In this paper, we investigate the Möbius geometry of curves in $ \mathbb{R}^4 $. First, using moving frame methods we construct a complete system of Möbius invariants for regular curves in $ \mathbb{R}^4 $ by the isometric invariants. Second, we completely classify the Möbius homogeneous curves in $ \mathbb{R}^4 $ up to a Möbius transformation of $ \mathbb{R}^4 $.
Citation: Tongzhu Li, Ruiyang Lin. Classification of Möbius homogeneous curves in $ \mathbb{R}^4 $[J]. AIMS Mathematics, 2024, 9(8): 23027-23046. doi: 10.3934/math.20241119
In this paper, we investigate the Möbius geometry of curves in $ \mathbb{R}^4 $. First, using moving frame methods we construct a complete system of Möbius invariants for regular curves in $ \mathbb{R}^4 $ by the isometric invariants. Second, we completely classify the Möbius homogeneous curves in $ \mathbb{R}^4 $ up to a Möbius transformation of $ \mathbb{R}^4 $.
| [1] |
Z. Guo, H. Li, Conformal invariants of submanifolds in a Riemannian space and conformal rigidity theorems on Willmore hypersurfaces, J. Geom. Anal., 28 (2018), 2670–2691. https://doi.org/10.1007/s12220-017-9928-7 doi: 10.1007/s12220-017-9928-7
|
| [2] |
T. Z. Li, J. Qing, C. P. Wang, Möbius curvature, Laguerre curvature and Dupin hypersurface, Adv. Math., 311 (2017), 249–294. https://doi.org/10.1016/j.aim.2017.02.024 doi: 10.1016/j.aim.2017.02.024
|
| [3] |
C. P. Wang, Moebius geometry of submanifolds in $S^n$, Manuscripta Math., 96 (1998), 517–534. https://doi.org/10.1007/s002290050080 doi: 10.1007/s002290050080
|
| [4] | U. Hertrich-Jeromin, Introduction to Möbius differential geometry, Cambridge University Press, 2003. https://doi.org/10.1017/CBO9780511546693 |
| [5] | R. Sulanke, Curves of constant curvatures in Möbius geometry, Math. Notebook, 2010. |
| [6] |
M. Magliaro, L. Marri, M. Rigoli, On the geometry of curves and conformal geodesics in the Möbius space, Ann. Glob. Anal. Geom., 40 (2011), 133–165. https://doi.org/10.1007/s10455-011-9250-8 doi: 10.1007/s10455-011-9250-8
|
| [7] |
K. F. Aksoyak, A new type of quaternionic frame in $\mathbb{R}^4$, Int. J. Geom. Methods Mod. Phys., 16 (2019), 1950084. https://doi.org/10.1142/S0219887819500841 doi: 10.1142/S0219887819500841
|
| [8] | Y. Long, Conformal invariants of curves in $3$-dimensional space, Adv. Math., 36 (2007), 693–709. |
| [9] |
Op. Nave, Modification of semi-analytical method applied system of ODE, Mod. Appl. Sci., 14 (2020), 75. https://doi.org/10.5539/mas.v14n6p75 doi: 10.5539/mas.v14n6p75
|
| [10] |
R. Sulanke, Submanifolds of the Möbius space, Ⅱ Frenet formulas and curves of constant curvatures, Math. Nachr., 100 (1981), 235–247. https://doi.org/10.1002/mana.19811000114 doi: 10.1002/mana.19811000114
|
| [11] | A. F. Beardon, The geometry of discrete groups, Springer-Verlag, 1983. https://doi.org/10.1007/978-1-4612-1146-4 |
| [12] | B. O'Neil, Semi-Riemannian geometry, Academic Press, 1983. |
Tongzhu Li, Ruiyang Lin. Classification of Möbius homogeneous curves in $ \mathbb{R}^4 $[J]. AIMS Mathematics, 2024, 9(8): 23027-23046. doi: 10.3934/math.20241119
DownLoad: