Geometric Hydrodynamics in Open Problems — Problem 9

Definitions and mathematical statement

One conventional interpretation of the notation is the following: if $M$ is a smooth manifold equipped with a smooth volume form $\mu$, then \[ \mathscr{D}_{\mu}(M)=\{\varphi\in\operatorname{Diff}(M):\varphi^{*}\mu=\mu\}, \] where $\operatorname{Diff}(M)$ denotes the group of smooth diffeomorphisms of $M$. This is a conditional explanation of the notation; the supplied record does not specify the manifold, volume structure, regularity, group component, or boundary conditions intended.

For a Riemannian manifold $(X,g)$, local symmetry means that each point $p$ admits a local isometry $s_p$ fixing $p$ with differential $({\,\mathrm{d}} s_p)_p=-\operatorname{Id}_{T_pX}$. Applying this definition in an infinite-dimensional setting requires a specified manifold structure and Riemannian metric.

Source question (Khesin, Misiolek, Modin, 2022, Problem 9). How "far" is $\mathscr{D}_{\mu}(M)$ from being an infinite-dimensional (locally) symmetric Riemannian space?

The supplied question is qualitative: it specifies neither a Riemannian metric on $\mathscr{D}_{\mu}(M)$ nor a meaning of "far." Consequently, it does not determine a numerical quantity, a quantified assertion, or a unique criterion to establish. A precise formulation requires these choices and clarification of whether local symmetry, global symmetry, or both are intended.

Short English statement

The question asks how the volume-preserving diffeomorphism group differs from an infinite-dimensional symmetric or locally symmetric Riemannian space. The supplied excerpt leaves the metric, underlying geometric hypotheses, and meaning of "far" unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $800810$, source code catalogue entry, title: Geometric Hydrodynamics in Open Problems --- Problem 9. The registry attributes the problem to Khesin, Misiolek, and Modin, with proposed year 2022, and labels it open; this is a supplied status label, not an independently verified assessment.
• Khesin, Misiolek, Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 9. Supplied source URL: https://arxiv.org/abs/2205.01143. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these provenance claims have not been independently verified.