Paper explores folding patterns of curved creases preserving their geometric properties.
problem Investigating rigid-ruling folding motions of curved crease-rule patterns.
method Deriving conditions for rigid-ruling foldability and analyzing combinations of creases.
result Constant fold-angle creases are only compatible with other constant fold-angle creases.
The task of translating between programming languages differs from the challenge of translating natural languages in that programming languages are designed with a far more rigid set of structural and grammatical rules. Previous work has used a tree-to-tree encoder/decoder model to take advantage of the inherent tree s…
The paper studies circle packings using renormalization and subdivision rules.
problem Characterizing and proving properties of circle packings with specific subdivision rules.
method Iterations of skinning maps on Teichmüller spaces, renormalization theory, subdivision rules.
result Uniformly contracting renormalization operator and geometric inflexibility of circle packings.
Study of flat ribbons constructed along curves in 3D space.
problem Determine the conditions for a ruled structure to form a flat ribbon.
method Investigate the ruled structure of flat ribbons and calculate energy bounds.
result There exists a well-defined flat ribbon only up to an initial condition.
A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus g≥1 is itself a ruled surface over a curve of genus g. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case g≥2.
The four-dimensional sphere is uniquely rigid in terms of scalar curvature.
problem Proving the uniqueness of the four-dimensional sphere in terms of scalar curvature.
method Combining harmonic map heat flow and Ricci flow to rule out non-isometric maps.
result A smooth map of non-zero degree from a four-dimensional manifold to the unit four-sphere is an isometry.
For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to po…
Projective structures are mostly rigid at the boundary but some are not.
problem Boundary rigidity of projective structures.
method Investigation of projective structures on manifolds with boundary.
result Existence of non-rigid projective structures and characterization of them.
We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?
We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spac…
We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.
The study of rigidity theorems on 4-manifolds with boundary.
problem Understanding topological restrictions on 4-manifolds with boundary.
method Introducing new conformal and smooth invariants, studying Weyl functional, and analyzing the expansion of a smooth Riemannian metric near the boundary.
result Established several conformally invariant rigidity theorems for 4-manifolds with boundary.
Study shows Einstein structures on 4-manifolds are rigid.
problem Rigidity of Einstein structures in four dimensions.
method Examined deformations of the round four-sphere and analyzed self-dual structure of Einstein manifolds.
result Any deviation from the standard metric of the round four-sphere breaks the Einstein condition.
Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.
problem Rigidity and existence of discrete conformal structures on surfaces with boundary.
method Axiomatic framework and classification of discrete conformal structures.
result Extends results by Guo-Luo and Guo to a general context.
Projective rigidity of circle packings on complex surfaces proved.
problem Proving rigidity of circle packings on complex projective surfaces.
method Proved projective rigidity through triangulations and complex projective structures.
result Space of circle packings is projectively rigid on complex projective surfaces.
The study proves rigidity for mixed Hodge structures and applies to curve families.
problem Rigidity of period maps for mixed Hodge structures.
method Holomorphic bisectional curvature approach.
result Establishes rigidity in various cases, including curve families.
In this paper, we consider non developable ruled surface with spacelike ruling, timelike ruling, respectively. We give the relations between the structure functions with the curvature and torsion of the striction line of the timelike and spacelike non developable ruled surfaces. Also, we have calculated the gaussian an…
Study semilinear equations on weighted manifolds to prove rigidity.
problem Prove rigidity of weighted manifolds via classification of semilinear equations.
method Classify positive solutions at the Sobolev-critical exponent, proving rigidity and weight triviality.
result Existence of positive solutions implies rigidity and weight triviality under certain curvature 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.
Rigorous model for 2-gerbes simplifies calculations in physics.
problem Constructing explicit geometric string structures.
method Defined a rigid model for bundle 2-gerbes and connections, proving equivalence to existing models.
result Chern-Simons bundle 2-gerbe can be rigidified and described via geometric string structures.
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
Unified framework for rigidity results on (κ,μ)-manifolds.
problem Rigidity of metrics on (κ,μ)-manifolds. method Study of deviations preserving bi-Legendrian structure, orthogonalizing canonical structure.
result Unified rigidity results in both Riemannian and semi-Riemannian categories.
Paper proves rigidity of discrete conformal structures on polyhedral surfaces.
problem Rigidity of discrete conformal structures on polyhedral surfaces.
method Variational principles.
result Proves Glickenstein's conjecture on the rigidity of discrete conformal structures.
Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.
problem Understanding the structure of Busemann spaces with measures.
method Analyzing geodesic completeness and non-collapse assumptions.
result Rigidity and structure theorems for Busemann spaces with MCP.
This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains.
Counterexample shows ADC contact structures can't have isomorphic cohomologies.
problem Rigidity of ADC contact structures and cohomology isomorphisms.
method Provided a counterexample to show non-isomorphic cohomologies.
result ADC contact structures do not have isomorphic integral cohomologies.
Model learns molecular structures from graphs without explicit rules.
problem Learning molecular structures from graphs without explicit rules.
method Adapted Transformer model for undirected molecular graphs.
result Transformer model can learn complex molecular structures.
We establish Bochner-type formulas for operators related to CR automorphisms and spherical CR structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.
Paper recovers uncertainty from dynamic valuation rules.
problem Recovering latent uncertainty from observable valuation rules.
method Developed procedures to identify and characterize uncertainty structures from valuation rules.
result Valuation rules contain sufficient information to identify and recover uncertainty structures.
The classical Liouville Theorem on conformal transformations determines local conformal transformations on the Euclidean space of dimension ≥3. Its natural adaptation to the general framework of Riemannian structures is the 2-rigidity of conformal transformations, that is such a transformation is fully determined…
New method for flexible tubes and structures, enabling rigid-foldability.
problem Creating flexible tubes with rigid-foldability.
method Discrete, semi-discrete, and smooth construction of surfaces (T-hedra and profile-affine surfaces).
result Unified treatment of continuous flexible structures composed of tubes.
To a hyperbolic manifold one can associate a canonical projective structure and ask whether it can be deformed or not. In a cusped manifold, one can ask about the existence of deformations that are trivial on the boundary. We prove that if the canonical projective structure of a cusped manifold is infinitesimally proje…
We show that derivations of the differential structure of a subcartesian space satisfy the chain rule and have maximal integral curves.
We study the moduli space of quaternionic Kaehler structures on a compact manifold of dimension 4n (n>2) from a point of view of Riemannian geometry, not twistor theory. Then we obtain a rigidity theorem for quaternionic Kaehler structures of nonzero scalar curvature by observing the moduli space.
Two exceptional flag manifolds' complex structures are studied, proving rigidity for one.
problem Proving rigidity of complex structures on two exceptional flag manifolds.
method Homogeneous Kähler manifolds under G2 group, relating to sphere complex structures. result Rigidity of complex structure proved for one manifold.
Study evaluates multivariate forecasting scoring rules and proposes new copula-based ones.
problem Evaluating and improving multivariate probabilistic forecasting methods.
method Analysis and comparison of existing scoring rules, development of copula-based scoring rules, simulation studies, and real data analysis.
result Proposed copula-based scoring rules provide strong distinction between models with correct and incorrect dependency structures.
Study second-order obstruction to nearly G2 structure deformations.
problem Proper nearly G2 structure rigidity on Aloff-Wallach space. method Second-order obstruction analysis, building on Alexandrov and Semmelmann work.
result Proves rigidity for nearly G2 structure on N(1,1). Following a survey of the abstract boundary definition of Scott and Szekeres, a rigidity result is proved for the smooth case, showing that the topological structure of the regular part of this boundary in invariantly defined.
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤π, possibly with boundary consisting of totally geodesic hyperbo…
Survey on harmonic maps in non-smooth spaces, focusing on rigidity.
problem Rigidity phenomena in non-smooth spaces.
method Regularity theory of harmonic maps to non-smooth targets.
result Generalizations of Margulis superrigidity and holomorphic rigidity of Teichmüller space.
Study on mappings in Carnot groups, proving rigidity results.
problem Understanding mappings in Carnot groups and proving rigidity.
method Structural results for Sobolev mappings, proving rigidity or regularity.
result Establishes partial rigidity and partial regularity theorems.
Study constructs H-space on metric spaces, providing rigidity criteria.
problem Understanding diffeomorphism actions on spaces of metrics.
method Constructs H-space multiplication on R+(M) for nullcobordant manifolds. result Provides rigidity criterion for diffeomorphism group action.
A new method for non-rigid point set registration reduces computational complexity.
problem Efficiently registering non-rigid point sets with large numbers of points.
method Structured Analytic Coherent Point Drift (Analytic-CPD) reformulates CPD for structured analytic mappings.
result Analytic-CPD reduces computational complexity by controlling the deformation model's dimensionality.
Combining deep neural networks with structured logic rules is desirable to harness flexibility and reduce uninterpretability of the neural models. We propose a general framework capable of enhancing various types of neural networks (e.g., CNNs and RNNs) with declarative first-order logic rules. Specifically, we develop…
Study proves structure results for homogeneous spaces supporting specific equations.
problem Proving structure results for homogeneous spaces supporting specific equations.
method Analyzing homogeneous spaces with non-constant solutions to two general classes of equations involving the Hessian and an invariant 2-tensor.
result Generalizes rigidity results for gradient Ricci solitons and warped product Einstein metrics.
Stable generalized complex structures on certain surfaces are constant.
problem Existence of stable generalized complex structures on ruled surfaces.
method Analysis of sphere bundles over surfaces of genus ≥2.
result Stable generalized complex structures on these surfaces are of constant type.
For 3-dimensional hyperbolic cone structures with cone angles θ, local rigidity is known for 0≤θ≤2π, but global rigidity is known only for 0≤θ≤π. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most π do not degenerate in defo…