Invariant complex structures on the Euclidean motion group as deformations of Calabi-Eckmann structures

We classify, up to equivalence, all left-invariant complex structures on the three-dimensional Euclidean rigid motion group $SE(3)\mathrm{.}$ Our approach is based on the correspondence between complex structures on a real Lie algebra $\mathfrak{e}(3)$ of the motion group and complex Lie subalgebras of the complexification of $\mathfrak{e}(3)\mathrm{.}$ The resulting structures form a one-parameter family $J_{\phi }\mathrm{.}$ We further establish a connection with the classical Calabi-Eckmann complex structures on $S^3\times S^3\mathrm{.}$ More precisely, we construct a one-parameter deformation of Lie brackets connecting $\mathfrak{s}\mathfrak{o}(3)\oplus \mathfrak{s}\mathfrak{o}(3)$ to $\mathfrak{e}(3),$ for which each endomorphism $J_{\phi }$ remains integrable. Consequently, every left-invariant complex structure on $SE(3)$ arises as a contraction limit of Calabi-Eckmann structures. The same family of endomorphisms also defines complex structures on $\mathfrak{s}\mathfrak{o}(3,1)\mathrm{.}$