34 research outputs found
Deforming a Lie algebra by means of a two form
We consider a vector space V over K=R or C, equipped with a skew symmetric
bracket [.,.]: V x V --> V and a 2-form omega:V x V --> K. A simple change of
the Jacobi identity to the form
[A,[B,C]]+[C,[A,B]]+[B,[C,A]]=omega(B,C)A+omega(A,B)C+omega(C,A)B opens new
possibilities, which shed new light on the Bianchi classification of
3-dimensional Lie algebras.Comment: An error consisting in overlooking few types in the Bianchi
classification is correcte
On a certain formulation of the Einstein equations
We define a certain differential system on an open set of . The system
locally defines a Lorentzian 4-manifold satisfying the Einstein equations. The
converse statement is indicated and its details are postponed to the
furthcoming paper.Comment: 7 pages, Late
Projective vs metric structures
We present a number of conditions which are necessary for an n-dimensional
projective structure (M,[nabla]) to include the Levi-Civita connection nabla of
some metric on M. We provide an algorithm, which effectively checks if a
Levi-Civita connection is in the projective class and, in the positive, which
finds this connection and the metric. The article also provides a basic
information on invariants of projective structures, including the treatment via
Cartan's normal projective connection. In particular we show that there is a
number of Fefferman-like conformal structures, defined on a subbundle of the
Cartan bundle of the projective structure, which encode the projectively
invariant information about (M,[nabla])
Elliptic fibrations associated with the Einstein spacetimes
Given a conformally nonflat Einstein spacetime we define a fibration over
it. The fibres of this fibration are elliptic curves (2-dimensional tori) or
their degenerate counterparts. Their topology depends on the algebraic type of
the Weyl tensor of the Einstein metric. The fibration is a double branched
cover of the bundle of null direction over the spacetime and is equipped with
six linearly independent 1-forms which satisfy certain relatively simple system
of equations.Comment: 15 pages, Late