arXiv research

Symmetries of Einstein-Weyl manifolds can be extended from boundary surfaces.

In this paper we relate the Fefferman-Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tract…

Embedding theorem for tractor bundles applied to conformal geometry.

On a Finsler manifold $(M,L)$, we consider the change $L\longrightarrow\bar{L}(x,y)=e^{σ(x)}L(x,y)+β(x,y)$, which we call a $β$-conformal change. This change generalizes various types of changes in Finsler geometry: conformal, $C$-conformal, $h$-conformal, Randers and generalized Randers changes. Under this change, we …

We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal …

We discuss in this paper the conformal geometry of bi-invariant metrics on compact semisimple Lie groups. For this purpose we develop a conformal Cartan calculus adapted to this problem. In particular, we derive an explicit formula for the holonomy algebra of the normal conformal Cartan connection of a bi-invariant met…

Defines a new conformally invariant Yang-Mills type energy for 6-manifolds.

We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely a…

We treat a non-normal Fefferman-type construction based on an inclusion $\SL(n+1)\embed\Spin(n+1,n+1)$. The construction associates a split signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. For $n\geq 3$ the induced conformal Cartan connection is shown to be normal if and only if it…

Let $R$ be the $hh$-curvature associated with the Chern connection or the Cartan connection. Adopting the pulled-back tangent bundle approach to the Finslerian Geometry, an intrinsic characterization of $R$-Einstein metrics is given. Finslerian metrics which are locally conformally $R$-Einstein are classified.

In this paper, we identify the Bott connection on the natural foliation of the projective sphere bundle of a Finsler manifold to the Chern connection of this manifold. As a consequence, the symmetrization of the Bott connection turns out to be the Cartan connection of the Finsler manifold. Following Liu-Zhang \cite{Liu…

We introduce the notion of a conformally Fedosov structure and construct an associated Cartan connection. When an appropriate curvature vanishes, this allows us to construct a family of natural differential complexes akin to the BGG complexes from parabolic geometry.

The purpose of this note is to provide yet another example of the link between certain conformal geometries and ordinary differential equations, along the lines of the examples discussed by Nurowski in math.DG/0406400. In this particular case, I consider the equivalence problem for 3-plane fields D on 6-manifolds M tha…

We show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes the main Theorem in [DM] where the same result was proved for the special cases of holomorphic affine connections and holomorphic conformal structu…

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p+1, assuming p <= q; further, if this maximal degree is atta…

The paper studies conformal and projective structures using 2-frame bundles.

Lightlike manifolds studied via Cartan geometries in Lorentz-Minkowski spacetime.

The paper develops the Finsler-like geometry on the 1-jet space for the jet conformal Minkowski (JCM) metric, which naturally extends the Minkowski metric in the Chernov-Pavlov framework. To this aim there are determined the nonlinear connection, distinguished (d-) Cartan linear connection, d-torsions and d-curvatures.…

We study the relations between the projective and the almost conformally symplectic structures on a smooth even dimensional manifold. We describe these relations by a single almost conformally symplectic connection with totally trace--free torsion sharing the geodesics (up to parametrization) with the projective class.…

The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing rep…

This paper extends complex Cartan geometry results to noncompact and non-Kähler manifolds.

New classification of hypersurfaces with conformal variations.

We present a number of conditions which are necessary for an n-dimensional projective structure (M,[nabla]) to include the Levi-Civita connection nabla of some metric on M. We provide an algorithm, which effectively checks if a Levi-Civita connection is in the projective class and, in the positive, which finds this con…

For a conformal manifold, we describe a new relation between the ambient obstruction tensor of Fefferman and Graham and the holonomy of the normal conformal Cartan connection. This relation allows us to prove several results on the vanishing and the rank of the obstruction tensor, for example for conformal structures a…

We introduce in this paper normal twistor equations for differential forms and study their solutions, the so-called normal conformal Killing forms. The twistor equations arise naturally from the canonical normal Cartan connection of conformal geometry. Reductions of its holonomy are related to solutions of the normal t…

Around 1923, Elie Cartan introduced affine connections on manifolds and definedthe main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be relate…

This paper extends geometric structure theory to infinite type structures.

For a generic distribution of rank two on a manifold $M$ of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly constructed a pseudo--Riemannian metric on $M$ of split signature. We prove tha…

In [5], P. Lecomte conjectured the existence of a natural and conformally invariant quantization. In [7], we gave a proof of this theorem thanks to the theory of Cartan connections. In this paper, we give an explicit formula for the natural and conformally invariant quantization of trace-free symbols thanks to the meth…

In this paper, we consider the problem of building a conformal boundary, embedding a pseudo-Riamnnian manifold as an open subset of a bigger one. We get first results about conformal maximality. We also show that in dimension $\geq 3$, there are rigidity properties for the topological boundary of such a conformal embed…

A geodesic circle in Finsler geometry is a natural extension of that in a Euclidean space. In this paper, we apply Lie derivatives and the Cartan $Y$-connection to study geodesic circles and (infinitesimal) concircular transformations on a Finsler manifold. We characterize a concircular vector field with some PDEs on t…

We define what we call morphisms of Cartan connections. We generalize the main theorems on Cartan connections to theorems on morphisms. Many of the known constructions involving Cartan connections turn out to be examples of morphisms. We prove some basic results concerning completeness of Cartan connections. We provide…

Motivated by the rich geometry of conformal Riemannian manifolds and by the recent development of geometries modeled on homogeneous spaces $G/P$ with $G$ semisimple and $P$ parabolic, Weyl structures and preferred connections are introduced in this general framework. In particular, we extend the notions of scales, clos…

For $(M,[g])$ a conformal manifold of signature $(p,q)$ and dimension at least three, the conformal holonomy group $\mathrm{Hol}(M,[g]) \subset O(p+1,q+1)$ is an invariant induced by the canonical Cartan geometry of $(M,[g])$. We give a description of all possible connected conformal holonomy groups which act transitiv…

The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullb…

Starting from a real analytic conformal Cartan connection on a real analytic surface $S$, we construct a complex surface $T$ containing a family of pairs of projective lines. Using the structure on $S$ we also construct a complex $3$-space $Z$, such that $Z$ is a twistor space of a self-dual conformal $4$-fold and $T$ …

Local equivalence shown between specific distributions and flat Cartan distribution.

The aim of the present paper is to establish a global theory of conformal changes in Finsler geometry. Under this change, we obtain the relationships between the most important geometric objects associated to $(M,L)$ and the corresponding objects associated to $(M,\tilde{L})$, $\widetilde{L}=e^{σ(x)} L$ being the Finsl…

Reinterprets Schrödinger equation using Cartan connection for geometric investigation.

This paper extends Jacobi field theory to Jacobi curves and their curvatures.

New geometrization of gravitational wave phase space.

We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the …

Study symplectification of rank 2 distributions and their connections.

The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physi…

We construct polynomial conformal invariants, the vanishing of which is necessary and sufficient for an $n$-dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold. We also construct invariants which give necessary and sufficient conditions for a metric to be conformally relate…

All parabolic geometries, i.e. Cartan geometries with homogeneous model a real generalized flag manifold, admit highly interesting classes of distinguished curves. The geodesics of a projective class of connections on a manifold, conformal circles on conformal Riemannian manifolds, and Chern--Moser chains on CR--manifo…

This text is the extended version of a talk given at 6th Meeting of Integrable Systems and Quantum Filed Theory at Peyresq hold from June 10 2006 to June 17, 2006 at Peyresq, France. The goal of this lecture is to give a brief introduction to Cartan-Kähler's theory. As examples to the application of this theory, we cho…

Constructs stochastic processes on sub-Riemannian manifolds using Cartan connections.