For two immersions, \(f:M\rightarrow {\mathbb E}^m\), \(h:N\rightarrow {\mathbb E}^n\), into Euclidean spaces, the tensor product immersion, \(f\otimes h:M\times N\rightarrow {\mathbb E}^{mn}\), was introduced and studied by B.-Y. Chen in [Bull. Inst. Math., Acad. Sin. 21, No. 1, 1–34 (1993; Zbl 0829.53052)].
In the present paper the authors study the tensor product immersion of a space curve \(\alpha\) in Lorentzian 3-space, \({\mathbb E}^3_1\), and a Lorentzian plane curve \(\beta\) in \({\mathbb E}^2_1\). They show that such an immersion is a minimal surface in \({\mathbb E}^6_3\) if and only if it is of one of five types. In this classification either \(\alpha\) and \(\beta\) are both lines through the origin or, in the other four cases, they are circles or orthogonal hyperbolas. The paper also studies the questions of \(\alpha\otimes\beta\) being a totally real or a complex immersion with respect to the pseudo-Hermitian structure on \({\mathbb E}^6_3\). In the first case \(\alpha\) lies in a pseudosphere or in a pseudohyperbolic space and \(\beta\) is an orthogonal hyperbola centered at the orign. In the latter case \(\alpha\) must be a line through the origin.