Paper classifies Randers metrics based on Ricci curvature properties.
problem Investigating isotropic projective Ricci curvature in Randers metrics.
method Classification of Randers metrics based on isotropic projective Ricci curvature properties.
result Randers metric of isotropic projective Ricci curvature is reversible if and only if it is of square projective Ricci curvature.
Investigates projective Finsler metrizability for non-isotropic SODEs.
problem Conditions for SODE solutions to be geodesics of a Finsler metric.
method Examined the Rapcsák system extended with curvature conditions, computed compatibility conditions.
result Found a class of non-isotropic sprays for which SODEs are projectively metrizable.
Study on spherical Finsler metrics with isotropic curvature rigidity.
problem Characterizing and understanding spherically symmetric Finsler metrics with isotropic E-curvature. method Provided the correct formula for mean Berwald curvature, established differential equations for projective and dual flatness, and derived a rigidity result.
result Rigidity result on spherically symmetric Finsler metrics with isotropic E-curvature. Study uses isotropic projection to link plane and null curves in Minkowski space.
problem Understanding the geometry of null curves in Minkowski 3-space.
method Utilizes isotropic projection of Laguerre geometry to establish a correspondence.
result Alternative description of plane curves congruent to a given one.
The paper extends Ricci curvature in Finsler geometry and finds conditions for specific metrics.
problem Characterizing and finding conditions for specific metrics in Finsler geometry.
method Introducing weighted projective Ricci curvature and analyzing Randers and Kropina metrics.
result Conditions for metrics to have weighted projective Ricci flat curvature.
New sprays of constant curvature introduced; conditions for metrizability given.
problem Conditions for sprays of isotropic curvature to be metrizable.
method Introducing sprays of constant curvature and investigating metrizability conditions.
result Sprays of isotropic curvature are not necessarily of constant curvature.
The paper classifies almost isotropic Kähler manifolds, proving their properties.
problem Classifying almost isotropic Kähler manifolds.
method Analyzing Jacobi operators and their ranks.
result Almost isotropic Kähler manifolds of dimension d=2n≥4 are isometric to complex projective space, complex hyperbolic space, or totally geodesically foliated by leaves isometric to Cn−1. Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projec…
We study the moduli space of CR-projective complex foliated tori. We describe it in terms of isotropic subspaces of Grassmannian and we show that it is a normal complex analytic space.
This work extends holomorphic surface representations to isotropic space.
problem Representing minimal surfaces in simply isotropic space with degenerate metrics.
method Developed new forms of Weierstrass and Björling representations for isotropic minimal surfaces.
result Holomorphic representations of isotropic minimal surfaces are achieved.
The article interprets Darboux's surfaces using triality in differential geometry.
problem Understanding Darboux's surfaces through triality.
method Geometric interpretation of Darboux's surfaces using triality.
result Darboux's surfaces are related to totally isotropic surfaces in a 6D projective quadric.
Study loxodromes and geodesics on rotational surfaces in pseudo-isotropic space.
problem No study on loxodromes in pseudo-isotropic space I_p^3.
method Define pseudo-isotropic angles, derive equations for space-like and time-like loxodromes and geodesics on rotational surfaces.
result Equations for space-like and time-like loxodromes and geodesics on rotational surfaces in pseudo-isotropic space.
Estimates index of harmonic maps from surfaces to complex projective spaces.
problem Estimating the index of harmonic maps from surfaces to complex projective spaces.
method Estimating dimensions of spaces of holomorphic sections of line bundles.
result Improved lower bounds on the index of harmonic maps.
The paper classifies spherically symmetric sprays and their curvature properties.
problem Understanding spherically symmetric sprays and their curvature.
method Established a canonical form and derived classification for projectively flat sprays.
result Derived explicit forms for sprays with isotropic and zero curvature.
Log-conformal projective pairs restrict to simple geometric structures.
problem Characterizing pairs of projective manifolds with logarithmic conformal tensors.
method Analyzing the nefness and triviality of KX+Δ to deduce geometric properties. result Pairs of projective manifolds with logarithmic conformal tensors are restricted to simple geometric structures.
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.
The paper classifies surfaces with isotropic circles through each point.
problem Classifying surfaces with isotropic circles through each point.
method Using isotropic circles as Euclidean circles and applying Skopenkov and Krasauskas' methods.
result Surfaces containing two isotropic circles through each point have a specific parametrization.
Finsler metrics of scalar flag curvature play an important role to show the complexity and richness of general Finsler metrics. In this paper, on an n-dimensional manifold M we study the Finsler metric F=F(x,y) of scalar flag curvature K=K(x,y) and discover some equations K should be satis…
Study proves rigidity of harmonic maps from 2-torus to complex projective space.
problem Rigidity of isotropic harmonic maps from a 2-torus to a complex projective space.
method Proves rigidity through holomorphic embeddings and complete linear systems.
result Ensures rigidity of harmonic bands in condensed matter physics.
In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic RP2m in CPn. Then it minimizes volume among the isotropic submanifolds in the same Z/2 homology class in CPn (but not among all submanifolds in this…
Study projective symmetries in Finsler spaces, showing reductions and constant flag curvature.
problem Exploring projective symmetries in Finsler spaces and their properties.
method Analyzing algebraic sub-algebras and curvature invariants of projective vector fields.
result Closed Finsler spaces with negative Ricci curvature reduce to Killing vector fields.
The paper classifies Willmore 2-spheres in Sn.
problem Classifying Willmore 2-spheres in Sn. method Construction of a harmonic sequence in the real Grassmannian over the Lorentz space, derived from the harmonic conformal Gauss map.
result Any Willmore 2-sphere is either totally isotropic or strictly k-isotropic. The paper explores curvature properties of specific metric spaces.
problem Investigating curvature properties of Cartan spaces with mth root metrics.
method Analyzing isotropic mean Berwald and Landsberg curvatures, and examining conformal β-changes.
result Conditions for conformal β-change of m-th root metrics to be locally dually flat and locally Minkowskian.
The paper proves the existence of isotropic complete solutions for generalized Hamilton-Jacobi equations.
problem Existence of isotropic complete solutions for generalized Hamilton-Jacobi equations.
method Analyzes symplectic manifolds, Hamiltonian vector fields, and fibrations to prove the existence of isotropic complete solutions.
result Proves the existence of isotropic complete solutions around almost every point of the manifold.
This article shows that every non-isotropic harmonic 2-torus in complex projective space factors through a generalised Jacobi variety related to the spectral curve. Each map is composed of a homomorphism into the variety and a rational map off it. The same ideas allow one to construct (pluri)-harmonic maps of finite ty…
The paper proposes using non-isotropic distances for more accurate trace link recommendation.
problem Time-consuming and error-prone creation and maintenance of trace links.
method Geometric viewpoint on semantic similarity using non-linear similarity measures.
result Non-isotropic distances improve trace link recommendation accuracy.
New symplectic groups defined for Lie subgroups of algebras.
problem Defining new symplectic groups for Lie subgroups of algebras.
method Introducing symplectic group Sp2(G,σ) for Lie subgroups G of associative algebras A with anti-involution σ. result New realizations of spin groups as Sp2(G,σ) for suitable subgroups G. The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Ri…
Classifies surfaces in isotropic geometry with constant curvature.
problem Classifying surfaces with constant curvature in isotropic geometry.
method Classifies surfaces under the condition that at least one translating curve lies in a plane.
result Classification of surfaces in isotropic geometry with constant curvature.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic S-curvature and are either Minkowskian or Riemannian. Study of invariant surfaces in isotropic and pseudo-isotropic geometries.
problem Prescribed curvature for invariant surfaces in singular metrics.
method Analysis of one-parameter subgroups of isotropic rigid motions, computation of fundamental forms and curvatures.
result Generalization of revolution and helicoidal surfaces to singular metrics.
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
Introduces symplectic groups over noncommutative algebras and their geometric actions.
problem Understanding symplectic groups over noncommutative algebras.
method Introducing symplectic groups Sp2(A,σ) over noncommutative algebras and constructing geometric spaces. result New insights into structure theory of classical Lie groups and construction of symmetric spaces.
The paper classifies surfaces in isotropic space satisfying Weingarten conditions.
problem Classifying surfaces in isotropic space with specific curvature conditions.
method Analyzing rotational surfaces in isotropic 3-space with Weingarten conditions.
result Classification of surfaces of linear Weingarten type in isotropic space.
Based on a self-contained, coordinate-free exposition of the necessary concepts and tools of spray and Finsler geometry (with detailed proofs), we derive new results among others on the consequences of the direction-independence of the Landsberg tensor and the stretch tensor of a Finsler manifold. We show that an at le…
The paper classifies pseudo-isotropic and lightlike pseudo-isotropic Lagrangian surfaces.
problem Characterizing Lagrangian surfaces in complex pseudo spaces.
method Analyzing properties of Lagrangian surfaces in complex pseudo spaces.
result Classification of Lagrangian surfaces based on pseudo-isotropy.
New isotropic tori found in complex space, not Hamiltonian isotopic.
problem Finding non-Hamiltonian isotopic isotropic tori in complex space.
method Analyzing isotropic tori in Cm for m>n≥2. result At least two exact isotropic n-tori in Cm are not Hamiltonian isotopic. In this paper, we find a condition on (α,β)-metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.
Study physical work done by isotropic vector forces along isotropic curves.
problem Investigate physical work done by isotropic vector forces.
method Analyze forces represented by isotropic vectors acting along isotropic curves on a manifold with specific metric structures.
result Calculate the work done by isotropic vector forces along isotropic curves.
Study isotropic Riemannian maps and helices along them.
problem Understanding Riemannian maps and their associated helices.
method Presented isotropic Riemannian maps and characterized helices along them.
result Characterization of helices along Riemannian maps.
We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…
The paper develops RM frames for isotropic and pseudo-isotropic spaces and characterizes spherical curves.
problem Differential geometry of curves in isotropic and pseudo-isotropic 3-spaces.
method Developing rotation minimizing frames and applying them to spherical curves in isotropic and pseudo-isotropic spaces.
result Characterization of spherical curves via linear equations involving curvatures and RM frames.
Study of surfaces in isotropic and pseudo-isotropic spaces using Gauss map and shape operator.
problem Differential geometry of surfaces in degenerate metric spaces.
method Introducing isotropic Gauss map and shape operator, computing curvature tensors and geodesics.
result New curvature tensor not vanishing identically, related to relative Gaussian curvature.
The paper classifies surfaces with constant curvature in isotropic space.
problem Classifying surfaces with constant curvature in isotropic space.
method Classification based on constant Gaussian and mean curvature.
result Non-existence result for surfaces with H/K=const.
Study isotropic curves on complex quadric with geometric relations.
problem Characterize isotropic curves on the complex quadric.
method Analyze geometric properties and relations of isotropic curves.
result Discovers relations between isotropic curves and surfaces in spaceforms.
In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald man…
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
Projection pursuit model improves Gaussian process regression for high-dimensional data.
problem Scalability issues with traditional Gaussian process models in high dimensions.
method Additive Gaussian process regression with dimension expansion and gradient descent.
result The proposed method approximates more complex functions and outperforms traditional models.