Geometric Invariant Theory for Homogeneous Spaces

Definitions and mathematical statement

A smooth homogeneous space is a smooth manifold $M$ equipped with a transitive smooth action of a Lie group $G$; choosing a point identifies $M$ with $G/H$, where $H$ is its closed stabilizer subgroup. A homogeneous Riemannian space additionally carries a $G$-invariant Riemannian metric. These definitions describe the differential-geometric setting suggested by the workshop context; the supplied question does not explicitly restrict its homogeneous spaces to this setting.

Geometric invariant theory studies algebraic group actions, their invariants, orbit structure, and quotient constructions. The source specifies no acting algebraic group, representation or variety, base field, or stability notion.

Research question. What additional tools from geometric invariant theory can be applied to obtain new information about the geometry of homogeneous spaces?

This is an open-ended research request, not a specified universally quantified conjecture. The source does not identify the previously applied tools relative to which "additional" is measured, the class of homogeneous spaces under consideration, or the geometric conclusions sought. The workshop context mentions Einstein metrics, Ricci solitons, and Ricci flow under symmetry assumptions, but supplies no restriction of the question to any one of these topics. Consequently, a more specific theorem statement or criterion for a complete answer cannot be extracted without additional information.

Short English statement

Identify further applications of geometric invariant theory that yield geometric information about homogeneous spaces. The supplied record leaves both the tools and the desired conclusions unspecified, so this remains a research direction requiring clarification rather than a precise conjecture.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002696$, titled Geometric Invariant Theory for Homogeneous Spaces, in the Oberwolfach Reports Open Problems collection; exact source problem code: catalogue entry. 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 literature-source label is "Oberwolfach Report 45/2014." The citation's parenthetical year 2015 differs from the report identifier and proposed year 2014; this discrepancy is retained unresolved.
• Supplied literature reference: Lauret, Einstein Solvmanifolds are Standard. https://annals.math.princeton.edu/2010/172-3/p09.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that geometric invariant theory has yielded major structural results for Einstein solvmanifolds and homogeneous solitons, while the request for further tools and new geometric consequences has no complete answer. This is supplied dated curation derived from the OWR material, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.