The paper examines *-Ricci tensor properties on Sasakian manifolds.
problem Analyzing *-Ricci tensor on Sasakian manifolds.
method Examining φ-confomally flat and confomally flat *-η-Einstein Sasakian manifolds, *-Ricci symmetric condition, and *-Ricci soliton.
result Investigates properties of *-Ricci tensor on Sasakian manifolds.
The paper bounds eigenvalues of specific operators on certain manifolds.
problem Bounding eigenvalues of Paneitz and third-order boundary operators on locally conformally flat manifolds.
method Proof based on conformal equivalence to canonical models, showing injectivity of developing maps, and explicit computations on canonical models.
result Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds.
In this paper, we prove that every confomal minimal immersion of an open Riemann surface into Rn for n≥5 can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into R5. One …
The detailed analysis of the generalised Weierstrass representation of surfaces of revolution and their deformations induced by the modified Korteweg--de Vries (mKdV) equations is done. In particular, it is shown that these deformations preserve tori. The geometric meaning of the potential of surface is discussed and t…
The paper studies geometric PDEs for flatness on Riemannian manifolds.
problem Understanding flatness in geometric PDEs.
method Study geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness.
result Introduce new Theorems about flatness in Differential Geometry.
Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.
problem Characterizing projectively flat metrics on Hopf manifolds.
method Partitioning projectively flat metrics into classes based on Chern scalar curvature sign and proving properties of each class.
result Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.
We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…
The paper characterizes complex Finsler metrics that are projectively flat or dually flat.
problem Characterizing complex Finsler metrics with specific geometric properties.
method Proving conditions for projective flatness and dually flatness in terms of Minkowski metrics.
result Strongly convex complex Finsler metrics are projectively flat or dually flat if and only if they come from Minkowski metrics.
Study flat manifolds' collapsed limits as flat orbifolds.
problem Understanding collapsed limits of flat manifolds.
method Analyzing totally geodesic foliations and Gromov-Hausdorff limits.
result Identify collapsed limits as flat orbifolds and provide criteria for singularity.
New method constructs holonomic immersions from flat submanifolds.
problem Creating holonomic immersions from flat submanifolds.
method Ribaucour transformation and principal coordinate system.
result Holonomic immersions can be constructed using Ribaucour transformation.
Study calculates intersection forms of almost-flat 4-manifolds.
problem Understanding the intersection forms of almost-flat 4-manifolds.
method Calculation of intersection forms for all 4-dimensional almost-flat manifolds.
result Intersection forms of all 4-dimensional almost-flat manifolds have been calculated.
Flat systems of up to 2 dimensions have flat subsystems.
problem Characterizing flat subsystems in flat systems of differential dimension 2.
method Analyzing subsystems of a flat system of differential dimension at most 2.
result Flat subsystems of a flat system of differential dimension at most 2 exist and can have independent time-uniform outputs.
Study flat manifolds and their collapse using Teichmüller theory.
problem Understanding the collapse of flat manifolds and orbifolds.
method Algebraic description of Teichmüller and moduli spaces, study of boundaries.
result Every closed flat orbifold can be obtained by collapsing closed flat manifolds, and collapsed limits of 3-manifolds are classified.
Two conjectures on Ricci-flat metrics verified under specific conditions.
problem Properties of Ricci-flat metrics on complex manifolds.
method Analyzing conjectures and verifying them under specific conditions.
result Conjectures verified for certain types of metrics on complex surfaces.
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
Classifies spin structures on 4D almost-flat manifolds.
problem Determining spin structures on almost-flat manifolds.
method Utilised the canonical orthogonal representation of fundamental groups.
result 15 out of 127 orientable families are non-spin.
Flatness of manifolds with open flat subsets proven using bipolar comparisons.
problem Conditions for flatness in Riemannian manifolds.
method Using (3,3)-bipolar comparisons and open flat subsets.
result Flatness of manifolds proven under specific conditions.
Defines new bi-flat structures from integrable systems and flat coordinates.
problem Creating new bi-flat structures from integrable systems.
method Combining Frölicher-Nijenhuis bicomplex with Lauricella bi-flat structures.
result Defines multi-parameter families of Lauricella bi-flat structures.
New examples of special geometric solitons found.
problem Finding new types of geometric solitons.
method Constructing specific examples of Bach-flat gradient Ricci solitons.
result Examples of solitons that are neither conformally flat nor Einstein.
We find a normal form for two-input flat discrete-time systems.
problem No comparable normal form exists for flat continuous-time systems.
method State- and input transformations to achieve a triangular structure.
result A systematic parameterization of system variables by the flat output and its shifts.
This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. Th…
The paper proves that linearization along trajectories preserves flatness in discrete-time systems.
problem The relation between nonlinear and linear time-varying systems.
method Linearization along trajectories of a flat discrete-time system.
result The linearized system is flat, and a flat output can be derived.
The Yamabe flow on flat manifolds converges to a scalar flat metric.
problem Analyzing the convergence of Yamabe flow on asymptotically flat manifolds.
method Yamabe flow starting from an asymptotically flat manifold, convergence analysis.
result The flow converges to an asymptotically flat, scalar flat metric under certain conditions.
Study of flat metrics on orbifolds and their moduli spaces.
problem Understanding flat metrics on orbifolds and their moduli spaces.
method Analysis of Teichmüller spaces and mapping class groups.
result Moduli space of flat metrics on orbifolds is a very good orbifold under certain conditions.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. This paper introduces PSI-flatness to better understand ReLU neural networks' flatness and generalization.
problem Existing flatness definitions fail to account for ReLU neural networks' Positively Scale-Invariant (PSI) property.
method Formalizes PSI-flatness on basis path values, proving its relation to generalization.
result Minimums with balanced basis path values are flatter and generalize better.
Study links' flat-virtual diagrams to create link invariants.
problem Equivalence and invariants of flat-virtual diagrams.
method Maps from links in thickened surfaces to flat-virtual links.
result Investigation of flat-virtual diagrams' equivalence and invariants.
Lower bound found for flat norm minimizers' reach.
problem Finding the minimum flat norm for boundary shapes.
method Quantitative lower bound calculation.
result Established a lower bound on reach.
Investigate AR-Finsler metrics for local dual flatness and projective flatness.
problem Locally dually and projectively flat AR-Finsler metrics
method Derive necessary and sufficient conditions and a compatibility relation.
result Establish a rigidity result for AR-Finsler metrics.
Study flat connections on Courant algebroids using Lie groups.
problem Flatness conditions on Courant algebroids.
method Metric generalized connections, flatness condition analysis.
result Existence of compact simple Lie groups as building blocks for flat transitive Courant algebroids.
New flat surfaces found in 3D sphere space.
problem Constructing flat surfaces in 3D sphere.
method Using Ribaucour transformations and flat torus theory.
result Families of complete flat surfaces in S3 determined by parameters. Paper introduces normalized flat minima to address scale dependence in neural network optimization.
problem Scale dependence in existing flat minima definitions affects generalization studies.
method PAC-Bayesian analysis to introduce normalized flat minima, free from scale dependence.
result Normalized flat minima provides better hierarchy in hypothesis class and improved generalization.
FP-BMA improves generalization by encouraging flat posteriors in Bayesian Model Averaging.
problem Lack of flat posterior in approximate Bayesian inference methods hinders effective Bayesian Model Averaging.
method Proposes Flat Posterior-aware Bayesian Model Averaging (FP-BMA) and Flat Posterior-aware Bayesian Transfer Learning schemes.
result FP-BMA successfully captures flat posteriors, improving generalization performance.
We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of Lorentzian flat Lie algebras to those with trivial center or those with degenerate ce…
The article classifies G2-structures with conformally flat metrics.
problem Identifying G2-structures with specific geometric properties.
method Classifying closed G2-structures with conformally flat metrics.
result Any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples.
Notes on flat pseudo-Riemannian manifolds, focusing on their characterization and properties.
problem Characterizing flat pseudo-Riemannian manifolds and their properties.
method Survey of basic concepts in affine and Riemannian geometry, characterization of flat manifolds, and analysis of Lie groups.
result Characterization and properties of flat pseudo-Riemannian Lie groups and their metrics.
Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. These flat plumbing basket surface can be presented by a sequential presentation known as flat plumbing basket code first found by Furihata, Hirasawa and Kobayashi. The minimum number of flat plumbings to obtain…
A Lorentzian flat Lie group is a Lie group G with a flat left invariant metric μ with signature (1,n−1)=(−,+,…,+). The Lie algebra g=TeG of G endowed with ⟨,⟩=μ(e) is called flat Lorentzian Lie algebra. It is known that the metric of a flat Lorentzian Lie group is geodesical…
Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
problem Understanding the rigidity of Ricci-flat manifolds with specific curvature decay.
method Analyzing the gradient of the Green function and using curvature decay conditions.
result Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
Decomposes flat nonlinear discrete-time systems into simpler components.
problem Flatness of nonlinear discrete-time systems.
method Coordinate transformations and feedback, using flow-box and Frobenius theorems.
result Flatness of a discrete-time system can be checked algorithmically.
We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-Kähler Vaisman manifolds.
Connected sum affects crossing numbers of flat virtual knots.
problem Understanding how connected sum impacts the crossing numbers of flat virtual knots.
method Analyzing minimal crossing diagrams and using super-additivity properties.
result Crossing number of flat virtual knots is super-additive under connected sum.
This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.
problem Classifying fibrations of compact flat orbifolds.
method Developed theory for classifying fibrations up to affine equivalence.
result Classified fibrations of compact flat 2-orbifolds.
Classifies flat knots up to 8 crossings using Lyndon words.
problem Classifying flat knots up to a certain number of crossings.
method Using matchings on Lyndon words and various knot invariants.
result Distinguished all flat knots up to 7 crossings except for five pairs.
Investigates how flatness of loss curve relates to generalization in machine learning models.
problem Understanding why flatness correlates with generalization in machine learning models.
method Relates flatness to interpolation from representative data, derives notions of representativeness and feature robustness.
result Derives a novel relative flatness measure that correlates with generalization and solves reparameterization issues.
Detect spacetime curvature with event causality measurements.
problem Detecting spacetime curvature without rulers and clocks.
method Prove spacetime non-flatness through causal relations.
result Sixteen measurements verify non-flatness of non-conformally flat spacetimes.
Study describes flat metric moduli spaces on 4D manifolds.
problem Understanding flat metrics on 4D closed manifolds.
method Algebraic and topological description of moduli spaces.
result Algebraic and topological description of moduli spaces of flat metrics.
Short introduction to discrete flat fronts in hyperbolic space with a Weierstrass representation proof.
problem Understanding discrete flat fronts in hyperbolic space.
method Proving a Weierstrass representation for discrete flat fronts.
result Any discrete flat front in the mixed area sense admits a Weierstrass representation.