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

Part of: Global differential geometry Local 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: c0nstc@yahoo.com)
MIRCEA CRASMAREANU
Affiliation:
Faculty of Mathematics, University ‘Al. I. Cuza’, 700506 Iaşi, Romania (email: mcrasm@uaic.ro)
*
✉For correspondence; e-mail: c0nstc@yahoo.com
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Abstract

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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. 

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