arXiv research

Necessary and sufficient conditions for a Riemannian product to be conformally equivalent to an Einstein manifold are given. Such spaces which are complete are characterized.

We study surfaces with decorations and prove uniformization in non-Euclidean geometries.

This article classifies closed G2-structures such that the induced metric is conformally flat. It is shown that any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples. In particular, it follows from the classification that any closed G2-structure inducing a metric t…

The aim of this paper is to classify the cohomogeneity one conformal actions on the three-dimensional essential Riemannian spaces, up to orbit equivalence. Among other results, the representations of all connected Lie groups acting with cohomogeneity one or zero within the full conformal group of a given three-dimensio…

Solves symplectic and conformal symplectic group actions equivalence problem.

We investigate the Fefferman spaces of conformal type which are induced, via parabolic geometry, by the quaternionic contact (qc) manifolds introduced by O.Biquard. Equivalent characterizations of these spaces are proved: as conformal manifolds with symplectic conformal holonomy of the appropriate signature; as pseudo-…

Liouville's theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding …

New algebraic approach classifies conformally superintegrable systems in arbitrary dimensions.

The conformal geometry of surfaces in the conformal space $\mathbf Q^n_1$ is studied. We classify the space-like surfaces in $\mathbf Q^n_1$ with vanishing conformal form up to conformal equivalence.

A new characterization of conformal transformations is given. By use of this, the general form of conformal transformation on two-dimensional Minkowski space is given and its conformal structure is analyzed.

We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invari…

Study geometrically measures to decide if modular companions are conformally equivalent.

We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.

Study finds Stäckel equivalence for superintegrable systems via invariant quadrics.

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 …

New boundary and point constraints for controlling conformal surfaces.

We describe the local conformal geometry of a Lorentzian spin manifold $(M,g)$ admitting a twistor spinor $φ$ with zero. Moreover, we describe the shape of the zero set of $φ$. If $φ$ has isolated zeros then the metric $g$ is locally conformally equivalent to a static monopole. In the other case the zero set consists o…

Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid …

The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well de…

Let $x: \mathbf{M} \rightarrow {\mathbb Q}^{n+1}_1$ be a regular space-like hypersurface in the conformal space ${\mathbb Q}^{n+1}_1$. We classify all those hypersurfaces with parallel Blaschke tensor in the conformal space up to the conformal equivalence.

Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.

Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.

The paper establishes a connection between force-free fields and conformally geodesic fields.

In this paper a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group $\mathbb H_3$, or its conformal classification coincides with the metric one…

Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal sectional curvatures are locally equivalent to canal hypersurfaces in Euclidean spa…

Study on discrete Gaussian curvature for polyhedral surfaces.

An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension $n \ne…

Compact 3D Cotton-parallel manifolds are always conformally flat.

We study a CR analogue of the Ahlfors derivative for conformal immersions of Stowe [23] that generalizes the CR Schwarzian derivative studied earlier by the second-named author [21]. This notion possesses several important properties similar to those of the conformal counterpart and provides a new invariant for spheric…

We give an equivalent definition of compact locally conformally hyperkähler manifolds in terms of the existence of a nondegenerate complex two-form with natural properties. This is a conformal analogue of Beauville's theorem stating that a compact Kähler manifold admitting a holomorphic symplectic form is hyperkähler.

We construct pairs of conformally equivalent isospectral Riemannian metrics $φ_1 g$ and $φ_2 g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres a…

We give results about the L^2 kernel and the spectrum of the Dirac operator on a complete Riemannian manifold which is conformally equivalent to the interior of a Riemannian manifold with nonempty boundary.

We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely 2^$ for graphs with n+1 singularities. We also give an explicit description of the family of entire maximal graphs with a f…

Equivalences between conformal foliations on Euclidean $3$-space, Hermitian structures on Euclidean $4$-space, shear-free ray congruences on Minkowski $4$-space, and holomorphic foliations on complex $4$-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued …

We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the $σ_2$-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.

The class of the hypercomplex pseudo-Hermitian manifolds is considered. The flatness of the considered manifolds with the 3 parallel complex structures is proved. Conformal transformations of the metrics are introduced. The conformal invariance and the conformal equivalence of the basic types manifolds are studied. A k…

The abstract discusses transformations on statistical and semi-Weyl manifolds with torsion.

Unified approach to conformal and modular invariants on surfaces.

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Ja…

The study proves a strong parametric h-principle for minimal surfaces.

We consider the space of tensor densities on the n-dimensional sphere with degree lambda (or, equivalently, of conformal densities with degree lambda). This space is a module over the group of diffeomorphisms, and consequently over the Lie algebra of vector fields, on the sphere, and we first prove that as a module ove…

New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.

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 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…

An integral stability estimate is proved for refraction coefficients of two conformal metrics in a plane domain in terms of its travel times. No assumption on absence of conjugate points of geodesics is made.

The paper examines the blow-up of Ricci curvatures in conformal metrics.

We study the spectral geometry of the conformal Jacobi operator on a 4-dimensional Riemannian manifold (M,g). We show that (M,g) is conformally Osserman if and only if (M,g) is self-dual or anti self-dual. Equivalently, this means that the curvature tensor of (M,g) is given by a quaternionic structure, at least pointwi…

The article classifies curvature functions on compact manifolds with boundaries.