We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé’s system of equations. We show that the symmetry group of the Lamé’s system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dependents variables. We obtain the solutions which are invariant under the acti...

The standard approach to initial data for both analytic and numerical computations of black hole collisions has been to use conformally-flat initial geometry. Among other advantages, this choice allows the simple superposition of holes with arbitrary mass, location and spin. The conformally flat restriction, however, is inappropriate to the study of Kerr holes, for which the standard constant-t...

In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally ...

We study the semi-Riemannian geometry of foliation F an indefinite locally conformal Kähler (l.c.K.) manifold M, given by Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is Lee form M). If M conformally flat then every leaf shown to be a totally geodesic hypersurface in space sectional curvature c/4, carrying c-Sasakian structure. As corollary result together with version de Rham decom...

Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension n ≥ 6. We prove that ‖u‖ L2 ∗ (M,g) ≤ K 2 ∫ M { |∇gu| 2 + c(n)Rgu 2 } dvg + A‖u‖ 2 L2n/(n+2)(M,g), for all u ∈ H(M), where 2 = 2n/(n − 2), c(n) = (n − 2)/[4(n − 1)], Rg is the scalar curvature, K −1 = inf ‖∇u‖L2(Rn)‖u‖ −1 L2n/(n−2)(Rn) and A > 0 is a constant depending on (M, g) only. The inequality is sharp in the...

We shall give an axiomatic construction of Wess-Zumino-Witten actions valued in G = SU(N), N ≥ 3. It is realized as a functor WZ from the category of conformally flat four-dimensional manifolds to the category of line bundles with connection that satisfies, besides the axioms of a topological field theory, the axioms which abstract the characteristics of Wess-Zumino-Witten actions. To each conf...