Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.

Geometric integrability of hyperbolic and multi-dimensional second order PDEs

Abstract: We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution of multi-dimensional second order partial differential equations (PDEs). We apply the geometric techniques to find the largest integrable distributions containing the contact distribution associated to second order PDEs and hence the solution of the PDEs. We also discuss Darboux-integrable hyperbolic second order PDEs in the plane and their relationship with our technique.

On non-negatively curved metrics on open five-dimensional manifolds

Abstract: Let $V^n$ be an open manifold of non-negative sectional curvature with a soul $\Sigma$ of co-dimension two. The universal cover $\tilde N$ of the unit normal bundle $N$ of the soul in such a manifold is isometric to the direct product $M^{n-2}\times R$. In the study of the metric structure of $V^n$ an important role plays the vector field $X$ which belongs to the projection of the vertical planes distribution of the Riemannian submersion $\pi:V\to\Sigma$ on the factor $M$ in this metric splitting $\tilde N=M\times R$. The case $n=4$ was considered in [GT] where the authors prove that $X$ is a Killing vector field while the manifold $V^4$ is isometric to the quotient of $M^2\times (R^2,g_F)\times R$ by the flow along the corresponding Killing field. Following an approach of [GT] we consider the next case $n=5$ and obtain the same result under the assumption that the set of zeros of $X$ is not empty. Under this assumption we prove that both $M^3$ and $\Sigma^3$ admit an open-book decomposition with a bending which is a closed geodesic and pages which are totally geodesic two-spheres, the vector field $X$ is Killing, while the whole manifold $V^5$ is isometric to the quotient of $M^3\times (R^2,g_F)\times R$ by the flow along corresponding Killing field.

Stability of Periodic Linear Systems and the Geometry of Lie Groups

Abstract: Publisher Summary This chapter discusses the stability of periodic linear systems and the geometry of lie groups. In a study on the stability of linear periodic canonical systems, Gel’fand and Lidskii investigate certain aspects of the geometry of the group of symplectic matrices. They show that the interior of the subset of the 2 n x 2 n symplectic matrices ( Sp ( n )), which consists of those matrices similar to orthogonal ones, has 2 n connected components—a particular component being characterized by the distribution of Floquet multipliers of first and second type in the sense of Krein. Krein's work establishes that for Hill's equation, one of these connected components plays a particularly important role. The chapter describes the geometric content of the Liapunov–Krein theorems in the context of the theory of Lie groups. Many results on the stability of Hill's equation have an interpretation in terms of a variational problem of the above type on the symplectic group. The boundedness, asymptotic stability is properties of linear systems of differential equations that are preserved under tensoring and reduction.

Sliding mode control for a generalization of the Caginalp phase-field system

Abstract: In the present paper, we present and solve the sliding mode control (SMC) problem for a second-order generalization of the Caginalp phase-field system. This generalization, inspired by the theories developed by Green and Naghdi on one side, and Podio-Guidugli on the other, deals with the concept of thermal displacement, i.e., a primitive with respect to the time of the temperature. Two control laws are considered: the former forces the solution to reach a sliding manifold described by a linear constraint between the temperature and the phase variable; the latter forces the phase variable to reach a prescribed distribution $\varphi^*$. We prove existence, uniqueness as well as continuous dependence of the solutions for both problems; two regularity results are also given. We also prove that, under suitable conditions, the solutions reach the sliding manifold within finite time.