Study of extended Bianchi-Cartan-Vranceanu spaces in seven dimensions.
problem Understanding geometric properties of extended Bianchi-Cartan-Vranceanu spaces.
method Investigation of Levi-Civita connection, Ricci curvatures, Killing fields, and geodesics in seven-dimensional EBCV spaces.
result Characterization of geometric properties in extended Bianchi-Cartan-Vranceanu spaces.
Researchers found explicit formulae for special solitons in 3D spaces.
problem Understanding Ricci solitons in specific 3D Lorentzian spaces.
method Analyzing homogeneous Ricci solitons on Bianchi-Cartan-Vranceanu spaces.
result Explicit formulae for Ricci solitons were derived.
The paper extends Bour's theorem to BCV spaces with constant mean curvature helicoids.
problem Generalizing Bour's theorem to helicoidal surfaces in BCV spaces.
method Two-parameter family representation of helicoidal surfaces.
result Characterization of helicoidal surfaces with constant mean curvature.
Study classifies triharmonic surfaces in 3D homogeneous spaces.
problem Classifying triharmonic surfaces in 3D homogeneous spaces.
method Classification through isoparametric and CMC surfaces.
result Complete classification of CMC r-harmonic Hopf cylinders in BCV-spaces.
Study on triharmonic curves in 3D spaces, proving their existence and classification.
problem Characterizing triharmonic curves in 3D homogeneous spaces.
method Analyzing curves with constant curvature in Riemannian manifolds, focusing on Frenet helices and space forms.
result Classification of triharmonic Frenet helices in space forms and Bianchi-Cartan-Vranceanu spaces.
We prove that a totally umbilical biharmonic surface in any 3 3 3 -dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if and only if it is a part of S 2 ( 1 / 2 ) S^2(1/\sqrt{2}) S 2 ( 1/ 2 ) in S 3 S^3 S 3 . We also give complete c…
The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.
problem Existence and classification of polyharmonic helices of order r.
method Analytical and geometric approaches, including Lie group theory and Euclidean sphere analysis.
result Complete classification of proper r-harmonic helices in Sol_3 and new examples in Bianchi-Cartan-Vranceanu spaces.
In this article we generalize the notion of constant angle surfaces in S^2 x R and H^2 x R to general Bianchi-Cartan-Vranceanu spaces, i.e. essentially to three-dimensional homogeneous spaces with a four-dimensional isometry group. We show that these surfaces have constant Gaussian curvature and we give a complete loca…
Biconservative hypersurfaces are hypersurfaces with conservative stress-energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3-dimensional Bianchi-Cartan-Vranceanu spaces,…
We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-Cartan-Vranceanu family. This gives a positive answer to a conjecture formulated in 2002. As a partial result, we prove that totally umbilical surfa…
The Bonnet problem is solved for specific Thurston geometries.
problem Determining when an isometric immersion can be continuously deformed through isometric immersions preserving principal curvatures.
method Study of Bonnet pairs in Bianchi--Cartan--Vrănceanu spaces and S o l 3 \mathrm{Sol}_3 Sol 3 ; use of differential equations. result Uniqueness of Bonnet mates in specific Thurston geometries.
Totally biharmonic hypersurfaces in space forms and 3D BCV spaces classified.
problem Characterizing totally biharmonic hypersurfaces in space forms and 3D BCV spaces.
method Analyzing geodesics and isoparametric properties to classify hypersurfaces.
result Classification of totally biharmonic hypersurfaces in space forms and 3D BCV spaces.
Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.
problem Characterizing homogeneous Lorentzian three-manifolds with a 4D isometry group.
method Explicit global coordinate description and proof of Ricci soliton properties.
result All special examples are non-gradient expanding Ricci solitons.
Study on extended weakly symmetric spaces, classifying and providing an example.
problem Understanding geometric properties of extended weakly symmetric spaces.
method Classification and presentation of a non-trivial example.
result Existence of extended weakly symmetric spaces established.
To give a Cartan calculus on the extended quantum 3d space, the noncommutative differential calculus on the extended quantum 3d space is extended by introducing inner derivations and Lie derivatives.
Study geodesic extendibility on metric spaces and map them to a half-space.
problem Geodesic extendibility on metric spaces.
method Explicit isometry between ( Σ ( X ) , d H ) (Σ(X),d_H) ( Σ ( X ) , d H ) and X i m e s R ≥ 0 X imes \mathbb{R}_{\ge 0} X im es R ≥ 0 . result Established group isometry between Iso(X,d) and Iso(Σ(X),d_H) for Hadamard spaces.
Extended Rank-One Theorem to special metric spaces.
problem Extending a theorem to new types of spaces.
method Applied to a new class of metric measure spaces.
result Rank-One Theorem proven for R C D ( K , N ) \mathrm{RCD}(K,N) RCD ( K , N ) spaces. Extends h-principle to stratified spaces using sheaf and jet theories.
problem Applying h-principle to stratified spaces.
method Developed new sheaf and bundle theories for stratified spaces, and proved the h-principle.
result Stratified continuous sheaves and homotopy fiber sheaves lead to the parametric h-principle.
Hodge decomposition extended to H 1 H^1 H 1 space on non-compact manifolds.
problem Extending Hodge decomposition to non-compact manifolds and Sobolev spaces.
method Generalization of Hodge decomposition to H 1 H^1 H 1 space on non-compact manifolds of nonpositive curvature. result Hodge decomposition established for H 1 H^1 H 1 space on non-compact manifolds of nonpositive curvature. We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric …
Classification extended for reducible symmetric spaces.
problem Classifying homogeneous foliations on symmetric spaces.
method Extended classification from irreducible to reducible symmetric spaces.
result Classification completed for all noncompact symmetric spaces.
Schwartz functions smoothly extend to real projective spaces.
problem Identifying functions that extend smoothly to compactifications.
method Showed a similar result for real projective spaces.
result Schwartz functions extend smoothly to real projective spaces.
Study path spaces and their homology, extending loop products and coproducts.
problem Understanding the homology of path spaces in closed manifolds.
method Morse-Bott theory and homology operations.
result Complete computation of extended loop product and coproduct on spheres.
Riemann extension for the anti Mach metric is derived, the solution of geodesic equations for the extended space are given, some properties for the extended space was studied and compared with the basic space and the constructions of a translation surface for the anti Mach metric in four dimension is established.
Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
Study extended Bogomolny equations on curved space with special boundary conditions.
problem Classify solutions to extended Bogomolny equations with gauge group SU(2).
method Relate solutions to holomorphic data via Kobayashi-Hitchin correspondence.
result Completely classify solutions to the extended Bogomolny equations.
Symplectic embedding extended to stratified spaces.
problem Symplectic embedding theorem for stratified spaces.
method Defined symplectic structure on stratified spaces, demonstrated embedding in complex projective space.
result Symplectic embedding theorem extended to stratified spaces.
This paper extends the C*-signature to non-Witt spaces using noncommutative geometric methods.
problem Extending the signature to non-Witt spaces with noncommutative geometric methods.
method Noncommutative geometric methods, combinatorial framework, and comparison with analytical signature.
result Constructing the C*-signature on non-Witt spaces.
Classifies and constructs all extendable automorphisms of closed surfaces over the 3-sphere.
problem Identifying extendable automorphisms of closed surfaces over the 3-sphere.
method Classification and construction of extendable automorphisms through embeddings and Heegaard surfaces.
result All extendable automorphisms of closed surfaces can be induced by automorphisms of the 3-sphere on Heegaard surfaces.
Extends Margulis Lemma to RCD(K,N) spaces.
problem Applying Margulis Lemma to new geometric structures.
method Improved Regularity Estimates for Regular Langrangian Flows.
result Margulis Lemma extended to RCD(K,N) spaces.
Diffeology extends differential geometry to complex spaces.
problem Handling singular and infinite-dimensional settings in differential geometry.
method Introduces diffeology as a new framework.
result Diffeology provides a natural and effective framework for complex spaces.
Extended Kalman Filter is shown to be a gradient descent in trajectory space.
problem Estimating state of dynamical systems from noisy measurements.
method Recovery of extended Kalman filter equations from Amari's natural gradient in trajectory space.
result Extended Kalman Filter is equivalent to natural gradient descent in trajectory space.
Extends functions on symmetric spaces to analytic functions.
problem Extending functions on symmetric spaces to analytic functions.
method Harmonic analysis on symmetric spaces and representation theory of groups.
result Proves Whitney type extension theorems for symmetric spaces.
The paper extends Thompson Sampling to infinite action spaces using information theory.
problem Addressing the limitation of finite action spaces in Thompson Sampling.
method Information-theoretic analysis, extending rate-distortion theory to infinite action spaces.
result Derives a near-optimal regret bound for bandits with infinite and continuous action spaces.
Extends Lipschitz functions while preserving local constants.
problem Extending Lipschitz functions on metric spaces while maintaining local constants.
method Extends Lipschitz functions on metric spaces while locally preserving the asymptotic Lipschitz constant.
result Sobolev spaces on metric measure spaces are invariant under isomorphism of mm-structures.
The study examines spaces of non-extendable quasimorphisms for group pairs.
problem Analyzing the space of non-extendable quasimorphisms for group pairs.
method Established a five-term exact sequence of cohomology relative to bounded subcomplex.
result Proved stable commutator length equivalent to stable mixed commutator length for certain pairs.
New findings on Frobenius structures on Kodaira manifolds.
problem Understanding Frobenius structures on Kodaira manifolds.
method Extended deformation theory and Frobenius structures.
result Frobenius structure on Kodaira manifolds is trivial on degree-2 component.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.
problem Extending isometry properties of Lipschitz metric to virtually free groups.
method Analyzing finite-index subgroups and their covers, identifying folding paths, and using deformation retraction.
result Existence of candidates for Lipschitz distance and deformation retraction of spine.
Extends biharmonic concepts to new hypersurfaces and curves.
problem Characterize new types of hypersurfaces and curves in Riemannian manifolds.
method Extend p p p -biharmonic maps and hypersurfaces to ( p , q ) (p,q) ( p , q ) -harmonic hypersurfaces and curves, providing new examples. result New explicit examples of proper ( p , q ) (p,q) ( p , q ) -harmonic hypersurfaces and curves in space forms. Extends Feller theory to non-locally compact spaces for stochastic equations.
problem Stochastic partial differential equations and fractional processes.
method Extended Feller processes and proofs of folklore results.
result No condition of generalized Feller semigroups can be dropped.
Extends graph degree theorem to simplicial closure of Auter space.
problem Connectivity of graphs in Auter space.
method Defines degree for simplicial closure, extends Hatcher-Vogtmann theorem.
result Simplicial closure of Auter space is (d-1)-connected for degree d.
Extended recurrent pseudo-Riemannian manifolds were introduced by Mileva Prvanovic'. We reconsider her work in the light of recent results and show that the manifold is conformally flat, and it is a space of quasi-constant curvature. We also show that an extended recurrent Lorentzian manifold, with time-like associated…
Method extends eigenfunction construction to non-symmetric spaces.
problem Constructing eigenfunctions on harmonic manifolds.
method Applying Sullivan's method to non-compact harmonic manifolds.
result Eigenfunctions constructed for non-symmetric spaces.
The paper extends higher signature invariants to stratified spaces and establishes a new surgery exact sequence.
problem Extending higher signature invariants to stratified spaces.
method Revisiting and extending the surgery exact sequence for stratified spaces.
result Established a natural transformation between the surgery exact sequence and a long exact sequence of K-theory groups for stratified spaces.
We give a conformal representation in terms of meromorphic data for a certain class of spacelike surfaces in the Lorentz-Minkowski 4-space L^4 whose mean curvature vector is either lightlike or zero at each point. This representation extends simultaneously the Weierstrass representation for minimal surfaces in Euclidea…
Extends harmonic maps compactification to punctured Riemann surfaces.
problem Compactifying Teichmüller spaces for punctured Riemann surfaces.
method Using harmonic maps rays to extend compactification.
result The compactification still coincides with Thurston's compactification.
In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces…