New findings show fundamental group is not audible in spherical space forms.
problem Isospectral spherical space forms with non-cyclic fundamental groups.
method Revisited and found new examples of spherical space forms.
result Fundamental group is not audible among spherical space forms.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study on generalized spherical surfaces in Euclidean spaces.
problem Characterizing and analyzing generalized spherical surfaces in Euclidean spaces.
method Investigation of generalized spherical curves and surfaces in Euclidean spaces, calculation of curvatures.
result Identification and characterization of different types of generalized spherical surfaces in Euclidean spaces.
Minimal surfaces in spherical space forms have limited genus.
problem Understanding minimal surfaces in spherical space forms.
method Analyzing orientable index one minimal surfaces with large fundamental groups.
result Surfaces have genus at most two, confirming a conjecture.
In this paper we consider the spherical slant helices in R 3 R^3 R 3 . More- over, we show how could be obtained to a spherical slant helix and we give some spherical slant helix examples in Euclidean 3-space.
Study spherical surfaces with conical points, proving systole inequality.
problem Characterize moduli spaces of spherical metrics with conical singularities.
method Analyze systole inequality and properness of forgetful map.
result Explicit systole inequality linking metric and conformal invariants.
Researchers find highest volumes for isospectral spherical orbifolds and space forms.
problem Finding the maximum volumes of isospectral spherical orbifolds and space forms.
method Analyzing isospectral properties and calculating volumes of spherical orbifolds and space forms.
result Highest volumes for specific dimensions and conditions of isospectral spherical orbifolds and space forms.
In this work, we studied the properties of the spherical indicatrices of involute curve of a space curve and presented some characteristic properties in the cases that involute curve and evolute curve are slant helices and helices, spherical indicatrices are slant helices and helices and we introduced new representatio…
Study on metrics with positive scalar curvature on spherical space forms.
problem Determining metrics with positive scalar curvature on spherical space forms.
method Analysis of moduli spaces of metrics with positive scalar curvature on topological spherical space forms.
result Determination of the number of path components of moduli spaces for dimensions at least 5 and non-simply connected forms.
In this paper, we give definitions and characterizations of normal and spherical curves in the dual space. We show that normal curves are also spherical curves in D^3.
The paper explores integral formulas and behavior of spherical functions.
problem None explicitly stated, focuses on spherical functions.
method Integral formulas and behavior analysis at infinity.
result New integral formulas and results about behavior at infinity.
Study spherical images of modified vector fields on a unit sphere.
problem Understanding spherical images of modified vector fields.
method Analysis of spherical images using modified orthogonal vector fields and Darboux vector.
result Characterization of spherical indicatrices with modified orthogonal frame.
DELIMIT PyTorch enhances deep learning for diffusion imaging.
problem Applying deep learning to spherical diffusion imaging data.
method Added spherical harmonic interpolation and local convolution layers to PyTorch.
result Deep learning can now be applied conveniently to diffusion imaging data.
Timelike geometry of spherical simplices is shown to be isometric to vector spaces.
problem Characterizing the geometry of spherical simplices.
method Proved isometry to vector spaces with a timelike norm.
result Timelike spherical Hilbert geometry of simplices is isometric to a union of six copies of vector spaces.
New classification of spherical 2-Dupin submanifolds.
problem Characterizing spherical 2-Dupin submanifolds.
method Analyzing properties of submanifolds in space forms.
result Classification of 2-Dupin submanifolds in space forms.
The paper proves stability of inequalities for nearly spherical sets in various spaces.
problem Stability of geometric inequalities for nearly spherical sets.
method Deriving a quantitative quermassintegral inequality and applying it to derive stability results.
result Stability of geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets in R n + 1 \mathbb{R}^{n+1} R n + 1 and H n + 1 \mathbb{H}^{n+1} H n + 1 . Study local features of decorated representation spaces for spherical surfaces.
problem Local structure of moduli space of spherical surfaces with conical points.
method Analysis of decorated representation spaces of fundamental groups in SU(2).
result Smooth locus of decorated representation spaces is dense and connected.
Study spherical convex bodies using L p L_p L p -floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced L p L_p L p -floating areas and curvature entropy for spherical convex bodies. result Established isoperimetric inequalities and dual isoperimetric inequalities.
This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ( κ > 0 κ>0 κ > 0 ) and $H_\k^3$ ( κ < 0 κ<0 κ < 0 ), to the standard {\itshape spherical wav…
Locally classifies 4D spherical symmetric Finsler spaces.
problem Classifying 4D spherical symmetric Finsler spaces.
method Local classification of pseudo-Finsler Berwald structures.
result Six classes of non-Riemannian SO(3)-symmetric pseudo-Finsler Berwald functions.
Study null surfaces of pseudo-spherical curves in anti-de Sitter space.
problem Characterize null surfaces of pseudo-spherical spacelike framed curves in anti-de Sitter 3-space.
method Introduced nullcone fronts, classified singularities, defined Anti-de Sitter distance-squared functions.
result Relate singularities of nullcone fronts to those of framed curves.
String-net models explore non-spherical fusion categories, revealing new spin structures and representations.
problem Investigating string-net models in non-spherical fusion categories.
method String-net models associate vector spaces to surfaces in terms of graphs decorated by objects and morphisms of a pivotal fusion category.
result String-net spaces count r-spin structures and carry representations of the mapping class group.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
Study pseudo-spherical submanifolds with 1-type Gauss map in pseudo-spheres.
problem Characterize and classify pseudo-Riemannian submanifolds with 1-type pseudo-spherical Gauss map.
method Classify Lorentzian surfaces and pseudo-Riemannian submanifolds in pseudo-spheres with 1-type pseudo-spherical Gauss map.
result Classification of submanifolds with 1-type pseudo-spherical Gauss map in pseudo-spheres.
We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems.
Proves rigidity for higher rank three-manifolds without curvature constraints.
problem Rigidity of higher rank three-manifolds without curvature assumptions.
method Analyzes complete Riemannian three-manifolds of higher rank.
result Proves conditions for manifolds to be spherical or hyperbolic space forms.
Proposes spherical text embedding for better directional similarity.
problem Directional similarity is more effective but unsupervised text embeddings are typically learned in Euclidean space.
method Develops a spherical generative model and an efficient optimization algorithm for unsupervised word and paragraph embeddings.
result Achieves state-of-the-art performances on various text embedding tasks.
Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional symmetric spaces G ∞ / K ∞ = l i m → G n / K n G_\infty/K_\infty = \varinjlim G_n/K_n G ∞ / K ∞ = lim G n / K n . We use the representation t…
Characterizes curves on geodesic spheres and totally geodesic hypersurfaces in hyperbolic and spherical spaces.
problem Characterizing curves in curved spaces.
method Rotation minimizing frames and exponential maps.
result Characterizes geodesic spherical curves in hyperbolic and spherical spaces through linear equations.
We prove that M. Kramer's classification of list of spherical pairs coincides with that for weakly symmetric spaces by examining the linear isotropy representation of the corresponding homogeneous space associated to each pair.
Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.
problem Comparing eigenvalues on spherically symmetric manifolds to Euclidean balls.
method Examines Dirichlet Laplace eigenvalues on balls of spherically symmetric manifolds and Euclidean space.
result Eigenvalues on spherically symmetric manifolds are smaller for small radii, but larger for hyperbolic spaces.
Dunkl connections on complex plane don't preserve metrics.
problem Preserving metrics with Dunkl connections on \(\mathbb{C}^2\).
method Analysis of the topology of spherical tori with conical points.
result General Dunkl connections on \(\mathbb{C}^2\) do not preserve non-zero Hermitian forms.
Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.
problem Existence and uniqueness of spherical helicoidal surfaces in 3-sphere.
method Continuous function of distance to axis, spherical angular momentum of spherical curves.
result Existence and uniqueness theorem for spherical helicoidal surfaces in 3-sphere.
Simple geodesics on spherical tetrahedra identified for specific angles.
problem Identifying simple closed geodesics on regular tetrahedra in spherical space.
method Analyzing pairs of coprime integers (p,q) to find angles α1 and α2.
result Existence and non-existence of simple closed geodesics for specific angles.
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.
New identities for simplex volume in Euclidean space.
problem Calculating the volume of a spherically faced simplex.
method Using Cayley-Menger determinants and hypergeometric integrals.
result Derivation of two identities describing simplex volume.
Study shows spherical hyperbolic manifolds almost rigidly converge to hyperbolic space.
problem Almost rigidity of positive mass theorem for spherical hyperbolic manifolds.
method Intrinsic flat distance to prove convergence.
result Spherically symmetric asymptotically hyperbolic manifolds converge to hyperbolic space if mass limit is zero.
The study characterizes compact submanifolds with pinched Ricci curvature in Euclidean and spherical space forms.
problem Characterizing compact submanifolds with specific Ricci curvature bounds.
method Proving rigidity results for submanifolds with Ricci curvature bounded below by a function of mean curvature.
result Submanifolds are either isometric to the Einstein Clifford torus or have vanishing homology groups up to a certain degree.
The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.
problem Finding bounds on intersections and inflections for spherical polygons.
method Adapting smooth curve theorems to spherical polygons using discrete tools.
result Proves discrete analogs of four-vertex theorems for spherical polygons.
Study spherical curve flow with density in a specific geometric space.
problem Analyzing spherical curve flow with density in a specific geometric space.
method Carried out a systematic study of spherical curve flow by mean curvature with density in a 3D rotationally symmetric space with density.
result Conditioning of the flow behavior based on the parabolicity or hyperbolicity of the space.
Study degenerate Bianchi transformations for pseudo-spherical submanifolds in 5D space.
problem Characterize three-dimensional pseudo-spherical submanifolds with degenerate Bianchi transformations.
method Complete description through holonomically degenerate Bianchi transformations.
result Obtained a complete description of degenerate pseudo-spherical submanifolds.
In this work, we studied the properties of the spherical indicatrices of a Bertrand curve and its mate curve and presented some characteristic properties in the cases that Bertrand curve and its mate curve are slant helices, spherical indicatrices are slant helices and we also researched that whether the spherical indi…
The paper extends geometric inequalities for nearly spherical sets in various space forms.
problem Investigating weighted inequalities for nearly spherical sets in space forms.
method Generalizing and extending inequalities for nearly spherical sets in C 1 C^1 C 1 and W 2 , ∞ W^{2,\infty} W 2 , ∞ settings, with convex weight functions. result Quantitative stability estimates for weighted inequalities in R n + 1 \mathbb{R}^{n+1} R n + 1 and H n + 1 \mathbb{H}^{n+1} H n + 1 . In this paper, we show that small spherical soap bubbles in irreducible simply connected symmetric spaces of rank greater than one are constructed from the limits of a certain kind of modified mean curvature flows starting from small spheres in the Euclidean space of dimension equal to the rank of the symmetric space, …
Describes the space of spherical triangles on a smooth 3-manifold.
problem Understanding the geometric structure of spherical triangles.
method Analyzes the homotopy and analytic properties of the space.
result The space is a smooth 3-manifold embedded in R^6.
Spherical CNNs tackle 3D data analysis, especially spherical images.
problem Learning problems involving spherical images, like omnidirectional vision and molecular regression.
method Defined spherical cross-correlation, developed a generalized FFT for efficient computation.
result Demonstrated spherical CNNs' effectiveness in 3D model recognition and atomization energy regression.
Study floating bodies of polytopes, linking volume to flags.
problem Understanding floating bodies of polytopes in various spaces.
method Introducing flag simplices to connect metric and combinatorial structures.
result Weighted volume depends on complete flags of polytopes.
We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space E n = G / K E^n = G/K E n = G / K where G G G is the semidirect product R n ⋅ K R^n \cdot K R n ⋅ K of the translation group with a closed subgroup K K K of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K…