We construct a Kähler structure (J,Ω, G) on the space L(H) of oriented geodesics of hyperbolic 3-space H and investigate its properties. We prove that (L(H), J) is biholomorphic to P×P−∆, where ∆ is the reflected diagonal, and that the Kähler metric G is of neutral signature, conformally flat and scalar flat. We establish that the identity component of the isometry group of the metric G on L(H)...

We establish compactness of solutions to the Yamabe problem on any smooth compact connected Riemannian manifold (not conformally diffeomorphic to standard spheres) of dimension n 7 as well as on any manifold of dimension n 8 under some additional hypothesis. To cite this article: Y.Y. Li, L. Zhang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  2004 Académie des sciences. Published by Elsevier SA...

We classify conformally flat Riemannian 3−manifolds which possesses a free isometric S−action.

We present the Ricci-flat metric and its Kähler potential on the conifold with theO(N) isometry, whose conical singularity is repaired by the complex quadric surfaceQ = SO(N)/SO(N− 2)× U(1). ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] Introduction. Conformally invariant nonlinear sigma models withN = 2 supers...

A “conformal tensor” is constructed from the metric tensor gMN (or Vielbein e A M ) and is invariant against Weyl rescaling gMN → egMN (or eM → eeM ). Moreover, it vanishes if and only if the space is conformally flat, gMN = e ηMN (or e A M = eδ M ). In dimension four or greater the conformal tensor is the Weyl tensor. In three dimensions the Weyl tensor vanishes identically, while the Cotton t...

Abstract. This paper studies the two-component spinor form of massive spin2 potentials in conformally flat Einstein four-manifolds. Following earlier work in the literature, a nonvanishing cosmological constant makes it necessary to introduce a supercovariant derivative operator. The analysis of supergauge transformations of primary and secondary potentials for spin 3 2 shows that the gauge fre...

Here we will consider examples of conformally flat manifolds that are conformally equivalent to open subsets of the sphere S. For such manifolds we shall introduce a Cauchy kernel and Cauchy integral formula for sections taking values in a spinor bundle and annihilated by a Dirac operator, or generalized Cauchy-Riemann operator. Basic properties of this kernel are examined, in particular we exa...

We present a novel approach to creating flat maps of the brain. Our approach attempts to preserve the conformal structure between the original cortical surface in 3-space and the flattened surface. We demonstrate this with data from the human cerebellum. Our maps exhibit quasiconformal behavior and offer several advantages over existing approaches.

In this paper, we are interested in certain global geometric quantities associated to the Schouten tensor and their relationship in conformal geometry. For an oriented compact Riemannian manifold (M,g) of dimension n > 2, there is a sequence of geometric functionals arising naturally in conformal geometry, which were introduced by Viaclovsky in [29] as curvature integrals of Schouten tensor. If...