arXiv research

The study describes metrics geodesically compatible with Nijenhuis operators and their applications to integrable systems.

New maximal families of compatible Poisson structures derived from geodesically equivalent metrics.

In the literature different concepts of compatibility between a projective structure and a conformal structure on a differentiable manifold are used. In particular compatibility in the sense of Weyl geometry is slightly more general than compatibility in the Riemannian sense. An often cited paper [Ehlers-Pirani-Schild:…

Study compatible and associated metrics for contact-symplectic structures, showing geodesic integral curves and minimal leaf properties.

Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation …

In a space-time, a conformal structure is defined by the distribution of light-cones. Geodesics are traced by freely falling particles, and the collection of all unparameterized geodesics determines the projective structure of the space-time. The article contains a formulation of the necessary and sufficient conditions…

New equations reveal how cylinder power in progressive lenses depends on geodesic curvature.

The study defines Finsler metrics on special surfaces and constructs geodesic currents.

We study the curvature of a manifold on which there can be defined a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form. We also give a necessary curvature condition for the exis…

We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.

Study random walks on sub-Riemannian manifolds using retractions.

Study local control in a 7D quaternionic Heisenberg group.

Study of multiplicative connections in Lie groupoids.

Study of 3D trans-Sasakian manifolds using Newman--Penrose formalism.

The article classifies liftings of connections on differential manifolds for geodesic modeling.

This research connects Higgs bundles to projective structures via conformal limits.

Injectivity of geodesic X-ray transform on low-regularity manifolds.

Higher-order tangent bundles have geometric structures compatible with their iterated bundle structure.

In this paper we give a geometric proof of the Karpelevich's theorem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non compact type, has a totally geodesic orbit. In fact, this is equivalent to a well-known result of Mostow about existence of compatible Cartan decompositions.

Geodesic rays constructed in Kähler metrics with torus symmetry.

Maximal representations are studied using tree embeddings and geodesic currents.

We introduce a new approach for computing curvature of sub-Riemannian manifolds. Curvature is here meant as symplectic invariants of Jacobi curves of geodesics, as introduced by Zelenko and Li. We describe how they can be expressed using a compatible affine connection and induced tensors, without any restriction on our…

Generalizes Rips' result on hyperbolic spaces to metric spaces, showing collapses for tree metrics.

We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boun…

The paper explores conjugate points in Lorentzian spaces, comparing different definitions and proving related theorems.

In this work a proposal for definition of twistors on generic curved spaces is exposed and investigated. We consider superpositions of nearly autoparallel and nearly geodesic maps (nearly conformal maps, nc-maps) of (pseudo-)Riemannian spaces as generalizations of conformal transforms. We introduce the nearly autoparal…

We obtain explicitly all solutions of the SU(infinity) Toda field equation with the property that the associated Einstein-Weyl space admits a 2-sphere of divergence-free shear-free geodesic congruences. The solutions depend on an arbitrary holomorphic function and give rise to new hyperKahler and selfdual Einstein metr…

For a closed symplectic manifold $(M,ω)$ with compatible Riemannian metric $g$ we study the Sobolev $H^1$ geometry of the group of all $H^s$ diffeomorphisms on $M$ which preserve the symplectic structure. We show that, for sufficiently large $s$, the $H^1$ metric admits globally defined geodesics and the corresponding …

Develops radiant structures for statistical manifolds.

Let $M$ be a symplectic symmetric space, and let $\imath : M \to V$ be an extrinsic symplectic symmetric immersion, i.e., $(V, Ω)$ is a symplectic vector space and $\imath$ is an injective symplectic immersion such that for each point $p \in M$, the geodesic symmetry in $p$ is compatible with the reflection in the affi…

The abstract introduces a new concept called flagfolds to model multi-dimensional shapes.

We study the behavior of the Quillen metric for the family of Riemann surfaces with cusps when the additional cusps are created by degeneration. More precisely, in our previous paper, we've seen that the renormalization of the Quillen metric associated with a family of Riemann surfaces with cusps extends continuously o…

Study Riemannian geometry of maximal surface group representations in pseudo-hyperbolic space.

We classify both local and global Kähler structures admitting totally geodesic homothetic foliations with complex leaves. The main building blocks are related to Swann's twists and are obtained by applying Weinstein's method of constructing symplectic bundles to Kähler data. As a byproduct we obtain new classes of: hol…

We consider strict and complete nearly Kaehler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kaehler is locally a Riemannian product of homogenous nearly Kaehler spaces, twistor spaces over quate…

We are interested in the geometry of the group $\mathcal{D}_q(M)$ of diffeomorphisms preserving a contact form $θ$ on a manifold $M$. We define a Riemannian metric on $\mathcal{D}_q(M)$, compute the corresponding geodesic equation, and show that solutions exist for all time and depend smoothly on initial conditions. In…

New ODD metrics defined on manifolds with degeneracy conditions.

Let $\mathcal M= (M,\mathcal O_\mathcal M)$ be a smooth supermanifold with connection $\nabla$ and Batchelor model $\mathcal O_\mathcal M\congΓ_{ΛE^\ast}$. From $(\mathcal M,\nabla)$ we construct a connection on the total space of the vector bundle $E\to{M}$. This reduction of $\nabla$ is well-defined independently of …

Paper constructs exotic spacetimes with same physical properties.

We consider the Chern connection of a (conic) pseudo-Finsler manifold $(M,L)$ as a linear connection $\nabla^V$ on any open subset $Ω\subset M$ associated to any vector field $V$ on $Ω$ which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor $g$.…

Abstract commensurators linked to topological models of solenoids.

The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

Constructs graph manifolds with many Anosov flows.

Conformally quasi-recurrent (CQR)_n pseudo-Riemannian manifolds are investigated, and several new results are obtained. It is shown that the Ricci tensor and the gradient of the fundamental vector are Weyl compatible tensors (the notion was introduced recently by the authors and applies to significative space-times), (…

Notions of compatible and almost compatible pseudo-Riemannian metrics, which are motivated by the theory of compatible (local and nonlocal) Poisson structures of hydrodynamic type and generalize the notion of flat pencil of metrics, are introduced and studied.

Defines compatibility between Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.

New $3$-manifolds created from $4$-regular graphs with unique Eulerian cycles.