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arXiv · 2606.29317

Three-Dimensional Real Affine Lie Groups

T. Aït Aissa·S. El Bourkadi·M. W. Mansouri
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Abstract

We classify all left-invariant real affine connections in dimension three. Our approach reduces the three-dimensional problem to a two-dimensional one by decomposing each left-invariant affine connection into a two-dimensional part and an additional one-dimensional component. After characterizing all possible two-dimensional left-invariant affine connections, we return to the three-dimensional setting to obtain a simplified description of all three-dimensional left-invariant affine connections. We then explicitly solve the resulting simplified quadratic equations and perform a refined analysis up to isomorphism, leading to a complete classification. Furthermore, we determine several geometric and algebraic properties of these structures, including the Novikov, associative, radiant, and bi-symmetric conditions, as well as geodesic completeness.

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T. Aït Aissa, S. El Bourkadi, M. W. Mansouri. 2026-06-28. Three-Dimensional Real Affine Lie Groups. https://arxiv.org/abs/2606.29317

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