Biharmonic Curves in Solvable Lie Group

Year 2025, Volume: 18 Issue: 2, 230 - 242

Ji-eun Lee

https://doi.org/10.36890/iejg.1660936

Abstract

In this paper, we study biharmonic curves in the solvable Lie group $G(c_1,c_2)$ with Lorentzian metric. In particular, we find biharmonic almost Legendre curves in the solvable Lie group $G(c_1,c_2)$ with Lorentzian metric.

Keywords

Almost Legendre curve , biharmonic curve , para-$\beta$-Kenmotau manifold , solvable Lie group

References

• Chen, B. Y., Ishikawa, S.: Biharmonic surfaces in pseudo-Euclidean spaces. Mem. Fac. Sci. Kyushu Univ. Ser. A 45(2) , 323–347 (1991).
• Cho, J.T., Inoguchi, J., Lee, J.-E.: Biharmonic curves in 3-dimensional Sasakian space form. Ann. Math. Pura Appl. 186 , 685-701 (2007).
• Hawking, S.W., Ellis, G. F. R. : The large scale structure of space-time. Cambridge Monographs on Mathematical Physics 1. London: Cambridge University Press, 1973.
• Inoguchi, J.: Submanifolds with harmonic mean curvature vector field in contact 3-manifolds. Coll. Math. 100, 163-179 (2004).
• Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups. Chinese Ann. Math. B. 24, 73-84 (2003).
• Inoguchi, J.: Minimal surfaces in 3-dimensional solvable Lie groups II. Bull. Austral. Math. Soc. 73, 365-374 (2006).
• Inoguchi, J.: Biharmonic curves in Minkowki 3-space. Int. J. Math. Math. Sci. 2003(21), 1365-1368 (2003).
• Inoguchi, J., Lee, S.: A Weierstrass representation for minimal surfaces in Sol. Proc. Amer. Math. Soc. 136, 2209-2216 (2008).
• Inoguchi, J., Lee, J. E.: Slant curves in 3-dimensional almost contact metric geometry. Int. Electron. J. Geom. 8(2), 106–146 (2015).
• Lee, J. E.: Biharmonic spacelike curves in Lorentzian Heigenberg space. Commun. Korean Math. Soc., 33(4) , 1309–1320 (2018).
• Lee, J. E.: Biharmonic curves in 3-dimensional Lorentzian Sasakian space forms. Commun. Korean Math. Soc., 35(3) , 967–977 (2020).
• Lee, J. E.: Pointwise slant curves in pseudo-Hermitian geometry,.Mediterr. J. math. 19, Article number 130 (2022).
• Sasahara, T.: Biharmonic submanifolds in normalflat Lorentz 3-space forms. Bull. Aust. Math. Soc. 85, 422–432 (2012).
• Tricerri, F., Vanhecke, L.: Homogeneous Structures on Riemannian Manifolds. Lecture Notes Series, London Math. Soc. 52, Cambridge Univ. Press (1983).
• Wełyczko, J.: Slant curves in 3-dimensional normal almost paracontact metric manifolds. Mediterr. J. math. 11, 965–978 (2014).
• Wełyczko, J.: On Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Result. Math. 54, 377-387 (2009).