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We construct a natural conformally invariant one-form of weight $-2k$ on any $2k$-dimensional pseudo-Riemannian manifold which is closely related to the Pfaffian of the Weyl tensor. On oriented manifolds, we also construct natural…

In this paper we introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide here…

In these Notes, a comprehensive description of the universal fractal geometry of conformally-invariant scaling curves or interfaces, in the plane or half-plane, is given. The present approach focuses on deriving critical exponents…

Linear connections satisfying the Einstein metricity condition are important in the study of generalized Riemannian manifolds $(M,G=g+F)$, where the symmetric part $g$ of $G$ is a non-degenerate $(0,2)$-tensor, and $F$ is the skew-symmetric…

The conformal Einstein equations for a tracefree (radiation) perfect fluid are derived in terms of the Levi-Civita connection of a conformally rescaled metric. These equations are used to provide a non-linear stability result for de…

Tractor Calculus is a powerful tool for analyzing Weyl invariance; although fundamentally linked to the Cartan connection, it may also be arrived at geometrically by viewing a conformal manifold as the space of null rays in a Lorentzian…

A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity…

The conformal Willmore functional (which is conformal invariant in general Riemannian manifold $(M,g)$) is studied with a perturbative method: the Lyapunov-Schmidt reduction. Existence of critical points is shown in ambient manifolds…

In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions…

We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of…

Using a metric conformal formulation of the Einstein equations, we develop a construction of 4-dimensional anti-de Sitter-like spacetimes coupled to tracefree matter models. Our strategy relies on the formulation of an initial-boundary…

A discussion is given of the conformal Einstein field equations coupled with matter whose energy-momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the…

We define pure radiation metrics with parallel rays to be n-dimensional pseudo-Riemannian metrics that admit a parallel null line bundle K and whose Ricci tensor vanishes on vectors that are orthogonal to K. We give necessary conditions in…

An important operation in geometry processing is finding the correspondences between pairs of shapes. The Gromov-Hausdorff distance, a measure of dissimilarity between metric spaces, has been found to be highly useful for nonrigid shape…

We study a Fefferman-type construction based on the inclusion of Lie groups ${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a split-signature $(n,n)$-conformal spin structure to a projective structure of dimension…

It is well-known that the Einstein condition on warpedgeometries requires the fibres to be necessarily Einstein. However, exact warped solutions have often been obtained using one- and two-dimensional bases. In this paper, keeping the…

The paper introduces a general method to construct conformal measures for a local homeomorphism on a locally compact non-compact Hausdorff space, subject to mild irreducibility-like conditions. Among others the method is used to give…

Using variational considerations, we establish that there exists a new symmetric trace-free tensor conformal invariant of hypersurfaces embeddings in even dimensional conformal manifolds. This conformal invariant completes the family of…

We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time…

In this manuscript, a conformally invariant theory of gravitation in the context of metric measure space is studied. The proposed action is invariant under both diffeomorphism and conformal transformations. Using the variational method, a…