Invariant complex structures on the Euclidean motion group as deformations of Calabi-Eckmann structures
Abstract
We classify, up to equivalence, all left-invariant complex structures on the three-dimensional Euclidean rigid motion group SE(3). Our approach is based on the correspondence between complex structures on a real Lie algebra e(3) of the motion group and complex Lie subalgebras of the complexification of e(3). The resulting structures form a one-parameter family Jϕ. We further establish a connection with the classical Calabi-Eckmann complex structures on S3× S3. More precisely, we construct a one-parameter deformation of Lie brackets connecting so(3)⊕ so(3) to e(3), for which each endomorphism Jϕ 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 so(3,1).
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BibTeX
@misc{bokan2026invariant,
title = {{Invariant complex structures on the Euclidean motion group as deformations of Calabi-Eckmann structures}},
author = {Neda Bokan and Srdjan Vukmirovic},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20232},
eprint = {2610.20232},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20232},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Invariant complex structures on the Euclidean motion group as deformations of Calabi-Eckmann structures
AU - Neda Bokan
AU - Srdjan Vukmirovic
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.20232
UR - https://arxiv.org/abs/2610.20232
AB - We classify, up to equivalence, all left-invariant complex structures on the three-dimensional Euclidean rigid motion group SE(3). Our approach is based on the correspondence between complex structures on a real Lie algebra e(3) of the motion group and complex Lie subalgebras of the complexification of e(3). The resulting structures form a one-parameter family J_ϕ. We further establish a connection with the classical Calabi-Eckmann complex structures on S^3× S^3. More precisely, we construct a one-parameter deformation of Lie brackets connecting so(3)⊕ so(3) to e(3), for which each endomorphism J_ϕ 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 so(3,1).
ER -