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

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.