Tohoku Mathematical Journal

Biharmonic integral $\mathcal{C}$-parallel submanifolds in 7-dimensional Sasakian space forms

Abstract

We find the characterization of maximum dimensional proper-biharmonic integral $\mathcal{C}$-parallel submanifolds of a Sasakian space form and then we classify such submanifolds in a 7-dimensional Sasakian space form. Working in the sphere $\boldsymbol{S}^7$ we explicitly find all 3-dimensional proper-biharmonic integral $\mathcal{C}$-parallel submanifolds. We also determine the proper-biharmonic parallel Lagrangian submanifolds of $\boldsymbol{C}P^3$.

Article information

Source
Tohoku Math. J. (2), Volume 64, Number 2 (2012), 195-222.

Dates
First available in Project Euclid: 2 July 2012

https://projecteuclid.org/euclid.tmj/1341249371

Digital Object Identifier
doi:10.2748/tmj/1341249371

Mathematical Reviews number (MathSciNet)
MR2948819

Zentralblatt MATH identifier
1258.53059

Citation

Fetcu, Dorel; Oniciuc, Cezar. Biharmonic integral $\mathcal{C}$-parallel submanifolds in 7-dimensional Sasakian space forms. Tohoku Math. J. (2) 64 (2012), no. 2, 195--222. doi:10.2748/tmj/1341249371. https://projecteuclid.org/euclid.tmj/1341249371

References

• K. Arslan, R. Ezentas, C. Murathan and T. Sasahara, Biharmonic anti-invariant submanifolds in Sasakian space forms, Beiträge Algebra Geom. 48 (2007), 191–207.
• A. Arvanitoyeorgos, F. Defever, G. Kaimakamis and V. J. Papantoniou, Biharmonic Lorentz hypersurfaces in $E_1^4$, Pacific J. Math. 229 (2007), 293–305.
• A. Arvanitoyeorgos, F. Defever and G. Kaimakamis, Hypersurfaces of $E_s^4$ with proper mean curvature vector, J. Math. Soc. Japan 59 (2007), 797–809.
• C. Baikoussis and D. E. Blair, 2-type integral surfaces in $\boldsymbol{S}^5(1)$, Tokyo J. Math. 2 (1991), 345–356.
• C. Baikoussis and D. E. Blair, 2-type flat integral submanifolds in $\boldsymbol{S}^7(1)$, Hokkaido Math. J. 24 (1995), 473–490.
• C. Baikoussis, D. E. Blair and T. Koufogiorgos, Integral submanifolds of Sasakian space forms $\bar{M}^7$, Results Math. 27 (1995), 207–226.
• A. Balmuş, Biharmonic maps and submanifolds, Ph. D. Thesis, Geometry Balkan Press, Bucureşti, DGDS Monographs 10, 2009, http://www.mathem.pub.ro/dgds/mono/dgdsmono.htm.
• A. Balmuş, S. Montaldo and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math. 168 (2008), 201–220.
• A. Balmuş, S. Montaldo and C. Oniciuc, Properties of biharmonic submanifolds in spheres, J. Geom. Symmetry Phys. 17 (2010), 87–102.
• D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics 203, Birkhäuser, Boston, 2002.
• R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J. Math. 130 (2002), 109–123.
• R. Caddeo, S. Montaldo and P. Piu, Biharmonic curves on a surface, Rend. Mat. Appl. (7) 21 (2001), 143–157.
• B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (1996), 117–337.
• B. Y. Chen, Classification of marginally trapped Lorentzian flat surfaces in $\boldsymbol{E}^4_2$ and its application to biharmonic surfaces, J. Math. Anal. Appl. 340 (2008), 861–875.
• F. Dillen and L. Vrancken, $\mathcal{C}$-totally real submanifolds of $\boldsymbol{S}^7(1)$ with non-negative sectional curvature, Math. J. Okayama Univ. 31 (1989), 227–242.
• I. Dimitric, Submanifolds of $\boldsymbol{E}^m$ with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica 20 (1992), 53–65.
• J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.
• D. Fetcu, E. Loubeau, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of $\boldsymbol{C}P^n$, Math. Z. 266 (2010), 505–531.
• D. Fetcu and C. Oniciuc, Explicit formulas for biharmonic submanifolds in non-Euclidean 3-spheres, Abh. Math. Sem. Univ. Hamburg 77 (2007), 179–190.
• D. Fetcu and C. Oniciuc, Explicit formulas for biharmonic submanifolds in Sasakian space forms, Pacific J. Math. 240 (2009), 85–107.
• D. Fetcu and C. Oniciuc, Biharmonic hypersurfaces in Sasakian space forms, Differential Geom. Appl. 27 (2009), 713–722.
• D. Fetcu and C. Oniciuc, A note on integral $\mathcal{C}$-parallel submanifolds in $\boldsymbol{S}^7(c)$, Rev. Un. Mat. Argentina 52 (2011), 33–45.
• T. Ichiyama, J. Inoguchi and H. Urakawa, Bi-harmonic maps and bi-Yang-Mills fields, Note Mat. 28 (2008), 233–275.
• J. Inoguchi, Submanifolds with harmonic mean curvature in contact 3-manifolds, Colloq. Math. 100 (2004), 163–179.
• G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A7 (1986), 389–402.
• S. Montaldo and C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47 (2006), 1–22.
• H. Naitoh, Parallel submanifolds of complex space forms I, Nagoya Math. J. 90 (1983), 85–117.
• Y. L. Ou, On conformal biharmonic immersions, Ann. Global Anal. Geom. 36 (2009), 133–142.
• Y. L. Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. Math. 248 (2010), 217–232.
• S. Tanno, Sasakian manifolds with constant $\varphi$-holomorphic sectional curvature, Tôhoku Math. J. 21 (1969), 501–507.
• T. Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors, Publ. Math. Debrecen 67 (2005), 285–303.
• K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics 3, World Scientific Publishing Co., Singapore, 1984.
• W. Zhang, New examples of biharmonic submanifolds in $\boldsymbol{C}P^n$ and $\boldsymbol{S}^{2n+1}$, An. \c Stiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 57 (2011), 207–218.