Tensor Product Surfaces in R4 and Lie Groups

In this paper, we show that a hyperquadric M in R4 is a Lie group by using bicomplex number product. By means of the tensor product surfaces of Euclidean planar curves, we determine some special subgroup of this Lie group M. Thus, we obtain Lie group structure of tensor product surfaces of Euclidean planar curves. Moreover, we obtain left invariant vector fields of these Lie groups. We identify R4 with C2 and consider the left invariant vector fields on these group which constitute complex structure. By means of these, we characterize these Lie groups as totally real, complex or slant in R4 .