Conformal flatness and self-duality of Thurston-geometries

We show which Thurston-geometries in dimensions 3 and 4 admit invariant conformally flat or half-conformally flat metrics.

     

Relative index pairing and odd index theorem for even dimensional manifolds

We prove an analogue for even dimensional manifolds of the Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles. We show that the eta invariant appearing in this result coincides with the eta invariant by Dai and Zhang up to an integer. We also obtain the odd dimensional counterpart for manifolds with boundary of the relative index pairing by Lesch, Moscovici and Pflaum.     

Conformal vector fields on four–manifolds with negative scalar curvature

We prove a vanishing theorem for conformal vector fields on four–manifolds of negative scalar curvature. Our assumptions are conformally invariant, and in the case of locally conformally flat manifolds reduce to a sign condition on the Euler characteristic. Received April 14, 1998; in final form May 15, 1998     

Conformal Operators on Weighted Forms; Their Decomposition and Null Space on Einstein Manifolds

There is a class of Laplacian like conformally invariant differential operators on differential forms ${L^\ell_k}$ which may be considered as the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as factored polynomials in second-order differential operators. In the case that the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the ${L^\ell_k}$ in terms of the null spaces of mutually commuting second-order factors.     

Conformal metrics of constant mass

We give a new interpretation of the canonical metrics associated to scalar positive locally conformally flat structures introduced in a previous paper. This allows us to extend the definition to not necessarily locally conformally flat structures in low dimensions. Received May 1, 2000 / Accepted May 9, 2000 / Published online September 14, 2000     

The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group \mathbbZ_p{\mathbb{Z}_p}, where p is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p = n = 3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.     

On some conformally invariant fully nonlinear equationsSur certaines équations completement nonlinéaires invariantes par transformation conforme

We outline proofs of our results in [7] on Liouville type theorems, Harnack type inequalities, and existence and compactness of solutions to some conformally invariant fully nonlinear elliptic equations of second order on locally conformally flat Riemannian manifolds. Details will appear in [7]. To cite this article: A. Li, Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 305–310.     

An integral invariant from the view point of locally conformally Kähler geometry

In this article we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifold.     

Rigidité conforme des hémisphères {{\rm S}^4_+} et {{\rm S}^6_+}

Let (M, g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its Yamabe invariant is positive, then (M, g) is conformally isometric to the standard hemisphere. As an application and using a result of Hang and Wang (Comm Anal Geom 14(1):91–106, 2006), we prove a rigidity result for these hemispheres regarding the Min-Oo conjecture.     

Conformally Flat Algebraic Ricci Solitons on Lie Groups

The paper is devoted to the study of conformally flat Lie groups with left-invariant (pseudo) Riemannianmetric of an algebraic Ricci soliton. Previously conformally flat algebraic Ricci solitons on Lie groups have been studied in the case of small dimension and under an additional diagonalizability condition on the Ricci operator. The present paper continues these studies without the additional requirement that the Ricci operator be diagonalizable. It is proved that any nontrivial conformally flat algebraic Ricci soliton on a Lie group must be steady and have Ricci operator of Segrè type {(1... 1 2)} with a unique eigenvalue (equal to 0).     

A deformation of flat conformal structures

We consider deformations of flat conformal structures from a viewpoint of connected sum decomposition of conformally flat manifolds.

     

Examples of almost Einstein structures on products and in cohomogeneity one

Almost Einstein manifolds are conformally Einstein up to a scale singularity, in general. This notion comes from conformal tractor calculus. In the current paper we discuss almost Einstein structures on closed Riemannian product manifolds and on 4-manifolds of cohomogeneity one. Explicit solutions are found by solving ordinary differential equations. In particular, we construct three families of closed 4-manifolds with almost Einstein structure corresponding to the boundary data of certain unimodular Lie groups. Two of these families are Bach-flat, but neither (globally) conformally Einstein nor half conformally flat. On products with a 2-sphere we find an exotic family of almost Einstein structures with hypersurface singularity as well.     

On conformally flat hypersurfaces,curved flats and cyclic systems

It will be shown that suitable “Gauß maps” associated to a conformally flat hypersurface inS ⁿ⁺¹ (n≥3) yield normal congruences of circles having a whole 1-parameter family of conformally flat orthogonal hypersurfaces. However such a “cyclic system” is not uniquely associated to a conformally flat hypersurface. The key idea is to show that these Gauß maps are “curved flats” in a pseudo Riemannian symmetric space. Additionally, in this context some characterizations of 3-dimensional conformally flat hypersurfaces arise with a new flavour. The curved flat approach allows us to handle conformally flat hypersurfaces in the context of integrable system theory.     

On Conformally Flat Almost Contact Metric Manifolds

Mediterranean Journal of Mathematics - In this paper, we first focus on conformally flat almost $${C(\alpha)}$$ -manifolds. Moreover, we construct an example of a 3-dimensional conformally flat...     

Conformally parallel G 2 structures on a class of solvmanifolds

Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G₂-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G₂ structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding non-homogeneous Ricci-flat metrics with holonomy contained in G₂. In the process we also find a new metric with exceptional holonomy. Received: 20 September     

Orthogonal complex structures on domains in \mathbb R4{\mathbb {R}^4}

An orthogonal complex structure on a domain in is a complex structure which is integrable and is compatible with the Euclidean metric. This gives rise to a first order system of partial differential equations which is conformally invariant. We prove two Liouville-type uniqueness theorems for solutions of this system, and use these to give an alternative proof of the classification of compact locally conformally flat Hermitian surfaces first proved by Pontecorvo. We also give a classification of non-degenerate quadrics in under the action of the conformal group SO _°(1, 5). Using this classification, we show that generic quadrics give rise to orthogonal complex structures defined on the complement of unknotted solid tori which are smoothly embedded in . To Nigel Hitchin on the occasion of his 60th birthday. This work was partially supported by MIUR (Metriche Riemanniane e Varietà Differenziabili, PRIN05) and NSF Grant DMS-0503506.     

On Conformal Invariants Built from Spinor Fields

By means of a conformal covariant differentiation process we construct generating systems for conformally invariant symmetric (r, s)–spinors in an arbitrary curved space–time. Extending this method to conformally invariant linear differential operators acting on symmetric spinor fields some classes of such operators are derived.     

共形平坦黎曼流形上的Schouten张量

M是一个紧致的局部共形平坦黎曼流形,其上定义的Schouten张量是一个Codazzi张量.本文借助这个Codazzi张量引入Cheng和Yau的自伴算子,从而获得了局部共形平坦流形上的一些性质,改进了已有的结论.     

Generalized Willmore functionals and related variational problems

The purpose of this paper is to study the conformally invariant functionals of hypersurfaces in a Riemannian manifold and variational problems related to these functionals. A class of conformal invariants is presented and the variational problem related to this class of conformally invariant functionals is studied.     

Locally conformally homogeneous pseudo-Riemannian spaces

Locally homogeneous Riemannian spaces were studied in [1–4]. Locally conformally homogeneous Riemannian spaces were considered in [10]. Moreover, the theorem claiming that every such space is either conformally flat or conformally equivalent to a locally homogeneous Riemannian space was proved.In this article, we study locally conformally homogeneous pseudo-Riemannian spaces and prove a theorem on their structure. Using three-dimensional Lie groups and the six-dimensional Heisenberg group [11], we construct some examples showing the difference between the Riemannian and pseudo-Riemannian cases for such spaces.