Invariant Algebraic Surfaces for Generalized Raychaudhuri Equations

  • Claudia VallsEmail author


We consider a generalized Raychaudhuri equation,
$$\begin{array}{llll} \dot x = -\frac 1 2 x^2 -\alpha x -2(y^2 +z^2 -w^2)-2 \beta,\\ \dot y = -(\alpha +x) y -\gamma,\\ \dot z = -(\alpha +x) z -\delta,\\ \dot w = -(\alpha +x) w, \end{array}$$
where α, β, γ, δ are real parameters. This model has appeared in modern string cosmology. We study the algebraic invariants of this model for all values of the parameters \({\alpha,\beta,\gamma,\delta\in \mathbb{R}}\) . We prove that when γ = δ = 0 the system is integrable and for any other values of the parameters γ, δ, α, β we characterize all the invariant surfaces of this system. In particular we characterize all the polynomial and proper rational first integrals.


Arbitrary Function Homogeneous Polynomial Polynomial Vector Raychaudhuri Equation Invariant Surface 
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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Departamento de MatemáticaInstituto Superior Técnico, Universidade Técnica de LisboaLisboaPortugal

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