Carnot metrics, dynamics and local rigidity

This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.

Digital optimization of car’s hinged components

The purpose of this work is to reduce the weight of the car’s hinged structures (front and rear doors, tailgate and lock reinforcements) by using various types of optimization which involve removing material, changing thicknesses, and topography from parts (topology, topography, parameter and multivariate) as well as the interpretation of optimization results for specific types of production while maintaining the rigidity and strength characteristics. This technique is necessary to reduce the time for product development and reduce the cost of manufacturing parts. This study addressed the issue of the maximum possible weight reduction while meeting all the requirements (strength calculations, stiffness calculations, modal analysis, analysis of resistance to external loads, local rigidity of the door attachment points - strength loadcases, stiffness loadcases, modal, denting, IPI) and technological limitations (Envelope requirements, casting technology). The final design solution was required to contain visible changes from the original structure, meet performance requirements, and obtain a feasibility study from the production department.

Rigidity theorems for holomorphic curves in a complex Grassmann manifold G(3,6)

In this paper, we prove some local rigidity theorems of holomorphic curves in a complex Grassmann manifold [Formula: see text] by moving frames. By applying our rigidity theorems, we also give a characterization of all homogeneous holomorphic two-spheres in [Formula: see text] classified by the second author.

Geodesic stretch, pressure metric and marked length spectrum rigidity

We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in [GL19] and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov geodesic flows. In turn, we also introduce a new pressure metric on the space of isometry classes, which reduces to the Weil–Petersson metric in the case of Teichmüller space and is related to the works [BCLS15, MM08].

Transversal local rigidity of discrete abelian actions on Heisenberg nilmanifolds

In this paper we prove a perturbative result for a class of ${\mathbb Z}^2$ actions on Heisenberg nilmanifolds that have Diophantine properties. Along the way we prove cohomological rigidity and obtain a tame splitting for the cohomology with coefficients in smooth vector fields for such actions.

A splitting of local rigidity of Clifford–Klein forms of homogeneous spaces of completely solvable Lie groups

In this paper, we discuss local rigidity of Clifford–Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. We split the property of local rigidity into two conditions: vertical rigidity and horizontal rigidity. By this separation, we discuss local rigidity, in particular, Baklouti’s conjecture.