The paper defines generalized s-manifolds and explores their polars and antipodal sets.
problem Understanding polars and antipodal sets in generalized s-manifolds.
method Introduced generalized s-manifolds and provided a method to construct them. Studied polars and antipodal sets.
result Extended results on compact symmetric spaces to generalized s-manifolds.
Study curvature properties of w.a. S-manifolds with new conditions.
problem Curvature properties of weak almost S-manifolds.
method Partial Ricci flow, additional conditions, f-(κ,μ)-nullity.
result Characterize S-manifolds as limits of w.a. S-manifolds.
Defines s-manifolds and s-manifolds with corners for symplectic applications.
problem Creating a framework for singular manifolds with useful properties.
method Introducing categories of stratified manifolds and manifolds with corners.
result Fundamental classes and transverse fibre products in s-manifolds.
The paper characterizes Legendre curves on trans-S-manifolds.
problem Characterizing Legendre trajectories on trans-S-manifolds.
method Obtained curvature characterizations and classified Legendre curves.
result Classified Legendre curves with linearly dependent Frenet frame fields.
We expound some results about the relationships between the Jacobi operators with respect to null vectors on a Lorentzian S-manifold M and the Jacobi operators with respect to particular spacelike unit vectors on M. We study the number of the eigenvalues of such operators in a φ-null Osserman Lorentzi…
We introduce a new general class of metric f-manifolds which we call (nearly) trans-S-manifolds and includes S- manifolds, C-manifolds, s-th Sasakian manifolds and generalized Kenmotsu manifold studied previously. We prove their main properties and we present many examples which justify their study.
We investigate L-sectional curvature of S-manifolds with respect to the Rieman- nian connection and to certain semi-symmetric metric and non-metric connections naturally related with the structure, obtaining conditions for them to be constant and giving examples of S-manifolds in such conditions. Moreover, we calculate…
We present a compared analysis of some properties of indefinite almost S-manifolds and indefinite S-manifolds. We give some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields. We study the sectional and φ-sectional curvature of indefinite almost $\…
An η-Einstein condition is introduced in the context of indefinite g.f.f-manifolds, and a few Schur-type lemmas for indefinite S-manifolds are provided.
A condition of Osserman type, called φ-null Osserman condition, is introduced and studied in the context of Lorentz globally framed f-manifolds. An explicit example shows the naturalness of this condition in the setting of Lorentz S-manifolds. We prove that a Lorentz S-manifold with constant…
The main result we give in this brief note relates, under suitable hypotheses, the φ-null Osserman, the null Osserman and the classical Osserman conditions to each other, via semi-Riemannian submersions as projection maps of principal torus bundles arising from a Lorentzian S-manifold.
New spectral sequence for K-manifolds, computing cohomology and harmonic forms.
problem Computing cohomology and harmonic forms of K-manifolds. method Introducing a new spectral sequence and using it to generalize theorems from K-contact geometry. result Computed cohomology ring and harmonic forms of S-manifolds. Fintushel-Stern's knot surgery gave many pairs of exotic manifolds, which are homeomorphic but non-diffeomorphic. We show that if an elliptic fibration has two parallel, oppositely oriented vanishing circles (for example S2×S2 or Matsumoto's S4), then the knot surgery gives rise to standard manifolds. The …
In a metric g.f.f-manifold we study lightlike hypersurfaces M tangent to the characteristic vector fields, and owing to the presence of the f-structure, we determine some decompositions of TM and of a chosen screen distribution obtaining two distributions invariant with respect to the structure. We discuss the …
We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) mani…
We discuss two families of closed orientable three-dimensional manifolds which arise as cyclic generalizations of two hyperbolic icosahedral manifolds listed by Everitt. Everitt's manifolds are cyclic coverings of the lens space L3,1 branched over some 2-component links. We present results on covering properties, …
Study of CR-submanifolds in various Lorentzian manifolds.
problem Exploring CR-submanifolds in different Lorentzian structures.
method Analyzing properties and results of CR-submanifolds in LCS, LP-cosymplectic, S, and GKM manifolds.
result Obtained results on totally umbilical and geodesic CR-submanifolds.
New Spin(7)-instantons constructed on Joyce's manifold.
problem Constructing Spin(7)-instantons on Joyce's compact manifold. method Gluing non-flat connections on local model spaces to a flat connection on the Spin(7)-orbifold. result More than 20,000 new four-parameter families of Spin(7)-instantons. In his 1974 thesis, Martin Scharlemann constructed a fake homotopy equivalence from a closed smooth manifold f:Q -> S^3 x S^1 # S^2 x S^2 and asked whether the manifold Q itself is diffeomorphic to S^3 x S^1 # S^2 x S^2. Here we answer this question affirmatively.
We consider a homology sphere Mn(K1,K2) presented by two knots K1,K2 with linking number 1 and framing (0,n). We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of Mn(T2,3,K2) if n<2τ(K2) holds. We also give a formula of Ozsváth-Szabó's τ-invariant as…
Compact metric f-K-contact manifolds constructed via specific transformations.
problem Constructing all metric f-K-contact manifolds.
method Iteration of constructions of mapping tori, rotations, and type II deformations.
result Compact metric f-K-contact manifolds are derived from compact K-contact manifolds.
Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.
problem Understanding quandle modules and their connection to Lie-Yamaguti representations.
method Examine quandle modules over quandle spaces, focusing on geometric structures.
result Modules over quandle spaces are linked to representations of Lie-Yamaguti algebras.
We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ⊳. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry. Assume that Q is normal as an algebraic variety and that the a…
New connections found in higher-dimensional geometries with skew-torsion.
problem Finding metric connections with skew-torsion on metric f-manifolds. method Analyzing properties of Reeb vector fields and Nijenhuis tensor to construct a unique connection.
result A natural generalization of adapted connections in higher dimensions, including parallel skew-torsion.
The paper defines and analyzes f-biharmonic θα-slant curves in S-space forms.
problem Characterizing f-biharmonic θα-slant curves in S-space forms. method Provided a concise overview, derived a key equation, and analyzed it to establish conditions for θα-slant curves to be f-biharmonic. result Established necessary and sufficient conditions for θα-slant curves to be f-biharmonic. In this article a homotopy co-momentum map (à la Callies-Frégier-Rogers-Zambon) trangressing to the standard hydrodynamical co-momentum map of Arnol'd, Marsden and Weinstein and others is constructed and then generalized to a special class of Riemannian manifolds. Also, a covariant phase space interpretation of the coa…
In the present paper, we study globally framed f-manifolds in the particular setting of indefinite S-manifolds for both spacelike and timelike cases. We prove that if M=N⊥×fNT is a warped CR-submanifold such that N⊥ is φ?-anti-invariant and NT is φ?-invariant, then M is a CR-product. We…
The study examines properties of f-contact manifolds with constant curvature.
problem Investigating constant curvature in f-contact manifolds. method Analyzing (κ,μ)-nullity condition and f-sectional curvature. result An f-(κ,μ) manifold with constant f-sectional curvature implies specific conditions on μ and κ. An f-structure on a manifold M is an endomorphism field φ satisfying φ3+φ=0. We call an f-structure {\em regular} if the distribution T=kerφ is involutive and regular, in the sense of Palais. We show that when a regular f-structure on a compact manifold M is an almost §-structure, as defined by Dugg…
Investigates special metrics in hypercomplex geometry.
problem Characterizing and understanding special hyperhermitian metrics.
method Characterization of hypercomplex structures with Obata holonomy, investigation of quaternionic Gauduchon and balanced metrics, incompatibility results, and introduction of Einstein-type conditions.
result Joyce's manifolds always admit special metrics.
We study Brownian motion and stochastic parallel transport on Perelman's almost Ricci flat manifold M=M×SN×I, whose dimension depends on a parameter N unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for N→∞ …
The study classifies compact Cauchy horizons in vacuum spacetimes.
problem Classifying compact Cauchy horizons in vacuum spacetimes.
method Complete classification theorem based on topology and null generators.
result Different cases of compact Cauchy horizons with specific manifolds and spacetime properties.
The paper presents counterexamples to LS-category conjectures and constructs maps between manifolds.
problem Counterexamples to LS-category conjectures for manifolds and their squares.
method Construction of manifolds and maps to analyze LS-category properties.
result Shows that mcatLS(M2imesM3)≥4 and reduces Rudyak's conjecture. Paper defines generalized braids and proves their subgroup status.
problem Understanding the structure of generalized braids and knots.
method Defined generalized braid theories and computed their generating sets.
result Quasitoric normal generalized braids form a subgroup of normal generalized braid group.
Defines a new Poisson structure for generalized Sasakian spaces.
problem No specific problem stated; focuses on new structure definition.
method Defines a canonical Poisson structure on generalized contact metric spaces.
result Shows distinction between generalized Sasakian and coKähler structures.
Improved image generation through iterative flow matching to reduce hallucinations.
problem Hallucinations in image generation models.
method Iterative flow matching to refine and correct paths in generative models.
result Enhanced generative modeling with reduced unrealistic images.
Plug-and-play multimodal controller improves class-conditional image generation.
problem Generating class-conditional images from user-specified labels.
method Introduces a `multimodal controller` to generate multimodal data without additional learning parameters.
result Multimodal controlled generative models produce higher quality class-conditional images and novel modalities.
Framework generates personalized insulin treatment strategies using deep models.
problem Developing optimal personalized treatment strategies for diabetes patients.
method Combines deep generative time series models with decision theory.
result Demonstrated improved personalized insulin treatment strategies for diabetes patients.
OptiGAN uses GAN and RL to optimize sequence generation for specific goals.
problem Challenging in sequence generation tasks to generate sequences with specific desired goals.
method Integrates GAN and RL to optimize desired goal scores using policy gradients.
result Achieves higher desired scores in text and real-valued sequence generation.
Survey on deep models for graph generation.
problem Improving fidelity of generated graphs.
method Taxonomy and comparison of deep generative models.
result Advances in deep generative models for graph generation.
Improves deep generative models to generate images of any size.
problem Fixed-sized output images from deep generative models.
method Integrates spatial noise vectors into fully convolutional neural networks.
result Theoretical interpretation of infinite spatial generation using spatial stochastic processes.
Develops a unified theory of Yang-Mills and GR using generalized principal bundles.
problem Combining Yang-Mills theories and General Relativity into a single framework.
method Using generalized principal bundle theory, the authors develop a new approach to field theories.
result Recover General Relativity within the framework of generalized principal connections.
Meta-CoTGAN improves adversarial text generation by preventing mode collapse.
problem Mode collapse in adversarial text generation.
method Meta-Cooperative Training Paradigm with a language model.
result Meta-CoTGAN effectively slows down mode collapse and improves generation quality and diversity.
Generative models can still learn from contaminated data, but with limitations.
problem How much contamination can generative models tolerate?
method Characterized robustness under contaminated enumerations, proving generation is achievable for all countable collections if contamination fraction converges to zero.
result Generation under contamination is achievable for all countable collections if contamination fraction converges to zero, but dense generation is strictly less robust.
Generative AI tasks analyzed for text, images, audio, video, code, and molecules.
problem What is the core question when using generative AI?
method Survey of generative model families, probabilistic framework, game-theoretic setup, post-training modifications, socially responsible considerations.
result Generative AI is a distinct machine learning task with connections to prediction, compression, and decision-making.
Defines Kahler angle for a broader context.
problem Generalizing results about Kahler angle.
method Provides a general definition of Kahler angle.
result Generalized results about Kahler angle.
Established a generalized Boothby-Wang theorem in contact geometry.
problem Generalized contact structures and their properties.
method Courant reduction methods and construction of principal bundles.
result Induced symplectic foliation on leaf space under certain conditions.
The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry à la Hitchin). The generalized twistor space associated to such a manifold is defined as the bundle of generalized complex structures on the tangent spaces of the…