## Arkiv för Matematik

• Ark. Mat.
• Volume 51, Number 2 (2013), 197-221.

### Biharmonic PNMC submanifolds in spheres

#### 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 $\mathbb{S}^{n}$.

#### Note

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.

#### Article information

Source
Ark. Mat., Volume 51, Number 2 (2013), 197-221.

Dates
Revised: 5 March 2012
First available in Project Euclid: 1 February 2017

https://projecteuclid.org/euclid.afm/1485907217

Digital Object Identifier
doi:10.1007/s11512-012-0169-5

Mathematical Reviews number (MathSciNet)
MR3090194

Zentralblatt MATH identifier
1282.53047

Rights

#### Citation

Balmuş, Adina; Montaldo, Stefano; Oniciuc, Cezar. Biharmonic PNMC submanifolds in spheres. Ark. Mat. 51 (2013), no. 2, 197--221. doi:10.1007/s11512-012-0169-5. https://projecteuclid.org/euclid.afm/1485907217

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