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Essential Killing fields of parabolic geometries: projective and conformal structures

Andreas Čap, Karin Melnick (2013)

Open Mathematics

We use the general theory developed in our article [Čap A., Melnick K., Essential Killing fields of parabolic geometries, Indiana Univ. Math. J. (in press)], in the setting of parabolic geometries to reprove known results on special infinitesimal automorphisms of projective and conformal geometries.

Geometrical aspects of the covariant dynamics of higher order

D. Opris, I. D. Albu (1998)

Czechoslovak Mathematical Journal

We present some geometrical aspects of a higher-order jet bundle which is considered a suitable framework for the study of higher-order dynamics in continuous media. We generalize some results obtained by A. Vondra, [7]. These results lead to a description of the geometrical dynamics of higher order generated by regular equations.

Geometry of third order ODE systems

Alexandr Medvedev (2010)

Archivum Mathematicum

We compute cohomology spaces of Lie algebras that describe differential invariants of third order ordinary differential equations. We prove that the algebra of all differential invariants is generated by 2 tensorial invariants of order 2, one invariant of order 3 and one invariant of order 4. The main computational tool is a Serre-Hochschild spectral sequence and the representation theory of semisimple Lie algebras. We compute differential invariants up to degree 2 as application.

Homogeneous Cartan geometries

Matthias Hammerl (2007)

Archivum Mathematicum

We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely algorithmic. As an example of an homogeneous parabolic geometry we treat a conformal structure on the product of two spheres.

Homogeneous systems of higher-order ordinary differential equations

Mike Crampin (2010)

Communications in Mathematics

The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation...

Infinitesimal automorphisms and deformations of parabolic geometries

Andreas Čap (2008)

Journal of the European Mathematical Society

We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de Rham sequence associated to a certain linear connection on the adjoint tractor bundle. For regular normal geometries, this description can be related to the underlying geometric structure using the machinery of BGG sequences. In the locally flat case, this leads to a deformation complex, which generalizes the well known complex for locally conformally...

Non-associative geometry and discrete structure of spacetime

Alexander I. Nesterov, Lev Vasilʹevich Sabinin (2000)

Commentationes Mathematicae Universitatis Carolinae

A new mathematical theory, non-associative geometry, providing a unified algebraic description of continuous and discrete spacetime, is introduced.

Non-degenerate hypersurfaces of a semi-Riemannian manifold with a semi-symmetric metric connection

Ahmet Yücesan, Nihat Ayyildiz (2008)

Archivum Mathematicum

We derive the equations of Gauss and Weingarten for a non-degenerate hypersurface of a semi-Riemannian manifold admitting a semi-symmetric metric connection, and give some corollaries of these equations. In addition, we obtain the equations of Gauss curvature and Codazzi-Mainardi for this non-degenerate hypersurface and give a relation between the Ricci and the scalar curvatures of a semi-Riemannian manifold and of its a non-degenerate hypersurface with respect to a semi-symmetric metric connection....

On almost pseudo-conformally symmetric Ricci-recurrent manifolds with applications to relativity

Uday Chand De, Avik De (2012)

Czechoslovak Mathematical Journal

The object of the present paper is to study almost pseudo-conformally symmetric Ricci-recurrent manifolds. The existence of almost pseudo-conformally symmetric Ricci-recurrent manifolds has been proved by an explicit example. Some geometric properties have been studied. Among others we prove that in such a manifold the vector field ρ corresponding to the 1-form of recurrence is irrotational and the integral curves of the vector field ρ are geodesic. We also study some global properties of such a...

On Galilean connections and the first jet bundle

James Grant, Bradley Lackey (2012)

Open Mathematics

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order...

Currently displaying 21 – 40 of 86