No conformal product structures on compact manifolds with constant curvature.
problem Existence of conformal product structures on compact manifolds with constant curvature.
method Analyzing non-flat manifolds and irreducible, compact locally symmetric spaces of non-positive curvature.
result Compact non-flat manifolds with constant sectional curvature admit no conformal product structure.
New curvature equations obstruct integrability of complex structures.
problem Understanding curvature obstructions to integrability of complex structures.
method Direct approach using Nijenhuis tensor derivatives and curvature scalars.
result Certain complex structures cannot coexist with non-flat constant curvature metrics.
Study open Alexandrov spaces with nonnegative curvature, proving structural results.
problem Understanding open Alexandrov spaces with nonnegative curvature.
method Establishing structural results on open Alexandrov spaces.
result Structural results on open Alexandrov spaces with nonnegative curvature.
Study on G2-structures with negative Ricci curvature on closed and noncompact manifolds.
problem Existence and properties of closed G2-structures with negative Ricci curvature. method Analyzing existence and nonexistence of closed G2-structures with negative Ricci curvature on closed and noncompact manifolds. result No closed manifold admits a closed G2-structure with negative Ricci curvature. For noncompact manifolds, restrictions on lengths of geodesics are found. In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann-Christoffel curvature tensor, the same type structure given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various ge…
Modeling curvature-sensitive cells in visual cortex using manifold geometry.
problem Understanding how curvature influences cell function in the visual cortex.
method Developed a 4D manifold with canonical Engel structure to represent orientation, position, curvature, and scale.
result Characterized curvature-sensitive receptive profiles using left-invariant generators of the Engel structure.
The φ-sectional curvature of statistical structures on almost contact metric manifolds is always non-positive.
problem Analyzing the φ-sectional curvature of statistical structures on almost contact metric manifolds.
method Investigating the φ-sectional curvature induced by a statistical structure and deriving sufficient conditions.
result The φ-sectional curvature is always non-positive.
In this paper, we define almost paracontact and normal almost paracontact Finsler structures on a vector bundle and find some conditions for integrability of these structures. We define paracontact metric, para- Sasakian and K-paracontact Finsler structures and study some properties of these structures. For a K-paracon…
Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.
problem Whether mean curvature flow is a gradient flow on nondegenerate metric spaces of simple closed plane curves.
method Examined two nondegenerate metric spaces: uniformness-preserving and curvature-weighted structures.
result Mean curvature flow is not a gradient flow on either metric space.
Deform quantization recovers scalar curvature in complex structures.
problem Recovering scalar curvature in complex structures.
method Formal moment map construction on almost complex structures.
result Formal moment map deforms scalar curvature moment map in integrable cases.
Study the structure of Kähler foliations with negative Ricci curvature.
problem Characterize the structure of Kähler foliations with negative Ricci curvature.
method Prove a de Rham type theorem decomposition on the leaf space.
result Characterize each factor in the decomposition of the leaf space.
This paper connects graph curvature to community structure.
problem Understanding the relationship between network curvature and community formation.
method Defining curvature on networks and analyzing its relation to community structure.
result Apriori bounds on the curvature of intercommunity edges.
The projective curvature tensor P is invariant under a geodesic preserving transformation on a semi-Riemannian manifold. It is well known that P is not a generalized curvature tensor and hence it possesses different geometric properties than other generalized curvature tensors. The main object of the present paper …
Solves curvature problems on manifolds with negative curvature.
problem Prescribed curvature problems on closed manifolds with negative curvature.
method Investigates fully nonlinear prescribed curvature problems for modified Schouten tensor on closed Riemannian manifolds with negative curvature.
result Proves solvability of curvature problems under certain conditions.
The paper shows inequality and rigidity for manifolds with integral Ricci curvature.
problem Analyzing structures of manifolds with integral Ricci curvature.
method Using segment inequality and similar methods as in \cite{CC1}, derive almost rigidity structure results.
result Sharp Hölder continuity result holds in the limit space of manifolds with integral Ricci curvature bound.
We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold M there is a metric, such that th…
The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.
problem Discrete conformal structures on polyhedral surfaces and their rigidity.
method Parameterized combinatorial curvature, combinatorial α-Ricci flow, and flow extension through singularities.
result Existence and convergence of combinatorial α-Ricci flow for solving the Yamabe problem.
The Ricci flow preserves product structures with instantaneous curvature bounds.
problem Preserving product structures under Ricci flow with curvature constraints.
method Proving a constant ε exists such that if a solution splits as a product at time 0 and has bounded curvature, it splits for all time.
result A constant ε exists depending on dimension such that if a solution splits as a product at time 0 and has curvature bounded by ε/t, it splits for all time.
The paper finds and analyzes the Funk-Finsler structure in constant curvature spaces.
problem Investigating the Funk-Finsler metric in spaces of constant curvature.
method Explicitly computed S-curvature, Riemann curvature, Ricci curvature, and flag curvature. result The S-curvature and flag curvature of the Funk-Finsler metric in hyperbolic, spherical, and Euclidean spaces are bounded. The paper finds the Finsler structure of Apollonian weak metric on unit disc.
problem Understanding the Finsler structure of Apollonian weak metric on the unit disc.
method Analyzing the deformation of hyperbolic Poincaré metric by a closed 1-form.
result The Apollonian weak-Finsler structure has bounded below S-curvature and flag curvature K satisfying −∞<K<−1. The paper explores Kähler-like metrics on generalized flag manifolds.
problem Finding invariant almost Hermitian structures with specific scalar curvature properties.
method Investigating invariant almost Hermitian geometry on generalized flag manifolds, focusing on Kähler-like metrics.
result Examples of Kähler-like metrics satisfying s=2smC are provided. Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature fo…
The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.
problem Classifying flag manifolds with specific isotropy components and finding conditions for Kähler-like scalar curvature.
method Investigating invariant almost Hermitian structures on generalized flag manifolds with two or three irreducible components.
result Classification of flag manifolds admitting Kähler-like scalar curvature and conditions for such structures.
Local flatness theorem for paraquaternionic contact structures.
problem Local flatness of paraquaternionic contact manifolds.
method Defined paraquaternionic contact conformal curvature tensor and showed local flatness condition.
result Paraquaternionic contact conformal curvature vanishing implies local flatness.
Study 2D spaces with curvature, finding a graph structure.
problem Understanding the geometry of 2D spaces with curvature constraints.
method Analyzing spaces as unions of disks, identifying singular points.
result Obtained a graph structure of topological singular points.
The Funk-Finsler structure is constructed in hyperbolic models, including the Klein unit disc.
problem Constructing Funk-Finsler structures in hyperbolic models.
method Using Finsler isometries and explicit computations, the Funk-Finsler structure is constructed in various hyperbolic models.
result The Funk-Finsler structure in the Klein unit disc is a Randers metric.
Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.
problem Investigating properties of Kähler manifolds with specific curvature conditions.
method Conformal perturbation method.
result Established structure and splitting theorems for Kähler manifolds with non-negative mixed curvature.
Paper proposes LCP for structural encodings, outperforming existing methods.
problem Improving Graph Neural Networks performance through effective structural encodings.
method Geometric perspective, Local Curvature Profiles (LCP) for structural encodings, combining with global positional encodings, comparing with rewiring techniques.
result LCP significantly outperforms existing structural encodings and combining LCP with global positional encodings improves performance.
Study on surface geometry in Lie groups with CR structures.
problem Understanding surface curvature in Lie groups with CR structures.
method Defined Gauss and mean curvature in Tanaka-Webster geometry.
result Gave specific examples of surface curvature calculations.
The paper studies geometric structures of curvature radii on Riemannian manifolds.
problem Exploring geometric structures of curvature radii on Riemannian manifolds.
method The main method involves constructing a pair of global vector fields f1,f2 that encode intrinsic geometry. result Existence of sub-Riemannian manifolds associated with curvature radii and investigation of their properties.
Study symplectic scalar curvature on supermanifolds.
problem Define and analyze symplectic scalar curvature on supermanifolds.
method Introduced two families of odd super-Fedosov structures using graded symmetric and non-symmetric connections.
result Found non-trivial odd symplectic scalar curvature for the second family.
Constructs flows on manifolds with small curvature, proving Euclidean topology.
problem Geometric structure of manifolds with unbounded curvature.
method Distance like functions with integral hessian bound, Ricci flows.
result Manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors n-regular and small curvature concentration are topologically Euclidean. Extending the work of G. Székelyhidi and T. Brönnle to Sasakian manifolds we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex …
The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.
problem Finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows, including Ricci flow and Calabi flow, are applied to deform Glickenstein's discrete conformal structures.
result The solution of the combinatorial Ricci flow can be uniquely extended and converges exponentially fast for any initial value under certain conditions.
Study curvature properties of specific Riemannian manifolds with skew-circulant structures.
problem Investigate curvature of Riemannian manifolds with a particular tensor structure.
method Analyze 4D Riemannian manifolds with right skew-circulant tensor S, invariant under S and g, focusing on Ricci tensor and sectional curvatures.
result Obtained properties of curvature tensors and sectional curvatures for specific manifolds.
Statistical manifolds with constant curvature are projectively flat and symmetric.
problem Characterizing statistical manifolds with constant curvature.
method Analyzing the curvature and projective flatness properties of statistical manifolds.
result Statistical manifolds with constant curvature are projectively flat and symmetric.
Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.
problem Understanding Weyl curvature pinching on 4-manifolds.
method Analyzing harmonic and pinched self-dual Weyl curvature, proving anti-self-duality.
result Proves anti-self-duality for compact 4-manifolds with pinched self-dual Weyl curvature.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersufaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in t…
In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the curvature satisfies some inequalities.
The study defines a canonical nilpotent structure for certain collapsed manifolds.
problem Understanding the structure of collapsed Riemannian manifolds.
method Analyzes the nilpotent structure of manifolds with bounded Ricci curvature and Reifenberg local covering geometry.
result A canonical nilpotent structure can be defined and uniquely determined over regular limit spaces.
Develops scalar curvature in generalized Kahler geometry and shows constant scalar curvature on compact Lie groups.
problem Defines scalar curvature in generalized Kahler geometry.
method Introduces scalar curvature in terms of pure spinors formalism and develops a moment map framework.
result Scalar curvature is given by the moment map, generalizing results from ordinary Kahler geometry.
In 5D, integrability is linked to curvature constraints of subconformal structures.
problem Dispersionless integrability in 5D partial differential equations.
method Relating integrability to curvature constraints of subconformal structures.
result In 5D, integrability is characterized by the vanishing of a certain curvature of the subconformal structure.
Using spinc structure we prove that Kähler-Einstein metrics with nonpositive scalar curvature are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Moreover if all infinitesimal complex deformation of the complex structure are integrable, th…
Study left invariant spray structures on Lie groups, calculating curvature and geodesics.
problem Understanding curvature and geodesics in left invariant spray structures on Lie groups.
method Use invariant frames and canonical bi-invariant Berwald spray structure to analyze left invariant spray structures.
result Established correspondence between geodesics and inverse integral curves of spray vector fields.
Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.
problem Collapsing geometry of Riemannian manifolds with Ricci curvature constraints.
method Locally bounded Ricci covering geometry and Ricci flow smoothing techniques.
result Volume collapsed Calabi-Yau manifolds admit Ricci-flat Kähler metrics and compatible Killing structures.
Cheeger and Gromov showed that F-structures are related to collapse with a double-sided curvature bound. We define fibered F-structures and extend some of the Cheeger-Gromov results to the setting of collapse with a lower bound on the curvature operator.
We obtain a class of locally symmetric Kaehler Einstein structures on the cotangent bundle of a Riemannian manifold of negative sectional curvature. Similar results are obtained in the case of a Riemannian manifold of positive sectional curvature. The obtained class of Kaehler Einstein structures depends on one essenti…