We provide a new construction method for r-harmonic submanifolds in spheres, i.e. proper isometric r-harmonic immersions into spheres. Furthermore, we give new examples of r-harmonic spheres in spheres, possibly of high codimension. We determine index and nullity of these isometric r-harmonic immersions when $r=2$. Finally, we provide the normal index of these maps when $r=3$.
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Balado-Alves et al. (2024) studied this question.
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