First, we study rectifying curves via the dilation of unit speed curves on the unit sphere S 2 in the Euclidean space E 3. Then we obtain a necessary and sufficient condition for which the centrode d(s) of a unit speed curve α(s) in E 3 is a rectifying curve to improve a main result of [4]. Finally, we prove that if a unit speed curve α(s) in E 3 is neither a planar curve nor a helix, then its dilated centrode β(s) = ρ(s)d(s) , with dilation factor ρ , is always a rectifying curve, where ρ is the radius of curvature of α .
We consider an n-dimensional compact Riemannian manifold (M, g) and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ > 0, together with an upper bound on the energy of the vector field ξ, implies that M is isometric to the n-sphere S n (λ). We also introduce the notion of ϕ-analytic conformal vector fields, study their properties, and obtain a characterization of n-spheres using these vector fields.
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