In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Fin…
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The paper characterizes surfaces in 4D space forms with flat normal connection.
problem Characterizing surfaces in 4D space forms with specific geometric properties.
method Analyzing linearly dependent conditions and using properties of sectional curvature.
result Characterizations of space-like and time-like surfaces with flat normal connection.
On CR Submanifolds of Maximal CR Dimension with Flat Normal Connection of a Complex Projective Spacemath.DG
Paper studies CR submanifolds in complex projective space with flat normal connection.
problem Existence and properties of CR submanifolds with flat normal connection.
method Investigation of umbilical normal vector and application to non-existence proof.
result Non-existence of certain CR submanifolds of maximal CR dimension.
The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying rectifying and normal curves in Lorentzian n-space.
method Introducing a position vector field and defining rectifying and normal curves based on this field.
result Comprehensive characterization and classification of rectifying and normal curves.
The paper studies polar normalizations of skew ruled surfaces in 3D space.
problem Understanding the properties and invariants of polar normalized skew ruled surfaces.
method Determination of invariants and analysis of Tchebychev and support vector fields.
result Special polar normalizations lead to degenerate curves.
Study on immersions with flat normal bundle in curved spaces.
problem Behavior of isometric immersions with negative curvature.
method Investigation of second fundamental form growth in space forms.
result Second fundamental form grows exponentially if normal bundle is flat.
The study identifies surfaces with Maslovian normal bundles.
problem Characterizing surfaces with specific geometric properties.
method Proving equivalence to round spheres, cylinders, or cones.
result Surfaces with Maslovian normal bundles are limited to specific shapes.
Real-Smooth Hypersurfaces in math.CV
Formal normal form created for real-smooth hypersurfaces.
problem Real-smooth hypersurfaces in complex spaces.
method Iterative normalization procedure.
result Formal normal form constructed for a large class of hypersurfaces.
Study on ruled surfaces and their Laplace normal vector field.
problem Characterizing ruled surfaces with degenerate Laplace normal image.
method Analyzing relatively normalized ruled surfaces in .
result Determined ruled surfaces and relative normalizations leading to degenerate Laplace normal image.
The study examines stability of triharmonic hypersurfaces in space forms.
problem Stability of triharmonic hypersurfaces in space forms.
method Derivation of general stability statements, focus on specific cases of constant mean curvature in Euclidean and hyperbolic spaces, and analysis of small proper triharmonic hyperspheres and Clifford tori.
result Triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations, while in hyperbolic space they are stable.
The paper studies special normalizations of ruled surfaces in 3D Euclidean space.
problem Investigating relative normalizations of skew ruled surfaces.
method Investigates new formulae for Pick invariant, relative curvature, mean curvature, and relative metric.
result Determines ruled surfaces that make the surface an improper or proper relative sphere.
We prove a Berger type theorem for the normal holonomy group (i.e., the holonomy group of the normal connection) of a full complete complex submanifold of the complex projective space. Namely, if the normal holonomy does not act transitively, then the submanifold is the complex orbit, in the complex projective space, o…
In this paper we study lightlike surfaces of Minkowski 3- space such that they have degenerate or non-degenerate planar normal sections. We first show that every lightlike surface of Minkowski space has degenerate planar normal sections. Then we study lightlike surfaces with non-degenerate planar normal sections a…
New normalizing flows in hyperbolic space improve posterior modeling for hierarchical data.
problem Limited flexibility of existing normalizing flows in Euclidean space for hierarchical data.
method Elevated normalizing flows to hyperbolic spaces using coupling transforms and Wrapped Hyperboloid Coupling.
result Improved performance on density estimation and hierarchical graph data.
The paper explores Bertrand and framed curves in 3D space.
problem Characterizing Bertrand and framed curves in Euclidean 3-space.
method Analyzing curves where tangent, normal, or binormal lines match another curve's lines.
result Conditions for the existence of Bertrand and framed curves are clarified.
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 study of geometric structures around transversals using deformation spaces.
problem Understanding local behavior of geometric structures around singular foliations.
method Using deformation spaces to study local behavior of geometric structures.
result Obtained normal form theorems around transversals for various geometric structures.
The study finds abundant normal generators for mapping class groups.
problem Understanding normal generation in mapping class groups.
method Analyzing restrictions on invariant subsurfaces and Teichmüller spaces.
result Reducible mapping classes can normally generate mapping class groups based on their asymptotic translation lengths.
2-stein submanifolds in space forms have constant curvature if normal connection is flat or codimension is 2.
problem Characterizing submanifolds with constant curvature in space forms.
method Analyzing submanifolds with flat normal connection or codimension 2.
result 2-stein submanifolds have constant curvature under specified conditions.
Defines weak normals for irregular curves in high-dimensional spaces.
problem Dealing with irregular curves in high-dimensional Euclidean spaces.
method Using sequences of inscribed polygonals and Gram-Schmidt procedure, introduces a relaxed notion of weak normals.
result Weak normals for irregular curves are the strong limit of approximating polygonals and agree with relaxed energy.
The study explores Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
problem Exploring Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
method Defining and investigating Bertrand and Mannheim curves of framed curves in 4D Euclidean space.
result Bertrand and Mannheim curves exist even for framed curves in 4D Euclidean space, contrary to regular curves.
New connections on symmetric spaces with invariant properties.
problem Understanding invariant connections on hermitian symmetric spaces.
method Introduced a class of -invariant connections on homogeneous bundles over hermitian symmetric spaces.
result Parameter space of connections is a normal variety with a canonical anti-holomorphic involution.
On the normal holonomy representation of spacelike submanifolds in pseudo-Riemannian space formsmath.DG
In this paper we study weakly irreducible holonomy representations of the normal connection of a spacelike submanifold in a pseudo-Riemannian space from. We associate screen representations to weakly irreducible normal holonomy groups and classify the screen representations having the Borel-Lichnérowicz property. In pa…
A new method learns latent space normalizing flow for approximate inference in generator models.
problem Approximate inference in generator models with complex posterior distributions.
method Jointly learns latent space normalizing flow and generator model using MCMC-based maximum likelihood.
result The short-run Langevin flow approximates the posterior and aligns with the normalizing flow prior.
Normal forms for equivariant maps in infinite dimensions established.
problem Establishing normal forms for equivariant maps in infinite-dimensional manifolds.
method Inspired by Lyapunov-Schmidt reduction and Kuranishi method, uses Slice Theorem for Fréchet manifolds.
result Abstract moduli spaces of equivariant maps are locally modeled on quotient by a compact group.
The study characterizes and analyzes spacelike surfaces with a canonical normal null direction in Minkowski 4-space.
problem Characterizing and analyzing spacelike surfaces with a specific null direction in Minkowski space.
method Using geometric properties, Gauss map, and a nonlinear partial differential equation, the study characterizes and analyzes these surfaces.
result Characterizations and properties of spacelike surfaces with a canonical normal null direction are obtained.
Fractal Flow enhances normalizing flows with interpretable latent space and hierarchical modeling.
problem High-dimensional density estimation and generative modeling challenges.
method Integrates topic modeling (LDA) and fractal strategy into normalizing flows.
result Achieves latent clustering, controllable generation, and superior estimation accuracy.
We study the normal holonomy group, i.e. the holonomy group of the normal connection, of a CR-submanifold of a complex space form. We complete the local classification of normal holonomies for complex submanifolds. We show that the normal holonomy group of a coisotropic submanifold acts as the holonomy representation o…
Study the geometry of projections in Krein spaces.
problem Geometric structure of -normal projections in Krein spaces.
method Analyzes the action of -unitary operators and the relationship between -normal and -selfadjoint projections.
result Connected components of -normal projections are analytic homogeneous spaces of -unitary operators.
Minimality of a Kind of Pseudo-Umbilical Totally Real Submanifolds in Non-Flat Complex Space Formsmath.DG
Minimal totally real submanifolds in complex space forms have special umbilical properties.
problem Characterizing minimal totally real submanifolds in complex space forms.
method Analyzing the position of umbilical normal vectors in the normal bundle.
result Pseudo-umbilical totally real submanifolds with flat normal connection in non-flat complex space forms are minimal.
We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…
The study preserves Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.
problem Preserving Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.
method Investigation of infinitely many generalized Wallach spaces (GWS) where Ricci curvature positivity is preserved.
result Infinite number of GWS where Ricci curvature positivity is preserved under the normalized Ricci flow.
Defines and analyzes generalized normal ruled surfaces of curves in 3D space.
problem Understanding the geometry of generalized normal ruled surfaces.
method Calculates Gaussian and mean curvatures to determine surface properties and examines curve conditions.
result Determines when surfaces are flat or minimal and identifies specific curve types.
Study on algebraic curves' invariants and vanishing criteria.
problem Vanishing criteria for Griffiths infinitesimal invariants of algebraic curves.
method Analysis of moduli space of smooth genus 4 curves, study of normal functions.
result Vanishing criteria for the Griffiths infinitesimal invariants of Ceresa normal function.
The study explores normal generators for mapping class groups and their properties.
problem Understanding normal generators for mapping class groups of surfaces.
method Examined the relation between normal generation and asymptotic translation lengths on Teichmüller space and curve graph.
result Discussed several open questions related to normal generators.
We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.
Normal forms and symplectic reduction for gauge field theory in infinite dimensions.
problem Understanding the structure of moduli spaces in gauge field theory.
method Establishing normal forms for equivariant maps and developing singular symplectic reduction in infinite dimensions.
result The reduced phase space decomposes into smooth manifolds each with a natural symplectic structure.
Holonomic property holds for Einstein submanifolds in space forms.
problem Holonomic property of submanifolds in space forms.
method Analyzing Einstein submanifolds with flat normal bundles in space forms.
result Holonomic property holds for Einstein submanifolds with flat normal bundles in space forms.
Study of normal and tangent maps to frontals.
problem Understanding geometric and dynamical properties of frontals.
method Geometrical and dynamical analysis of normal and tangent maps.
result Parallels of the tangent map to a frontal curve are right equivalent to the tangent map of a frontal curve under certain conditions.
The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be reg…
Riemannian flows model curved spaces better.
problem Normalizing flows misspecified on curved spaces.
method Riemannian continuous normalizing flows using ODE solutions.
result Improves modeling on spheres, torii, and hyperbolic spaces.
Proves minimum number of normals to curves in 3D space.
problem Finding the minimum number of normals to closed curves in 3D.
method Morse theory for squared distance function and self intersections of the focal surface.
result For generic curves, points have at least 6, 8, or 10 normals depending on knotting.
Classifies Einstein submanifolds with flat normal bundle and parallel mean curvature.
problem Classifying Einstein submanifolds in space forms.
method Extending previous results for isometric immersions of Riemannian manifolds with constant sectional curvature.
result Classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature.
In the present paper we study normal transport surfaces in four-dimensional Euclidean space which are the generalization of surface offsets in . We find some results of normal transport surfaces in of evolute and parallel type. Further, we give some examples of these ty…
Surfaces with Parallel Normalized Mean Curvature Vector Field in Euclidean or Minkowski 4-Spacemath.DG
Study surfaces with parallel mean curvature in 4D spaces.
problem Characterize surfaces with parallel normalized mean curvature in Euclidean or Minkowski 4-space.
method Introduced special isothermal parameters and described surfaces using invariant functions.
result Surfaces with parallel normalized mean curvature are uniquely determined by three invariant functions.
Study of a 3D system on Wallach spaces, finding interrelations with invariant metrics.
problem Understanding the dynamics of a 3D system on Wallach spaces.
method Analyzing the normalized Ricci flow on generalized Wallach spaces.
result Characterized interrelations between the normalized Ricci flow and invariant metrics on Wallach spaces.
Study ruled surfaces with non-zero Gaussian curvature in Euclidean space.
problem Characterize ruled surfaces with non-zero Gaussian curvature.
method Use relative normalizations and analyze properties of surfaces like Pick invariant and Tchebychev vector field.
result Determine specific properties of ruled surfaces with non-zero Gaussian curvature.
An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…