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The Lagrange–Charpit theory is a geometric method of determining a complete integral by means of a constant of the motion of a vector field defined on a phase space associated to a nonlinear PDE of first order. In this article, we establish this theory on the symplectic structure of the cotangent bundle T ∗ Q of the configuration manifold Q. In particular, we use it to calculate explicitly isotropic submanifolds associated with a Hamilton–Jacobi equation. [ABSTRACT FROM AUTHOR]

We find the first three most general Minkowski or Hsiung–Minkowski identities relating the total mean curvatures H i , of degrees i = 0 , 1 , 2 , 3 , of a closed hypersurface N immersed in a given orientable Riemannian manifold M endowed with any given vector field P. Then we specialize the three identities to the case when P is a position vector field. We further obtain that the classical Minkowski identity is natural to all Riemannian manifolds and, moreover, that a corresponding 1st degree Hsiung–Minkowski identity holds true for all Einstein manifolds. We apply the result to hypersurfaces with constant H 1 , H 2 . [ABSTRACT FROM AUTHOR]

We give a simple proof of the result of Tumanov and Zharnitsky (Int Math Res Not, 2006) that the set of period 4 orbits in planar outer billiard with piecewise smooth convex boundary has empty interior, provided that no four corners of the boundary form a parallelogram. We also obtain results on period 5 and 6 orbits. [ABSTRACT FROM AUTHOR]

We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree n associated to any given oriented Riemannian manifold M of dimension n + 1. The framework is that of the tangent sphere bundle of M. We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler–Lagrange differential systems. [ABSTRACT FROM AUTHOR]

Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold (M , g) and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold (N , h) , one can ask under what circumstances does the exterior differential system I for an isometric embedding M ↪ N have particularly nice solvability properties. In this paper we give a classification of all 2-dimensional metrics g whose isometric embedding system into flat Riemannian or pseudo-Riemannian 3-manifolds (N , h) is Darboux integrable. As an illustration of the motivation behind the classification, we examine in detail one of the classified metrics, g 0 , showing how to use its Darboux integrability in order to construct all its embeddings in finite terms of arbitrary functions. Additionally, the geometric Cauchy problem for the embedding of g 0 is shown to be reducible to a system of two first-order ODEs for two unknown functions—or equivalently, to a single second-order scalar ODE. For a large class of initial data, this reduction permits explicit solvability of the geometric Cauchy problem for g 0 up to quadrature. The results described for g 0 also hold for any classified metric whose embedding system is hyperbolic. [ABSTRACT FROM AUTHOR]

In this article, the prolongation structure technique is applied to a generalised inhomogeneous Gardner equation, which can be used to describe certain physical situations, such as the stratified shear flows in ocean and atmosphere, ion acoustic waves in plasmas with a negative ion, interfacial solitary waves over slowly varying topographies, and wave motion in a non-linear elastic structural element with large deflection. The Lax pairs, which are derived via the prolongation structure, are more general than the Lax pairs published before. Under the Painlevé conditions, the linear-damping coefficient equals to zero, the quadratic non-linear coefficient is proportional to the dispersive coefficient c( t), the cubic non-linear coefficient is proportional to c( t), leaving no constraints on c( t) and the dissipative coefficient d( t). We establish the prolongation structure through constructing the exterior differential system. We introduce two methods to obtain the Lax pairs: (a) based on the prolongation structure, the Lax pairs are obtained, and (b) via the Lie algebra, we can derive the Pfaffian forms and Lax pairs when certain parameters are chosen. We set d( t) as a constant to discuss the influence of c( t) on the Pfaffian forms and Lax pairs, and to discuss the influence of d( t) on the Pfaffian forms and Lax pairs, we set c( t) as another constant. Then, we get different prolongation structure, Pfaffian forms and Lax pairs. [ABSTRACT FROM AUTHOR]

Abstract: We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant. [Copyright &y& Elsevier]

Abstract: A geometric setting for constrained exterior differential systems on fibered manifolds with n-dimensional bases is proposed. Constraints given as submanifolds of jet bundles (locally defined by systems of first-order partial differential equations) are shown to carry a natural geometric structure, called the canonical distribution. Systems of second-order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds. As an important particular case, Lagrangian systems subjected to first-order differential constraints are considered. Different kinds of constraints are introduced and investigated (Lagrangian constraints, constraints adapted to the fibered structure, constraints arising from a (co)distribution, semi-holonomic constraints, holonomic constraints). [Copyright &y& Elsevier]

We deal with a Riemannian manifold M carrying a pair of skew symmetric conformal vector fields ( X, Y). The existence of such a pairing is determined by an exterior differential system in involution (in the sense of Cartan). In this case, M is foliated by 3-dimensional totally geodesic submanifolds. Additional geometric properties are proved. [ABSTRACT FROM AUTHOR]

We give a new proof of the generalized Minkowski identities relating the higher degree mean curvatures of orientable closed hypersurfaces immersed in a given constant sectional curvature manifold. Our methods rely on a fundamental differential system of Riemannian geometry introduced by the author. We develop the notion of position vector field, which lies at the core of the Minkowski identities. [ABSTRACT FROM AUTHOR]