We show that the E 11 representation that contains the space-time translation generators also contains the rank two and five totally anti-symmetric representations of A 10. By studying the behaviour of these latter A 10 representations under SL(32), which we argue is contained in the Cartan involution invariant sub-algebra of E 11 , we find that the rank two and five totally anti-symmetric repr...

Model of Riemann–Cartan gravity with varying signature metric is considered. The basic dynamical variables the formalism are vierbein, spin connection, and an internal in tangent space. corresponding action contains new terms, which depend on these fields. In general case determined dynamically. Minkowski preferred dynamically because configurations other signatures suppressed. We also discuss ...

Abstract : Let be a 2nd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates. We are going to solve this problem by an applicable method which has been recognized by R. Gardner, and classify them.   

Two special Finsler spaces have been introduced and investigated, namely R-recurrent Finsler space and concircularly recurrent Finsler space. The defining properties of these spaces are formulated in terms of the first curvature tensor of Cartan connection. The following three results constitute the main object of the present paper: (i) a concircularly flat Finsler manifold is necessarily of co...

A class of theories of gravitation that naturally incorporates preferred frames of reference is presented. The underlying space-time geometry consists of a partial parallelization of space-time and has properties of Riemann-Cartan as well as teleparallel geometry. Within this geometry, the kinematic quantities of preferred frames are associated with torsion fields. Using a variational method, i...

Abstract Through a new technique, we provide uniqueness, rigidity and non-existence results for compact minimal submanifolds of arbitrary dimension in large class Riemannian manifolds, which include between others, double-twisted warped products. Moreover, show that our can be applied particular, to space forms Cartan–Hadamard re-obtaining several classic different approach. Interesting applica...

We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipsch...

We define on-shell symmetries and characterize them for Lagrangian systems. The terms appearing in the variation of the Poincaré-Cartan form, which vanish because of field equations, are found to be strongly constrained if the space of solutions has to be preserved. The behaviour with respect to solution dragging is also investigated in order to discuss relations with the theory of internal sym...

This is a survey on recent joint work with A.R. Gover on the geometry of non–degenerate CR manifolds of hypersurface type. Specifically we discuss the relation between standard tractors on one side and the canonical Cartan connection, the construction of the Fefferman space and the ambient metric construction on the other side. To put these results into perspective, some parts of the general th...

Given a singular non-autonomous Lagrangian of higher order we construct a reduced evolution space of the same order and a local regular Lagrangian on it which is gauge equivalent to the original one. The general geometrical framework used is the theory of almost stable tangent geometry of higher order introduced in 17]. We show that higher order almost stable manifols can be endowed with a loca...