The paper finds unique equilibrium states for geodesic flows on certain surfaces.
problem Analyzing geodesic flows on surfaces without focal points.
method Proving the existence of unique equilibrium states for specific potentials.
result There are unique equilibrium states for certain potentials, including geometric multiples of scalar less than 1.
The geodesic flow on certain surfaces is shown to be ergodic.
problem Ergodicity of geodesic flows on surfaces without focal points.
method Analyzing geodesic flows on surfaces with no focal points, proving ergodicity under specific curvature conditions.
result The geodesic flow on the unit tangent bundle of a surface with no focal points is ergodic with respect to the Liouville measure.
Study focal surfaces of wave fronts with unbounded curvatures.
problem Characterizing singularities of focal surfaces near non-degenerate singular points.
method Characterizations based on types of singularities and geometrical properties of initial fronts.
result Investigation of Gaussian curvature behavior of focal surfaces.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.
In this note we show that a compact asymptotically harmonic manifold without focal points is either flat or a rank one locally symmetric space.
In this note, we consider the rigidity of the focal decomposition of closed hyperbolic surfaces. We show that, generically, the focal decomposition of a closed hyperbolic surface does not allow for non-trivial topological deformations, without changing the hyperbolic structure of the surface. By classical rigidity theo…
Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.
problem Investigating singularities of evolutes and focal surfaces of pseudo-spherical framed immersions.
method Introduced pseudo-spherical non-null framed curves, defined moving frames, and analyzed evolutes and focal surfaces.
result Evolutes of pseudo-spherical framed immersions are the sets of singular points of their focal surfaces.
Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.
problem Investigate differential geometry properties of framed curves, including singular points.
method Use Legendrian dualities to analyze focal surfaces and evolutes of hyperbolic framed curves.
result Show the relationship among focal surfaces, evolutes, and dual surfaces of evolutes.
The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.
problem Characterizing and understanding the singularities and geometric properties of surfaces formed by the singular loci of normal congruences of frontals with pure-frontal singular points.
method Characterizations of singularities in terms of geometric invariants of the initial frontal are provided for the normal ruled surface. Relations between certain singularities of focal surfaces and geometric properties of the frontal are also explored.
result Behavior of Gaussian curvature of focal surfaces of frontals with a 5/2-cuspidal edge is considered. We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by const…
The study counts geodesics on special manifolds without focusing points.
problem Counting geodesics on specific types of manifolds.
method Margulis-type asymptotic estimates and analysis of geodesic flow.
result The geodesic flow on these manifolds has a unique measure of maximal entropy with the Bernoulli property.
In this paper, we prove the existence of energy minimizers in each free homotopy class of maps between polyhedra with target space without focal points. Our proof involves a careful study of some geometric properties of riemannian polhyedra without focal points. Among other things, we show that on the relevant polyhedr…
The paper proves conditions for Anosov geodesic flows on non-compact manifolds.
problem Conditions for Anosov geodesic flows on non-compact manifolds.
method Analyzing sectional curvature and focal points to determine Anosov geodesic flows.
result A sufficient condition for Anosov geodesic flows on non-compact manifolds.
The paper classifies surfaces formed by quadrilateral gluings.
problem Classifying topological surfaces formed by quadrilateral gluings.
method Review of graphs embedded into surfaces, algorithms based on labeling schemes of fundamental polygons.
result Computing numbers of possible gluings for classification.
In this paper we consider on a complete Riemannian manifold M an immersed totally geodesic hypersurface $\Si$ existing together with an immersed submanifold N without focal points. No curvature condition is needed. We obtained several connectedness results relating the topologies of M and $\Si$ which depend on th…
Totally geodesic maps studied in manifolds without focal points.
problem Understanding maps with minimal energy in nonpositive curvature manifolds.
method Path-connectedness and energy minimization approach, avoiding geometric flows and Bochner identities.
result Totally geodesic maps are homotopic to energy-minimizing ones in nonempty classes.
Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
Study on focal surfaces of tubular surfaces in 3D space, focusing on their flatness and asymptotic properties.
problem Characterizing and understanding focal surfaces of tubular surfaces in 3D space.
method Defined tubular surfaces using Frenet and Darboux frames, analyzed their focal surfaces, and derived conditions for flatness.
result No minimal focal surface exists in 3D space for tubular surfaces.
Self-focal points on ellipsoids of dimension 3 or higher are rare.
problem Existence of self-focal points on Riemannian manifolds of dimension 3 or higher.
method Analyzing geodesics and umbilic points on ellipsoids of various dimensions.
result Ellipsoids of dimension 3 or higher with at least 4 distinct axes have no self-focal points.
Proves minimum number of normals to curves in 3D space.
problem Finding the minimum number of normals to closed curves in 3D.
method Morse theory for squared distance function and self intersections of the focal surface.
result For generic curves, points have at least 6, 8, or 10 normals depending on knotting.
The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the hölder continuity of local energy minimizers between Riemannian polyhedra with the target spaces without focal points. With this new result, we also complete ou…
We characterize singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.
Study convex functions on manifolds without focal points, deriving new spectral properties.
problem Convex functions on manifolds without focal points.
method Geometrically defined convex functions, spectral analysis.
result Spectrum is purely absolutely continuous on certain manifolds.
The paper solves the Dirichlet problem at infinity and defines Poisson boundaries for certain manifolds.
problem Existence of bounded harmonic functions on manifolds without conjugate points.
method Investigation of harmonic extensions and Poisson boundaries for specific types of manifolds.
result Harmonic extensions and Poisson boundaries defined for rank 1 manifolds without focal points.
We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal…
The space L of oriented lines, or rays, in R3 is a 4-dimensional space with an abundance of natural geometric structure. In particular, it boasts a neutral Kähler metric which is closely related to the Euclidean metric on R3. In this paper we explore the relationship between the focal se…
Study of families of lines on spheres and their focal sets.
problem Characterizing families of lines on spheres and their geometric properties.
method Analyzing submanifolds of TSn and their focal sets, using symplectic structures and sectional curvatures. result Derivation of formulas relating sectional curvatures of focal sets to differences in radii of curvature of generating hypersurfaces.
Study helicoidal surfaces from frontals, revealing geometric rigidity and stability of singularities.
problem Investigate helicoidal surfaces of frontals in Euclidean space.
method Using Legendre curves and framed surfaces, derive curvature expressions and analyze deformations.
result Singularities of curves persist under deformations, revealing geometric rigidity and stability.
The paper studies non-integer curvature flows and proves convergence to spheres under specific conditions.
problem Analyzing the convergence of non-integer curvature flows on rotationally symmetric surfaces.
method Spectral theory of singular Sturm-Liouville operators to construct an eigenbasis and prove convergence.
result The flow converges to a round sphere if the focal points coincide at the poles, otherwise to a non-round Hopf sphere.
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the a…
The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank~1. This conjecture has been proved by Z. Szabó \cite{Sz} for harmonic manifolds with compact universal cover. E. Damek and F. Ricci \cite{DR} provided examples showing that in the noncompact case the conj…
Study on focal locus of submanifolds in Finsler manifolds, showing regularity and smoothness.
problem Investigating the focal locus of submanifolds in Finsler manifolds.
method Using the normal exponential map and extending Warner's ideas, studying connected components and smoothness of focal time maps.
result Identified an open and dense subset where focal time maps are smooth, provided they are finite.
We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting.
The article proves a unique invariant measure for geodesic flows on certain rank 1 manifolds.
problem Existence and uniqueness of invariant measure for geodesic flows.
method Using Patterson-Sullivan measure and Busemann density.
result Geodesic flow on compact rank 1 manifolds has a unique invariant measure of maximal entropy.
Study of knotted defects in smectic liquid crystals using topological knot theory.
problem Understanding the topological structure of knotted defects in smectic liquid crystals.
method Investigation of screw and edge dislocations, focusing on their radial surface structure and knot fibration.
result Established a connection between smectic defects and knot theory, revealing the topological knotting of defects.
An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines CA, the set of points with Condition A in focal submanifolds. It turns out that the points in $…
The focal locus ΣX of an affine variety X is roughly speaking the (projective) closure of the set of points O for which there is a smooth point x∈X and a circle with centre O passing through x which osculates X in x. Algebraic geometry interprets the focal locus as the branching locus of the endpoi…
Study on unfolding maps of surfaces in 3D space, proving versality conditions.
problem Investigating the versality of rotation unfolding of folding maps for surfaces in R3. method Introducing and analyzing the rotation unfolding of folding maps, proving versality conditions in terms of geometry.
result Proved conditions for the rotation unfolding to be versal, showing diffeomorphic type of tangent plane locus.
Given a closed Riemannian manifold (M, g), there is a partition Σ_i of its tangent bundle TM called the focal decomposition. The sets Σ_i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note,…
This paper connects billiards in ellipses to focal billiards in ellipsoids.
problem Proving the existence of isometric counterparts between billiards in ellipses and focal billiards in ellipsoids.
method Continuous transition via isometric focal billiards in a fixed ellipsoid.
result Established the connection between planar and spatial billiards.
We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.
Focal loss improves deep neural networks' accuracy and calibration.
problem Miscalibration in deep neural networks.
method Using focal loss and temperature scaling to improve model calibration.
result Focal loss leads to state-of-the-art calibrated models without sacrificing accuracy.
The {\em focal curve} of an immersed smooth curve γ:s↦γ(s), in Euclidean space Rm+1, consists of the centres of its osculating hyperspheres. The focal curve may be parametrised in terms of the Frenet frame of γ (t,n1,...,nm), as $C_γ(s)=(γ+c_1{\bf n}_1+c_2{\bf n}_2+...+c_m{\bf n}…
Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces − minimal isoparametric hypersurfaces and focal submanifolds. By virtue of their rich structures, we find new series of minimizing cones. They are cones over focal submanifolds and cones over suitable products among th…
Algorithm selects variables and bandwidths for geographically weighted regression.
problem Estimating variable subsets and bandwidths for geographically weighted regression.
method Mathematical programming-based approach integrating variable selection and bandwidth estimation.
result Proposed algorithm provides stable spatially varying patterns with competitive explanatory power.
Focal loss reduces model curvature for better calibration.
problem Improving model confidence in classification problems.
method Geometric interpretation of focal loss to reduce curvature.
result Focal loss reduces the curvature of the loss surface, enhancing model calibration.
The normal map of curves is analyzed as a vector field on a cylinder.
problem Understanding the geometric properties of normal maps and their vector field interpretation.
method Interpreting critical points geometrically, studying Poincaré index, projecting to sphere, and analyzing winding and rotation indices.
result Counting theorems regarding winding and rotation indices of curves and their evolutes are proven.