Abstract
We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector fields in $\mathbb{S}^{n}$.
Then we investigate, for (not necessarily compact) proper-biharmonic submanifolds in $\mathbb{S}^{n}$, their type in the sense of B.-Y. Chen. We prove that (i) a proper-biharmonic submanifold in $\mathbb{S}^{n}$ is of 1-type or 2-type if and only if it has constant mean curvature f=1 or f∈(0,1), respectively; and (ii) there are no proper-biharmonic 3-type submanifolds with parallel normalized mean curvature vector fields in $\mathbb{S}^{n}$.
Funding Statement
The first author was supported by Grant POSDRU/89/1.5/S/49944, Romania. The second author was supported by Contributo d’Ateneo, University of Cagliari, Italy. The third author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-RU-TE-2011-3-0108.
Citation
Adina Balmuş. Stefano Montaldo. Cezar Oniciuc. "Biharmonic PNMC submanifolds in spheres." Ark. Mat. 51 (2) 197 - 221, October 2013. https://doi.org/10.1007/s11512-012-0169-5
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