We show that a hyperquadric M in \( \mathbb{R}_2^4 \) is a Lie group by using the bicomplex number product. For our purpose, we change the definition of tensor product. We define a new tensor product by considering the tensor product surface in the hyperquadric M. By using this new tensor product, we classify totally real tensor product surfaces and complex tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. By means of the tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve, we determine a special subgroup of the Lie group M. Thus, we obtain the Lie group structure of tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. Moreover, we obtain left invariant vector fields of these Lie groups. We consider the left invariant vector fields on these groups, which constitute a pseudo-Hermitian structure. By using this, we characterize these Lie groups as totally real or slant in \( \mathbb{R}_2^4 \).
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References
K. Arslan, R. Ezentas, I. Mihai, C. Murathan, and C. Özgür, “Tensor product surfaces of a Euclidean space curve and a Euclidean plane curve,” Beitr. Alg. Geom., 42, No. 2, 523–530 (2001).
F. Brickell and R. S. Clark, Differentiable Manifolds, Van Nostrand Reinhold, London (1970).
B. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit, Leuven (1990).
F. Decruyenaere, F. Dillen, L. Verstraelen, and L. Vrancken, “The semiring of immersions of manifolds,” Beitr. Alg. Geom., 34, 209–215 (1993).
K. Ilarslan and E. Nesovic, “Tensor product surfaces of a Lorentzian space curve and a Euclidean plane curve,” Kuwait J. Sci. Eng., 34, No. 2A, 41–55 (2007).
K. Ilarslan and E. Nesovic, “Tensor product surfaces of a Euclidean space curve and a Lorentzian plane curve,” Different. Geom. Dynam. Syst., 9, 47–57 (2007).
A. Karger and J. Novak, Space Kinematics and Lie Groups, Gordon and Breach (1985).
I. Mihai, R. Rosca, L. Verstraelen, and L. Vrancken, “Tensor product surfaces of Euclidean planar curves,” Rend. Sem. Mat. Messina. Ser. II, 18, No. 3, 173–185 (1993).
I. Mihai, I. Van de Woestyne, L. Verstraelen, and J. Walrave, “Tensor product surfaces of Lorentzian planar curves,” Bull. Inst. Math. Acad. Sinica, 23, 357–363 (1995).
I. Mihai, I. Van de Woestyne, L. Verstraelen, and J. Walrave, “Tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve,” Rend. Sem. Mat. Messina. Ser. II, 3, No. 18, 147–185 (1994–1995).
B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York (1983).
S. Özkaldı and Y. Yaylı, “Tensor product surfaces and Lie groups,” Bull. Malays. Math. Sci. Soc. (2), 33, No. 1, 69–77 (2010).
G. B. Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker (1990).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 3, pp. 307–317, March, 2012.
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Karakuş, S.Ö., Yayli, Y. Bicomplex number and tensor product surfaces in \( \mathbb{R}_2^4 \) . Ukr Math J 64, 344–355 (2012). https://doi.org/10.1007/s11253-012-0651-z
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DOI: https://doi.org/10.1007/s11253-012-0651-z