Related papers: A Geometry for Second-Order PDEs and their Integra…

Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. Hodge decomposition of the incompressible Euler's equation in terms of 1-forms yields a coupled PDE-ODE system. The $L^2$-orthogonal components are a…

In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…

We show that torsion-free four-dimensional $\mathrm{GL}(2)$-structures are flat up to a coframe transformation with a mapping taking values in a certain subgroup $H\subset\mathrm{SL}(4,\mathbb{R})$ which is isomorphic to a semidirect…

In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4-symmetric spaces. We first show that a 4-symmetric space $G/G_0$ can be embedded into the twistor space of the corresponding…

We show that Hertling-Manin F-manifolds provide the appropriate theoretical framework for studying the integrability of quasilinear systems of first-order evolutionary partial differential equations of the form ${\bf u}_t=X\circ {\bf u}_x$…

In this text we expound recent results by Idrisse Khemar on the construction of various geometric completely integrable systems generalizing the structure of Hamiltonian stationary Lagrangian surfaces (HSLS) discovered by F. H\'elein and P.…

For closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold M, where G-structures are considered as sections of the quotient bundle O(M)/G. Then, we deduce…

We study the fillability (or embeddability) of 3-dimensional $CR$ structures under the geometric flows. Suppose we can solve a certain second order equation for the geometric quantity associated to the flow. Then we prove that if the…

It is shown that the kinematic system describing planar non-steady motions of ideal fibre-reinforced fluids may be reduced to a single two-dimensional third-order partial differential equation in which time enters parametrically. A…

We give a topological interpretation of the free metabelian group, following the plan described in [Ver1,Ver2]. This interpretation is based on considering the Caley graph of a finitely generated group G as one-dimensional complex; its…

We classify the effective and transitive actions of a Lie group $G$ on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that $G$ is a closed, connected Lie subgroup of…

Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose controlling pair of conformal structures tends…

Paraconformal or $GL(2)$ geometry on an $n$-dimensional manifold $M$ is defined by a field of rational normal curves of degree $n-1$ in the projectivised cotangent bundle $\mathbb{P} T^*M$. Such geometry is known to arise on solution spaces…

Integrability condition of Hamiltonian perturbations of integrable Hamiltonian PDEs of hydrodynamic type up to the second order approximation is considered.

This paper analyzes two classes of second order level set PDE in periodic media in the parabolic scaling. First, we study fully nonlinear geometric operators under general assumptions in dimension $d = 2$ and prove that the associated…

We solve explicitly the geodesic equation for a wide class of (pseudo)-Riemannian homogeneous manifolds (G/H,m), including those with G compact, as well as non-compact semisimple Lie groups, under a simple algebraic condition for the metric…

For every group $G$, we introduce the set of hyperbolic structures on $G$, denoted $\mathcal{H}(G)$, which consists of equivalence classes of (possibly infinite) generating sets of $G$ such that the corresponding Cayley graph is hyperbolic;…

Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a…

We describe the $10$-dimensional space of $Sp(2)$-invariant $G_2$-structures on the homogeneous $7$-sphere $S^7=Sp(2)/Sp(1)$ as $\mathbb{R}^+\times Gl^+(3,\mathbb{R})$. In those terms, we formulate a general Ansatz for $G_2$-structures,…

This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are…