The study describes metrics geodesically compatible with Nijenhuis operators and their applications to integrable systems.
problem Geodesically compatible metrics and their applications to integrable systems.
method Describes metrics geodesically compatible with a gl-regular Nijenhuis operator and shows how these metrics relate to integrable PDE systems.
result Every metric geodesically compatible with a Nijenhuis operator gives a finite-dimensional reduction of an integrable PDE system.
Proves compatibility of light cones and projective structures.
problem Clarifying different concepts of compatibility between conformal and projective structures.
method Analyzes compatibility criteria introduced by Ehlers-Pirani-Schild and Trautman-Scholz.
result Proves that the compatibility criterion introduced by Ehlers-Pirani-Schild is correct.
New maximal families of compatible Poisson structures derived from geodesically equivalent metrics.
problem Constructing maximal families of compatible Poisson structures.
method Connecting geodesically equivalent metrics and compatible Poisson structures of hydrodynamic type.
result Maximal families of compatible Poisson structures of dimension (n+1)(n+2)/2 are constructed. Study compatible and associated metrics for contact-symplectic structures, showing geodesic integral curves and minimal leaf properties.
problem Characteristics foliations of metric contact-symplectic structures.
method Analysis of compatible and associated metrics, study of geodesic integral curves, and minimal leaf properties.
result Integral curves of the Reeb vector field are geodesics for any compatible metric, and associated metrics share a common volume element.
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.
problem Current understanding of cylinder power in progressive lenses is incomplete.
method Derived complete compatibility equations for spatially-varying curvature surfaces.
result Cylinder power depends on geodesic curvature, not just principal curvature.
The study defines Finsler metrics on special surfaces and constructs geodesic currents.
problem Defining and studying Finsler metrics on specific geometric structures.
method Defined compatible Finsler distances, studied geodesics, and constructed Liouville currents.
result Constructs a Liouville current for each metric, encoding curve lengths.
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.
problem Modeling random walks on sub-Riemannian manifolds.
method Use retractions to approximate normal geodesics and study convergence to Brownian motion.
result Convergence of geodesic random walks defined with different connections.
Study local control in a 7D quaternionic Heisenberg group.
problem Optimizing geodesics in a 7D quaternionic Heisenberg group.
method Matrix representation and analysis of sub-Riemannian structure symmetries.
result Impact of symmetries on geodesic optimality.
Study of multiplicative connections in Lie groupoids.
problem Defining and understanding multiplicative connections in Lie groupoids.
method Definition and study of multiplicative connections satisfying compatibility with the groupoid structure.
result Identification of the obstruction to the existence of a multiplicative connection.
Study of 3D trans-Sasakian manifolds using Newman--Penrose formalism.
problem Characterizing and understanding the geometry of 3D trans-Sasakian manifolds.
method Using Newman--Penrose formalism to encode the geometry of the structure vector field.
result Derivation of curvature and Laplacian identities for trans-Sasakian manifolds and their subclasses, including rigidity results.
The article classifies liftings of connections on differential manifolds for geodesic modeling.
problem Classifying liftings of connections on differential manifolds.
method Liftings of connections on frame bundles, induced and adjust liftings.
result Developed a method for geodesic modeling of differential equations.
This research connects Higgs bundles to projective structures via conformal limits.
problem Mapping Higgs bundles to projective structures.
method Using non-abelian Hodge correspondence and conformal limit.
result The family of connections in the conformal limit can be understood as complex projective structures.
Injectivity of geodesic X-ray transform on low-regularity manifolds.
problem Injectivity of geodesic X-ray transform on manifolds with low regularity.
method Calculus of differential and curvature operators on non-smooth structures.
result Injectivity of geodesic X-ray transform on simple Riemannian manifolds with C1,1-regularity. Higher-order tangent bundles have geometric structures compatible with their iterated bundle structure.
problem Connection towers and Sasaki metrics on higher-order tangent bundles
method Introduce the notion of a connection tower and study the geometric structures induced by such towers.
result Connection towers determine multiconnections, adapted splittings, and canonical vector bundle structures.
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.
problem Constructing geodesic paths in Kähler metrics with symmetry.
method Mixed polarization, complex structures, geodesic rays.
result Geodesic rays in the space of Kähler metrics.
Maximal representations are studied using tree embeddings and geodesic currents.
problem Maximal representations of surface groups in symplectic groups.
method Metric properties, geodesic currents, and tree embeddings.
result Translation length can be computed as intersection with a geodesic current.
New method for curvature computation in sub-Riemannian geometry.
problem Computing curvature in sub-Riemannian manifolds.
method Using compatible affine connections and induced tensors.
result Universal Bonnet-Myers theorem for sub-Riemannian geometry.
Generalizes Rips' result on hyperbolic spaces to metric spaces, showing collapses for tree metrics.
problem Understanding the contractibility of Vietoris-Rips complexes in metric spaces.
method Extending Rips' result using geodesic defect and apparent pairs gradient.
result Vietoris-Rips complexes collapse to subforests for finite 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.
problem Understanding conjugate points in Lorentzian geometry.
method Introducing and comparing different definitions of conjugate points in synthetic Lorentzian length spaces.
result All defined notions of conjugate points are compatible with the smooth spacetime setting.
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 H1 geometry of the group of all Hs diffeomorphisms on M which preserve the symplectic structure. We show that, for sufficiently large s, the H1 metric admits globally defined geodesics and the corresponding …
Develops radiant structures for statistical manifolds.
problem Statistical manifolds with radiant vector fields.
method Formulates Einstein equations for special statistical structures.
result Conelike radiant structures exist and have canonical normalizations.
Let M be a symplectic symmetric space, and let :M→V be an extrinsic symplectic symmetric immersion, i.e., (V,Ω) is a symplectic vector space and is an injective symplectic immersion such that for each point p∈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.
problem Modeling multi-dimensional shapes in a way that avoids going through higher dimensional spaces.
method Interpreting covariance matrices as nested subspaces and defining a Riemannian metric on the highest dimensional stratum.
result A Riemannian metric on the highest dimensional stratum allows for geodesics between subspaces of different dimensions.
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.
problem Characterize the geometry of maximal surface group representations in pseudo-hyperbolic space.
method Introduced a scalar product on the first cohomology group, leading to a Riemannian metric on the smooth locus.
result Found totally geodesic sub-varieties and orbifold structures in the space of representations.
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 Dq(M) of diffeomorphisms preserving a contact form θ on a manifold M. We define a Riemannian metric on Dq(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.
problem Defining metrics on manifolds with degeneracy conditions.
method Introducing ODD metrics that degenerate on submanifolds while maintaining compatibility.
result ODD metrics satisfy basic properties and induce metric space structures.
Let M=(M,OM) be a smooth supermanifold with connection ∇ and Batchelor model OM≅ΓΛE∗. From (M,∇) we construct a connection on the total space of the vector bundle E→M. This reduction of ∇ is well-defined independently of …
Paper constructs exotic spacetimes with same physical properties.
problem Whether two topologically identical manifolds can have different geometries.
method Computational approach to produce physical models on exotic spheres.
result Lorentzian metrics on homeomorphic but not diffeomorphic manifolds with same physical properties.
We consider the Chern connection of a (conic) pseudo-Finsler manifold (M,L) as a linear connection ∇V on any open subset Ω⊂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.
problem Understanding abstract commensurators through topological models.
method Relating homotopy equivalences of full solenoids to abstract commensurators.
result Isomorphism between homotopy equivalences and abstract commensurators in specific cases.
The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
problem Classifying Sasaki-Einstein orbits in compact Hermitian symmetric spaces.
method Examining orbits as CR submanifolds, proving total geodesy, and analyzing contact structures.
result Completely determine Sasaki-Einstein orbits.
The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.
problem Investigating compatibility conditions on spherically symmetric Finsler metrics.
method Using the inverse problem of calculus of variations, the paper focuses on Landsberg and Berwald types.
result All Landsberg spherically symmetric manifolds in higher dimensions are either Riemannian or have specific geodesic spray formulas.
Constructs graph manifolds with many Anosov flows.
problem Finding graph manifolds supporting multiple Anosov flows.
method Cutting geodesic flows, pulling back to finite covers, and gluing compatible pairs of flows.
result Constructs graph manifolds with at least n Anosov flows for any n.
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.
problem Generalizing compatibility between Poisson and pseudo-Riemannian metrics to Jacobi structures.
method Introduces and studies compatibility conditions for Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.
result Compatibility conditions are preserved under Poissonization and equivalent to Sasakian structures for contact pseudo-metrics.
New 3-manifolds created from 4-regular graphs with unique Eulerian cycles.
problem Creating compact 3-manifolds from specific graph structures. method Defining 3-manifolds via compatible Eulerian cycles in 4-regular graphs. result Each manifold in the class has a unique minimal ideal triangulation with n tetrahedra.