THE BSSN FORMULATION IS A PARTIALLY CONSTRAINED EVOLUTION SYSTEM

It is shown that the BSSN formulation of the Einstein equations, which forms the basis of most simulations in numerical relativity, explicitly uses the momentum constraints to evolve the conformal connection coefficients.

CONFORMAL CONNECTION AND EQUIVALENCE PROBLEM FOR THIRD ORDER ODEs

The equivalence problem for the third order ODEs solved by E. Cartan1 and S. S. Chern2 is reconsidered. We consider third order ODEs of the form y′′′ = F(x,y,y′,y′′) for which the Wunshman invariant I vanishes. All such ODEs split into equivalence classes with respect to the contact transformations of the variables. As shown by E. T. Newman3 and collaborators such equations are also in one-to-one correspondence with conformal classes of Lorentian three-metrics. We supplement Cartan-Chern-Newman results by providing explicit expressions for all the contact invariants of an ODE with I = 0. The invariants are explicitly written in terms of the function F and its partial derivatives. Explicit expression for the associated Cartan's O(2,3) connection is also given. The curvature of this conformal connection is reinterpreted in terms of the Cotton-York tensor of the Lorentzian three-metric associated with the equation.

Pseudo-Sasakian manifolds endowed with a contact conformal connection

Pseudo-Sasakian manifoldsM˜(U,ξ,η˜,g˜)endowed with a contact conformal connection are defined. It is proved that such manifolds are space formsM˜(K),K<0, and some remarkable properties of the Lie algebra of infinitesimal transformations of the principal vector fieldU˜onM˜are discussed. Properties of the leaves of a co-isotropic foliation onM˜and properties of the tangent bundle manifoldTM˜havingM˜as a basis are studied.