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We develop a method for describing invariant Monge-Amp\`ere equations in the sense of V. Lychagin and T. Morimoto (MAE) on a homogeneous contact manifold $N$ of a semisimple Lie group $G$, which is the contactification of the homogeneous symplectic manifold $M = G/H = \mathrm{Ad}_G Z \subset \mathfrak{g}$, where $M$ is the adjoint orbit of a splittable closed element $Z $ of the Lie algebra $\mathfrak{g} = \mathrm{Lie}(G)$. The method is then applied to a ten-dimensional semisimple orbit $M$ of the exceptional Lie group $\mathsf{G}_2$ and a complete list of mutually non-equivalent MAEs on $N$ is obtained.

In this paper we study $3^{\mathrm{rd}}$ order (system of) PDEs in two independent variables $x,y$ and one unknown function $u$ that are invariant with respect to the group of affine transformation $\mathrm{Aff}(3)$ of $\mathbb{R}^3=\{(x,y,u)\}$. After proving their relationship with the Fubini-Pick invariant, we derive the aforementioned PDEs by using a general method introduced in [D.V. Alekseevsky, J. Gutt, G. Manno, and G. Moreno: A general method to construct invariant {PDEs} on homogeneous manifolds. Communications in Contemporary Mathematics (2021)], which sheds light on some of their geometrical properties., Comment: 16 pages, 1 figure

For any second-order scalar PDE $\mathcal{E}$ in one unknown function, that we interpret as a hypersurface of a second-order jet space $J^2$, we construct, by means of the characteristics of $\mathcal{E}$, a sub-bundle of the contact distribution of the underlying contact manifold $J^1$, consisting of conic varieties. We call it the contact cone structure associated with $\mathcal{E}$. We then focus on symplectic Monge-Amp\`ere equations in 3 independent variables, that are naturally parametrized by a 13-dimensional real projective space. If we pass to the field of complex numbers $\mathbb{C}$, this projective space turns out to be the projectivization of the 14-dimensional irreducible representation of the simple Lie group $\mathsf{Sp}(6,\mathbb{C})$: the associated moment map allows to define a rational map $\varpi$ from the space of symplectic 3D Monge-Amp\`ere equations to the projectivization of the space of quadratic forms on a $6$-dimensional symplectic vector space. We study in details the relationship between the zero locus of the image of $\varpi$, herewith called the cocharacteristic variety, and the contact cone structure of a 3D Monge-Amp\`ere equation $\mathcal{E}$: under the hypothesis of non-degenerate symbol, we prove that these two constructions coincide. A key tool in achieving such a result will be a complete list of mutually non-equivalent quadratic forms on a $6$-dimensional symplectic space, which has an interest on its own., Comment: Accepted by Advances in Differential Equations

Let $M = G/H$ be an $(n+1)$-dimensional homogeneous manifold and $J^k(n,M)=:J^k$ be the manifold of $k$-jets of hypersurfaces of $M$. The Lie group $G$ acts naturally on each $J^k$. A $G$-invariant PDE of order $k$ for hypersurfaces of $M$ (i.e., with $n$ independent variables and $1$ dependent one) is defined as a $G$-invariant hypersurface $\mathcal{E} \subset J^k$. We describe a general method for constructing such invariant PDEs for $k\geq 2$. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup $H^{(k-1)}$ of the $(k-1)$-prolonged action of $G$. We apply this approach to describe invariant PDEs for hypersurfaces in the Euclidean space $\mathbb{E}^{n+1 }$ and in the conformal space $\mathbb{S}^{n+1}$. Our method works under some mild assumptions on the action of $G$, namely: A1) the group $G$ must have an open orbit in $J^{k-1}$, and A2) the stabilizer $H^{(k-1)}\subset G$ of the fibre $J^k\to J^{k-1}$ must factorize via the group of translations of the fibre itself., Comment: 16 pages; it reworks and expands the first part of the preprint "Invariant PDEs on homogeneous manifolds via the affine structure of the bundles of jet spaces", see arXiv:1907.06283; to appear in Communications in Contemporary Mathematics

In [Alekseevsky, Gutt, Manno, Moreno: "A general method to construct invariant PDEs on homogeneous manifolds", Communications in Contemporary Mathematics (2021)] the authors have developed a method for constructing $G$-invariant PDEs imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie groups $G$. In the present paper the method is applied to the case when $G=\mathsf{PGL}(n+1)$ or $G=\mathsf{Aff}(n+1)$ and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ or affine $\mathbb{A}^{n+1}$ space, respectively. The paper's main result is that projectively or affinely invariant PDEs with $n$ independent and one unknown variables are in one-to-one correspondence with $\mathsf{CO}(d,n-d)$-invariant hypersurfaces of the space of trace-free cubic forms in $n$ variables. Local descriptions are also provided., Comment: 14 pages, 1 figure; to appear in Proceedings of the Edinburgh Mathematical Society

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the first case we also discuss a distinguished contact cone structure, arising as the variety of minimal rational tangents. We discuss the repercussion of the aforementioned classification theorems for the geometry of quaternion-K\"ahler manifolds with positive scalar curvature and for the geometry of second-order PDEs imposed on hypersurfaces., Comment: 22 pages. Based on a talk given by the first aouthor at the workshop "Geometry of Lagrangian Grassmannians and nonlinear PDEs" held in Warsaw, September 5-9, 2016. Changes in v2: revised section 10

This paper contains a thorough introduction to the basic geometric properties of the manifold of Lagrangian subspaces of a linear symplectic space, known as the Lagrangian Grassmannian. It also reviews the important relationship between hypersurfaces in the Lagrangian Grassmannian and second-order PDEs., Comment: 28 pages. Designed as opening contribution to the poceeding volume of the workshop "Geometry of Lagrangian Grassmannians and nonlinear PDEs" held in Warsaw, September 5-9, 2016. Changes in v3: small typos, updated bibliography

In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n+1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a 30-hours Ph.D course given by two of the authors (GM and GM). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections., Comment: 48 pages, 10 figures

If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of a free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed curves. There exists a rigorous geometric way to formulate this sort of problems on smooth manifolds with boundary, which we review here in a friendly self-contained way. As an application, we study a particular free boundary values variational problem, the free-sliding Bernoulli beam., Comment: 12 pages

In his 1954 paper about the initial value problem for 2D hyperbolic nonlinear PDEs, P. Lax declared that he had "a strong reason to believe" that there must exist a well-defined class of "not genuinely nonlinear" nonlinear PDEs. In 1978 G. Boillat coined the term "completely exceptional" to denote it. In the case of $2^{\textrm{nd}}$ order (nonlinear) PDEs, he also proved that this class reduces to the class of Monge-Amp\`ere equations. We review here, against a unified geometric background, the notion of complete exceptionality, the definition of a Monge-Amp\`ere equation, and the interesting link between them., Comment: 14 pages, 3 figures, slightly revised, to appear on Banach Centre Publications

For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type ${\sf C}$) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}\mathfrak{g}$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomial of degree $\le d$ in the minors of $(u_{ij})$, with coefficients functions of the contact coordinates $x^i$, $u$, $u_i$ (e.g., Monge-Amp\`ere equations have degree 1). For $\mathfrak{g}$ of type ${\sf A}$ or ${\sf G}$ we show that this gives all invariant second-order PDEs. For $\mathfrak{g}$ of type ${\sf B}$ and ${\sf D}$ we provide an explicit formula for the lowest-degree invariant second-order PDEs. For $\mathfrak{g}$ of type ${\sf E}$ and ${\sf F}$ we prove uniqueness of the lowest-degree invariant second-order PDE; we also conjecture that uniqueness holds in type ${\sf D}$., Comment: to appear on "Communications in Contemporary Mathematics", 28 pages, extended and revised version of the original "Homogeneous 2nd order PDEs on the adjoint contact manifold of a simple complex Lie group"

We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, we use arbitrary supermanifolds by following the categorical approach to supermanifolds., Comment: 19 pages. Comments welcomed. Minor typos corrected

By studying the development of shock waves out of discontinuity waves, in 1954 P. Lax discovered a class of PDEs, which he called 'completely exceptional', where such a transition does not occur after a finite time. A straightforward integration of the completely exceptionality conditions allowed Boillat to show that such PDEs are actually of Monge-Ampere type. In this paper, we first recast these conditions in terms of characteristics, and then we show that the completely exceptional PDEs, with 2 or 3 independent variables, can be described in terms of the conformal geometry of the Lagrangian Grassmannian, where they are naturally embedded. Moreover, for an arbitrary number of independent variables, we show that the space of r-th degree sections of the Lagrangian Grassmannian can be resolved via a BGG operator. In the particular case of 1st degree sections, i.e., hyperplane sections or, equivalently, Monge-Ampere equations, such operator is a close analog of the trace-free second fundamental form., Comment: 21 pages, 2 figures, accepted on Journal of Geometry and Physics

In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order system of PDEs to a constrained first-order one, the symplectic structures naturally arising in the dynamics of a first-order Lagrangian theory, and the importance of the Poincar\'e-Cartan form for variational problems, are all well-established facts. However, their adequate combination corresponding to higher-order theories is missing in the literature. Here we obtain a consistent and truly finite-dimensional canonical formalism, as well as a higher-order version of the Poincar\'e-Cartan form. In our exposition, the rigorous global proofs of the main results are always accompanied by their local coordinate descriptions, indispensable to work out practical examples., Comment: 41 pages, updated references, comments are welcome

This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Amp\`ere equations, by using the so-called "meta-symplectic structure" associated with the 8D prolongation $M^{(1)}$ of a 5D contact manifold $M$. We write down a geometric definition of a third-order Monge-Amp\`ere equation in terms of a (class of) differential two-form on $M^{(1)}$. In particular, the equations corresponding to decomposable forms admit a simple description in terms of certain three-dimensional distributions, which are made from the characteristics of the original equations. We conclude the paper with a study of the intermediate integrals of these special Monge-Amp\`ere equations, herewith called of Goursat type.

We prove that the Cayley graph and the coset geometry of the von Dyck group $D(a,b,c)$ are linked by a vertex-to-edge duality., Comment: Accepted for publication on Mathematica Slovaca (23-10-2013); 12 pages, 7 figures; completely rewritten replacement of "On the free Burnside group as a factor of von Dyck group: a geometric perspective"

Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differential calculus over commutative algebras. An advantage of such an approach, based on the notion of sliceable structures on cylinders, is that the fundamental theorems of standard calculus are straightforwardly generalized to the context of families. As an example of that, we prove the universal homotopy formula., Comment: 19 pages. Accepted for publication on International Journal of Geometric Methods in Modern Physics (8-2-2013)

In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of $Y$, the manifold of dependent and independent variables underlying a given problem, as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over $Y$. Explicit examples of natural boundary conditions are obtained when $Y$ is an $(n+1)$--dimensional domain in $\R^{n+1}$, and the Lagrangian is first-order (in particular, the hypersurface area)., Comment: 24 pages. Accepted for publication on Mathematica Slovaca (18-2-2013)

A notion known as smooth envelope, or superposition closure, appears naturally in several approaches to generalized smooth manifolds which were proposed in the last decades. Such an operation is indispensable in order to perform differential calculus. A derivation of the enveloping algebra can be restricted to the original one, but it is a delicate question if the the vice-versa can be done as well. In a physical language, this would corresponds to the existence of a canonical connection. In this paper we show an example of an algebra which always possesses such a connection., Comment: 5 pages. Accepted for publication on Demonstratio Mathematica (19-3-2013)