Abstract:This paper investigates the algebraic and geometric consequences of the associativity of the symmetric part $U$ of the Levi-Civita connection on a pseudo-Riemannian Lie algebra $(\mathfrak{g}, \langle \cdot, \cdot \rangle)$. We demonstrate that in the Riemannian (positive-definite) setting, the associativity of $U$ is an extremely restrictive condition that forces the tensor to vanish identically, thereby recovering the class of bi-invariant metrics. In contrast, in the pseudo-Riemannian setting, we focus on the subclass where $(\mathfrak{g}, U)$ is associative and unimodular. As a primary result, we establish that every connected Lie group endowed with a left-invariant pseudo-Riemannian metric whose $U$-tensor is associative and unimodular is geodesically complete. Finally, we explore the families of 2-step nilpotent and almost-abelian Lie algebras. For the latter, we obtain some rigid structural classifications, showing that the paradigmatic models are the $3$-dimensional Heisenberg algebra with certain (anti)-Lorentzian metrics or a semi-direct extension involving a nondegenerate infinitesimal $\beta$-transformation on the canonical neutral space $W \oplus W^*$.
| Comments: | All comments or suggestions are most welcome. 14 pages |
| Subjects: | Differential Geometry (math.DG); Mathematical Physics (math-ph) |
| MSC classes: | 53C50 |
| Cite as: | arXiv:2605.24204 [math.DG] |
| (or arXiv:2605.24204v1 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24204 arXiv-issued DOI via DataCite (pending registration) |
From: Santiago Castañeda Montoya [view email]
[v1]
Fri, 22 May 2026 20:45:18 UTC (17 KB)
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