Canonical Left-Invariant Metrics on Lie Groups

Definitions and mathematical statement

Let $G$ be a Lie group with identity $e$, and write $\mathfrak g=T_eG$ for its Lie algebra. For $a\in G$, let $L_a:G\to G$ denote left translation, $L_a(x)=ax$. A Riemannian metric $g$ on $G$ is left-invariant if \[ L_a^*g=g\qquad\text{for every }a\in G. \] Write $\mathcal M_{\mathrm{left}}(G)$ for the set of these metrics. Each positive-definite inner product $h$ on $\mathfrak g$ determines a unique member of this set by \[ g_x(u,v)=h\left(({\,\mathrm{d}} L_{x^{-1}})_x u,({\,\mathrm{d}} L_{x^{-1}})_x v\right), \qquad x\in G,\quad u,v\in T_xG. \] The supplied question asks whether a given $G$ admits a best metric in $\mathcal M_{\mathrm{left}}(G)$, and, if so, how the geometry of that metric is related to the Lie-group structure of $G$.

Unresolved formulation. The source does not define "best": it supplies no optimization functional, ordering, distinguished geometric property, normalization, or equivalence relation. Consequently, it does not determine a predicate $B_G$ for which the intended existence question could be written as \[ \exists g\in\mathcal M_{\mathrm{left}}(G)\quad B_G(g). \] This display records the missing predicate, not a completed mathematical assertion. The source also does not specify whether the intended target concerns every Lie group or a restricted class, or which relationship between metric geometry and group structure is sought. No connectedness, simple connectedness, compactness, or solvability assumption is supplied. A precise theorem-level question therefore remains dependent on additional source clarification; none of these choices is imposed here.

Short English statement

The problem asks whether a Lie group has a distinguished left-invariant Riemannian metric and how its geometry reflects the group structure. Since the source leaves "best" and the desired relationship undefined, it describes a research question rather than a single precise proposition.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002693$, title "Canonical Left-Invariant Metrics on Lie Groups," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2014.
• Supplied source citation: Mini-Workshop: Einstein Metrics, Ricci Solitons and Ricci Flow under Symmetry Assumptions (2015), Oberwolfach Reports, pp. 2529--2568, DOI 10.4171/owr/2014/45. https://doi.org/10.4171/owr/2014/45. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: "Oberwolfach Report 45/2014." https://doi.org/10.4171/owr/2014/45.
• Supplied literature source: "Infinitesimal Maximal Symmetry and Ricci Solitons." https://arxiv.org/abs/2112.09733.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and states that "best metric" is insufficiently defined to give a single theorem. It also reports uniqueness and maximal-symmetry results for selected solvable Ricci-soliton classes, while reporting no established general canonical metric principle. The supplied background explicitly calls this dated literature triage and requests independent verification of time-sensitive status and cited claims.