Negative Ricci Curvature on Solvable Lie Algebras

Definitions and mathematical statement

The supplied record does not specify the ground field or dimension of the Lie algebra. The following definitions describe the usual finite-dimensional real Riemannian interpretation; this interpretation requires confirmation from the source.

For a finite-dimensional real solvable Lie algebra $\mathfrak g$, solvability means that the derived series $\mathfrak g^{(0)}=\mathfrak g$, $\mathfrak g^{(k+1)}=[\mathfrak g^{(k)},\mathfrak g^{(k)}]$ eventually vanishes. Its nilradical $\mathfrak n$ is its largest nilpotent ideal. Here a Lie algebra $\mathfrak h$ is nilpotent if its lower central series $\mathfrak h_1=\mathfrak h$, $\mathfrak h_{k+1}=[\mathfrak h,\mathfrak h_k]$ eventually vanishes. For $X\in\mathfrak g$, define $\operatorname{ad}X(Y)=[X,Y]$. Since $\mathfrak n$ is an ideal, the restriction $D_X=\left.\operatorname{ad}X\right|_{\mathfrak n}$ is an endomorphism and a derivation of $\mathfrak n$.

A positive-definite inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$ determines a left-invariant Riemannian metric on its simply connected Lie group. Write $\operatorname{Ric}_{\langle\cdot,\cdot\rangle}$ for its Ricci tensor at the identity. Strict negativity means \[ \operatorname{Ric}_{\langle\cdot,\cdot\rangle}(v,v)<0 \qquad\text{for every }v\in\mathfrak g\setminus\{0\}. \]

The supplied question asks whether, for every solvable Lie algebra $\mathfrak g$ with nilradical $\mathfrak n$, there is an equivalence of the following form: \[ \begin{aligned} &\exists\text{ a positive-definite inner product on }\mathfrak g \text{ with }\operatorname{Ric}<0\\ &\quad\Longleftrightarrow\quad \exists X\in\mathfrak g\setminus\mathfrak n \text{ such that }D_X\text{ satisfies linear inequalities}\\ &\hspace{7.8em}\text{determined solely by }\mathfrak n. \end{aligned} \] The intended characterization must therefore use inequalities depending only on the nilradical and work uniformly for solvable Lie algebras having that nilradical. However, the record gives neither the inequalities nor the quantities in which they are linear. In particular, it does not specify whether they concern matrix entries, eigenvalues, real parts of eigenvalues, or other data associated with $D_X$; nor does it specify strictness, finiteness, or invariance under changes of basis. Thus the displayed equivalence records the proposed characterization schematically, rather than defining a fully specified mathematical predicate. These missing specifications must be resolved before a precise conjecture can be stated.

No hypothesis $\mathfrak g\ne\mathfrak n$ is supplied. If $\mathfrak g=\mathfrak n$, the existential condition on the right has an empty domain; the draft retains this boundary case without adding an exclusion.

Short English statement

The question asks whether a solvable Lie algebra admits a positive-definite inner product with strictly negative Ricci curvature exactly when some element outside its nilradical induces a derivation satisfying linear inequalities depending only on that nilradical. The inequalities and their variables are unspecified, and the usual finite-dimensional real setting needs confirmation. The supplied label is partially solved; this is dated source curation, not an independently verified mathematical assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002699$, exact source problem code catalogue entry, title: Negative Ricci Curvature on Solvable Lie Algebras. Project registry locations: data/unsolvedmath/problems.json and catalog/problems.json. The supplied record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is identified.
• Supplied report 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 Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The record lists proposed year 2014, while its report title labels the workshop 2015; this discrepancy is unresolved.
• Supplied literature reference: Solvable Lie Groups of Negative Ricci Curvature. https://doi.org/10.48550/arXiv.1312.6803.
• Supplied literature reference: Solvable Extensions of Filiform Lie Groups. https://doi.org/10.1002/mana.201500019.
• Supplied literature reference: On Ricci Negative Lie Groups. https://doi.org/10.48550/arXiv.1912.06204.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that necessary and sufficient criteria are known for abelian, Heisenberg, and several filiform nilradicals, that moment-map convex cones describe broad families, and that the proposed characterization for arbitrary nilradicals remains open. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent checking.
• The registry supplies the statement label corrected_verified and reports a manual semantic, typographic, and LaTeX review describing the cleanup as meaning-preserving and self-contained. These are supplied provenance claims, not independent verification of the mathematical formulation. No cited source was visited for this draft, and no exact problem page or passage is supplied within the report's page range.