Differential Geometry Using Quaternions

Yıl 2024, Cilt: 17 Sayı: 2, 700 - 711

Sergio Giardino

https://doi.org/10.36890/iejg.1362006

Öz

This paper establishes the basis of the quaternionic differential geometry (HDG) initiated in a previous article.
The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and torsion concepts, differential forms, directional derivatives and the structural equations. The analogy between the quaternionic and the real geometries were obtained using a matrix representation of quaternions. The results evidences the quaternionic formalism as a suitable language to differential geometry that can be useful in various directions of future investigation.

Anahtar Kelimeler

Curves in Euclidean space, other special differential geometries, quaternion and other division algebras

Kaynakça

• [1] Giardino, S.: A primer on the differential geometry of quaternionic curves. Math. Methods Appl. Sci. 44 (18):14428–14436 (2021). https://doi.org/10.1002/mma.7709
• [2] Bharathi, K., Nagaraj, M.: Quaternion valued function of a real Variable Serret-Frenet formulae. Indian J. Pure Appl. Math. 18, 507–511 (1987).
• [3] Sivridag, A. I., Gunes, R., Keles, S.: The Serret-Frenet formulae for dual quaternion-valued functions of a single real variable. Mech. Mach. Theor. 29(5), 749–754 (1994). https://doi.org/10.1016/0094-114X(94)90116-3
• [4] Girard, P. R., Clarysse, P., Pujol, R., Wang, L., Delachartre, P.: Differential Geometry Revisited by Biquaternion Clifford Algebra. In: J. D. Boissonnat et al. (eds) Curves and Surfaces 2014. Lecture Notes in Computer Science 9213 Springer, Cham(2):47–64 (2015).
• [5] Aksoyak, F. K.: A new type of quaternionic frame in R4. Int. J. Geom. Meth. Mod. Phys. 16(6), 1959984 (2019). https://doi.org/10.1142/S0219887819500841
• [6] Coken, A. C., Tuna, A.: On the quaternionic inclined curves in the semi-Euclidean space E4 2”. Appl. Math. Comput. A155(2), 373–389 (2004). https://doi.org/10.1016/S0096-3003(03)00783-5
• [7] Gök, I., Okuyucu, O. Z., Kahraman, F., H. H. Hacisalihoglu, H. H.: On the quaternionic B2-slant helices in the Euclidean space E4. Adv. Appl. Clifford Algebras, 21, 707–719,(2011). https://doi.org/10.1007/s00006-011-0284-6
• [8] Gungor, m. A., Tosun, M.: Some characterizations of quaternionic rectifying curves. Differ. Geom. Dyn. Syst. 13, 89–100 (2011).
• [9] Kecilioglu, O., Ilarslan, K.: Quaternionic Bertrand curves in Euclidian 4−space. Bull. Math. Anal. Appl. 5(3), 27–38 (2013).
• [10] Bektas, O., Gurses, N. B., Yuce, S.: Osculating Spheres of a Semi Real Quaternionic Curve in E4 2 . Eur. J. Pure and Appl. Math. 7(1), 86–96 (2014). [11] Bektas, O., Gurses, N., Yuce, A.: Quaternionic osculating curves in Euclidean and semi-Euclidean space. J. Dyn. Sys. Geom. Theor. 14(1), 65–84 (2016). https://doi.org/10.1080/1726037X.2016.1177935
• [12] Ozturk, G., Kisi, I., Buyukkutuk, S.: Constant ratio quaternionic curves in Euclidean spaces. Adv. Appl. Clifford Algebras, 27:1659–1673 (2017). https://doi.org/10.1007/s00006-016-0716-4
• [13] Coken, A. C., Tuna Aksoy, A.: Null quaternionic Cartan helices in R3 v. Acta. Phys. Pol. A132(3-II), 896–899 (2017). https://10.12693/APhysPolA.132.896
• [14] Karadag, M., Sivridag, A. I.: Some characterizations for a quaternion-valued and dual variable curve. Symmetry, 11(2), 125 (2019). https://doi.org/10.3390/sym11020125
• [15] Kizilay, A., Yildiz, O. G., Okuyucu, O. Z.: Evolution of quaternionic curve in the semi-Euclidean space E4 2 . Math. Meth. Appl. Sci. 44(9), 7577-7587 (2021). https://doi.org/10.1002/mma.6374
• [16] Kahraman, T.: Differential equations of null quaternionic curves. Int. J. Appl. Comput. Math. 6(63), 6583–6592 (2020). https://doi.org/10.1007/s40819-020-00824-3
• [17] Soyfidan, T., Gungor, M. A.: On the quaternionic involute-evolute curves. Preprint arXiv:1311.0621[math.GT] (2013).
• [18] Hanif, M., Önder, M.: Generalized quaternionic involute-evolute curves in the Euclidean four-space E4. Math. Meth. Appl. Sci. 43(7), 4769–4780 (2020). https://doi.org/10.1002/mma.6231
• [19] Senyurt, S., Cevahir, C., Altun, Y,: On spatial quaternionic involute curve: a new view. Adv. Appl. Clifford Algebras, 18, 1815–1824 (2017). https://doi.org/10.1007/s00006-016-0669-7
• [20] Senyurt, S., Cevahir, C., Altun, Y,: On the Smarandache curves of spatial quaternionic involute curve. Proc. Natl. Acad. Sci. India A Phys. Sci. 1815–1824 (2019). https://doi.org/10.1007/s40010-019-00640-5
• [21] Hanif, M., Hou, Z. H.: Generalized involute and evolute curve-couple in Euclidean space. Int. J. Open Problems Compt. Math. 11(2), 28–39 (2018).
• [22] Aslan, S., Yayli, Y.: Split quaternions and canal surfaces in Minkowski 3−space. Int. J. Geom. 5(2), 51–61 (2016).
• [23] Aslan, S., Yayli, Y.: Canal surfaces with quaternions. Adv. Appl. Clifford Algebras, 26(2), 31–38 (2016). https://doi.org/10.1007/s00006- 015-0602-5
• [24] Aslan, S., Yayli, Y.: Quaternionic shape operator. Adv. Appl. Clifford Algebras, 27(2),2921–2931 (2017). https://doi.org/10.1007/s00006- 017-0804-0
• [25] Gök, I.: Quaternionic approach of canal surfaces constructed by some new ideas. Adv. Appl. Clifford Algebras, 27(2), 1175–1190 (2017). https://doi.org/10.1007/s00006-016-0703-9
• [26] Kocakusakli, E., Tuncer, O., Gök, I., Yayli, Y.: A new representation of canal surfaces with split quaternions in Minkowski 3−Space. Adv. Appl. Clifford Algebras, 27, 1387–1409 (2017). https://doi.org/10.1007/s00006-016-0723-5
• [27] Karakus, S. O.: Quaternionic approach on constant angle surfaces in S2 × R2. Appl. Math. e-not. 19, 497–506 (2019).
• [28] Canakci, Z., Tuncer, O. O., Gök, I., Y. Yayli, Y.: The construction of circular surfaces with quaternions. Asian-Eur. J. Math. 12(1), 1950091 (2019). https://doi.org/10.1142/S1793557119500918
• [29] Aslan, S., Bekar, M., Yayli, Y.: Ruled surfaces constructed by quaternions. J. Geom. Phys. 161,104048 (2021). https://doi.org/10.1016/j.geomphys.2020.104048
• [30] Tuncer, O. O.: Generalized tubes in pseudo-Galilean 3−space: Split semi-quaternionic representations and an application to magnetic flux tubes. Math. Meth. Appl. Sci. 45(3), 1468–1487 (2022). https://doi.org/10.1002/mma.7866
• [31] Berndt, J.: Real hypersurfaces in quaternionic space forms. Journal für die reine und angewandte Mathematik, 419(2), 9–26 (1991). https://doi.org/10.1515/crll.1991.419.9
• [32] Perez, J. D., Suh, Y. J.: Real hypersurfaces of quaternionic projective space satisfying ∇UiR = 0. Diff. Geom. Appl. 7(3), 211–217 (1997). https://doi.org/10.1016/S0926-2245(97)00003-X
• [33] Gentili, G., Gori, A., Sarfatti, G.: On compact affine curves and surfaces. J. Geom. Anal. 31, 1073–1092 (2021). https://doi.org/10.1007/s12220-019-00311-2
• [34] Ward, J. P.: Quaternions and Cayley Numbers. Springer Dordrecht (1997).
• [35] Garling, D. J. H.: Clifford algebras: an introduction. Cambridge Univ. Press (2011).
• [36] Vaz, J., da Rocha, R.: An introduction to Clifford algebras and spinors. Oxford University Press (2016).
• [37] Morais, J. P., Georgiev, S., Sprössig, W.: Real quaternionic calculus handbook. Birkhäuser (2014).
• [38] Hurwitz, A.: Ueber die Composition der quadratischen Formen von belibig vielen Variablen. Nachr. Gesell. Wiss. Göttingen, Math-Phys. Kl. 309-316 (1898).
• [39] Reese Harvey, F.: Spinors and calibrations. Academic Press (1990).
• [40] Gilbert. J. E., M. A. M. Murray, M. A. M.: Clifford algebras and Dirac operators in harmonic analysis. Cambridge Univ. Press (1991).