogle Scholar
[8]Călin, C., Crasmareanu, M. & Munteanu, M. I., ‘Slant curves in three-dimensional $f$-Kenmotsu manifolds’, J. Math. Anal. Appl. 394(1) (2012), 400–407.CrossRefGoogle Scholar
[9]Camci, C., Yayli, Y. & Hacisalihoglu, H. H., ‘On the characterization of spherical curves in three-dimensional Sasakian spaces’, J. Math. Anal. Appl. 342(2) (2008), 1151–1159.CrossRefGoogle Scholar
[10]Cho, J. T., Inoguchi, J.-I. & Lee, J.-E., ‘On slant curves in Sasakian 3-manifolds’, Bull. Aust. Math. Soc. 74(3) (2006), 359–367.CrossRefGoogle Scholar
[11]Cho, J. T. & Lee, J.-E., ‘Slant curves in contact pseudo-Hermitian $3$-manifolds’, Bull. Aust. Math. Soc. 78(3) (2008), 383–396.CrossRefGoogle Scholar
[12]Çiftçi, Ü., ‘A generalization of Lancret’s theorem’, J. Geom. Phys. 59(12) (2009), 1597–1603.CrossRefGoogle Scholar
[13]Dillen, F., Munteanu, M. I., Van der Veken, J. & Vrancken, L., ‘Classification of constant angle surfaces in a warped product’, Balkan J. Geom. Appl. 16(2) (2011), 35–47.Google Scholar
[14]Ferrández, A., Guerrero, J., Javaloyes, M. A. & Lucas, P., ‘Particles with curvature and torsion in three-dimensional pseudo-Riemannian space forms’, J. Geom. Phys. 56(9) (2006), 1666–1687.CrossRefGoogle Scholar
[15]Fetcu, D., ‘Biharmonic Legendre curves in Sasakian space forms’, J. Korean Math. Soc. 45(2) (2008), 393–404.CrossRefGoogle Scholar
[16]Izumiya, S. & Takeuchi, N., ‘New special curves and developable surfaces’, Turkish J. Math. 28(2) (2004), 153–163.Google Scholar
[17]Lee, J.-E., ‘On Legendre curves in contact pseudo-Hermitian $3$-manifolds’, Bull. Aust. Math. Soc. 81(1) (2010), 156–164.CrossRefGoogle Scholar
[18]O’Neill, B., Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, 103 (Academic Press, New York, 1983).Google Scholar
[19]Smoczyk, K., ‘Closed Legendre geodesics in Sasaki manifolds’, New York J. Math. 9 (2003), 23–47 (electronic).Google Scholar
[20]Tripathi, M. M., ‘A note on certain Legendre curves in a Kenmotsu manifold’, Ganita 51(1) (2000), 57–58.Google Scholar
[21]Welyczko, J., ‘On Legendre curves in three-dimensional normal almost contact metric manifolds’, Soochow J. Math. 33(4) (2007), 929–937.Google Scholar
[22]Welyczko, J., ‘On Legendre curves in $3$-dimensional normal almost paracontact metric manifolds’, Results Math. 54(3–4) (2009), 377–387.CrossRefGoogle Scholar