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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...

On geometry of curves of flags of constant type

Boris Doubrov, Igor Zelenko (2012)

Open Mathematics

We develop an algebraic version of Cartan’s method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of GL(W). Under some natural assumptions on the subgroup G and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure linear algebra. The scope...

On Jacobi fields and a canonical connection in sub-Riemannian geometry

Davide Barilari, Luca Rizzi (2017)

Archivum Mathematicum

In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [15]. We show why this connection is naturally nonlinear, and we discuss some of its properties.

On left invariant CR structures on SU ( 2 )

Andreas Čap (2006)

Archivum Mathematicum

There is a well known one–parameter family of left invariant CR structures on S U ( 2 ) ≅ S 3 . We show how purely algebraic methods can be used to explicitly compute the canonical Cartan connections associated to these structures and their curvatures. We also obtain explicit descriptions of tractor bundles and tractor connections.

On sectional curvature of a Riemannian manifold with semi-symmetric metric connection

Füsun Özen Zengin, S. Aynur Uysal, Sezgin Altay Demirbag (2011)

Annales Polonici Mathematici

We prove that if the sectional curvature of an n-dimensional pseudo-symmetric manifold with semi-symmetric metric connection is independent of the orientation chosen then the generator of such a manifold is gradient and also such a manifold is subprojective in the sense of Kagan.

On some flat connection associated with locally symmetric surface

Maria Robaszewska (2014)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

For every two-dimensional manifold M with locally symmetric linear connection ∇, endowed also with ∇-parallel volume element, we construct a flat connection on some principal fibre bundle P(M,G). Associated with - satisfying some particular conditions - local basis of TM local connection form of such a connection is an R(G)-valued 1-form build from the dual basis ω1, ω2 and from the local connection form ω of ▽. The structural equations of (M,∇) are equivalent to the condition dΩ-Ω∧Ω=0. This work...

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