Geometric Hydrodynamics in Open Problems — Problem 6

Definitions and mathematical statement

The source asks for a physical or fluid model realizing an $L^2$ left-invariant metric on the volume-preserving diffeomorphism group $\mathscr{D}_\mu(M)$. The geometry of $M$, boundary conditions, and regularity class are not specified. The following describes the metric condition conditionally, without selecting those missing hypotheses.

Given a smooth Riemannian manifold $(M,g)$ and a smooth positive volume density $\mu$, write \[ \mathscr{D}_\mu(M)=\{\varphi:M\to M:\ \varphi\text{ is a diffeomorphism and }\varphi^*\mu=\mu\}, \] with whatever regularity and boundary restrictions are intended. At the identity, the $L^2$ inner product on admissible divergence-free vector fields is \[ \langle u,v\rangle_{L^2}=\int_M g_x(u(x),v(x))\,{\,\mathrm{d}}\mu(x), \] provided the integral is finite. Its left-invariant extension is characterized by \[ G_\varphi(U,V)=\int_M g_x\!\left((D\varphi_x)^{-1}U(x),(D\varphi_x)^{-1}V(x)\right)\,{\,\mathrm{d}}\mu(x), \] where $U,V\in T_\varphi\mathscr{D}_\mu(M)$ are vector fields along $\varphi$. Equivalently, with $L_\eta(\varphi)=\eta\circ\varphi$, left invariance means \[ G_{\eta\circ\varphi}\left(T_\varphi L_\eta U,T_\varphi L_\eta V\right)=G_\varphi(U,V) \] for every admissible $\eta,\varphi,U,V$.

Problem. Describe a physical or fluid model whose configuration space is $\mathscr{D}_\mu(M)$ and whose kinetic-energy metric has this $L^2$ left-invariant form. Identify physically interesting systems exhibiting this invariance. The supplied text does not specify a criterion for an acceptable model, nor whether the request concerns one choice of $M$ or a general construction.

The supplied context also requests investigation of an unspecified preceding system for generic $S$ and $\rho(\omega)$, and the definition and study of a model with configuration space $SL(2)$, described as more "liquid-like." It states that in "all other cases" the preceding metric is neither left- nor right-invariant. These contextual requests cannot be formulated precisely from this record: the preceding system, the exceptional cases, $S$, $\rho$, $\omega$, the meaning of genericity, and the scalar field of $SL(2)$ are not supplied.

Short English statement

Find a physically meaningful model on the group of volume-preserving diffeomorphisms whose kinetic energy comes from an $L^2$ inner product transported by left translations. The source also suggests related investigations involving unspecified parameters and $SL(2)$. The geometric setting, boundary conditions, and criteria for a physical model remain unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $800807$, exact source problem code catalogue entry, title Geometric Hydrodynamics in Open Problems --- Problem 6. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Khesin, Misiolek, and Modin, Geometric Hydrodynamics in Open Problems (2022), Problem 6. Supplied source URL: https://arxiv.org/abs/2205.01143. The registry attributes the problem to these authors and gives proposed year 2022.
• Supplied provenance reports access on 2026-10-16 and extraction by arxiv_latex. These are source-record provenance claims; no independent source visit or verification is asserted here. The registry labels the problem open; this label is supplied metadata, not an independently verified literature assessment.