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The Bianchi equations in a fixed background spacetime
The causal structure of the spacetime
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adapted null frame algebra appropriate assume asymptotic background spacetime Bel-Robinson energy-type norms Bel-Robinson tensor Bianchi equations Cauchy problem causal structure Ch-Kl-Ni conserved constant covariant curvature tensor defined deformation tensors diffeomorphism double null foliation double null integral eikonal equation Einstein equations Einstein vacuum equations energy norms energy type norms equivariant estimates evolution equation finite GARC Symposium global existence hyperbolic inequality initial data set J.E.Marsden Killing or conformal Killing vector fields Kl-Ni Lecture notes linear wave equations log ft manifold metric g metric tensor Minkowski space Minkowski spacetime null condition null geodesics null hypersurfaces C(u null incoming null outgoing null Ricci coefficients obtain Partial Differential Equations proof properties Proposition 4.7 prove relative result Riemann norms Riemann tensor rotation vector fields S.Klainerman satisfy second derivatives second fundamental form Sobolev solve spacelike structure equations sufficiently small sup norms symmetries tangent Theorem traceless Weyl field Weyl tensor field