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Slant and Legendre curves in Bianchi-Cartan-Vranceanu geometry. (English) Zbl 1349.53111

Summary: We study Legendre and slant curves for Bianchi-Cartan-Vranceanu metrics. These curves are characterized through the scalar product between the normal at the curve and the vertical vector field and in the helix case they have a proper (non-harmonic) mean curvature vector field. The general expression of the curvature and torsion of these curves and the associated Lancret invariant (for the slant case) are computed as well as the corresponding variant for some particular cases. The slant (particularly Legendre) curves which are helices are completely determined.

MSC:

53D15 Almost contact and almost symplectic manifolds
53B25 Local submanifolds
53A55 Differential invariants (local theory), geometric objects
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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