Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.
problem Understanding non-linear Hopf manifolds and their properties.
method Holomorphic embeddings and LCK metrics.
result Non-linear Hopf manifolds admit LCK metrics.
Linear F-manifolds are studied with connections and dual spaces.
problem Understanding linear F-manifolds and their dual spaces.
method Developed systematic treatment and defined duality using connections.
result Defined compatibility conditions between linear F-manifolds and generalized tangent bundle.
The paper characterizes compatible linear connections on 3D Finsler manifolds.
problem Characterizing compatible linear connections on Finsler manifolds of dimension three.
method Intrinsic method to characterize compatible linear connections, focusing on indicatrices and Euclidean symmetries.
result If a compatible linear connection is not unique, indicatrices must be Euclidean surfaces of revolution.
The paper studies affine manifolds with linear foliations and their topological properties.
problem Characterizing the topology of affine manifolds with linear foliations.
method Analyzing the structure of affine manifolds and their foliations, proving topological properties.
result Compact affine manifolds with linear foliations are homeomorphic to the torus under certain conditions.
We introduce the Locally Linear Latent Variable Model (LL-LVM), a probabilistic model for non-linear manifold discovery that describes a joint distribution over observations, their manifold coordinates and locally linear maps conditioned on a set of neighbourhood relationships. The model allows straightforward variatio…
Study pin manifolds using Clifford linear Dirac operator and KO-theory.
problem Index theory on Pin manifolds.
method Clifford linear Dirac operator and differential KO-theory.
result Systematic treatment of index theory on Pin manifolds.
Defines smooth actions of a group on manifolds and vector spaces.
problem Representing the general linear group and its actions.
method Restricted functor of points and category theory.
result Smooth actions on Z2n-graded vector spaces and manifolds. Study shows solutions to certain equations form smooth manifolds.
problem Understanding moduli spaces of solutions to non-linear elliptic equations.
method Analyzes moduli spaces as derived log smooth manifolds.
result Moduli spaces of solutions are derived log smooth manifolds.
Study reveals how manifold geometry impacts linear regression solutions.
problem Impact of manifold geometry on linear regression solutions.
method Linear regression applied to manifold-structured data, focusing on extrinsic geometry.
result Linear regression does not have a unique solution on flat manifolds.
Paper solves complex equations on noncompact manifolds.
problem Establishing estimates and existence for fully non-linear equations.
method General class of fully non-linear equations on Kähler and Hermitian manifolds.
result Constructs complete Kähler metrics with prescribed volume forms.
Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibi\-li\-ty condition). By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Rie…
Gradient bounds and Liouville theorems for quasi-linear equations on manifolds with nonnegative Ricci curvature.
problem Establishing bounds and theorems for solutions to quasi-linear elliptic equations on compact manifolds with nonnegative Ricci curvature.
method Gradient bounds, Liouville-type theorems, local splitting theorem, Harnack-type inequality, ABP estimate.
result Gradient bounds and Liouville-type theorems for solutions to quasi-linear equations on compact manifolds with nonnegative Ricci curvature.
This is a review with examples concerning the concepts of affine (in particular, constant and linear) vector fields and fundamental vector fields on a manifold. The affine, linear and constant vector fields on a manifold are shown to be in a bijective correspondence with the fundamental vector fields on it of respectiv…
Study linear differential operators on special manifolds.
problem Analyzing elliptic differential operators on specific types of manifolds.
method Examining a linear elliptic differential operator of the form Δ + V - λ on quasi-asymptotically conical manifolds.
result Established an isomorphism theorem for these operators.
We introduce linear holonomy on Poisson manifolds. The linear holonomy of a Poisson structure generalizes the linearized holonomy on a regular symplectic foliation. However, for singular Poisson structures the linear holonomy is defined for the lifts of tangential path to the cotangent bundle (cotangent paths). The lin…
The paper splits manifolds using infinity harmonic functions with linear growth.
problem Splitting manifolds with specific harmonic functions.
method Analyzes manifolds with non-negative Ricci or sectional curvature, focusing on infinity harmonic functions with linear growth.
result Extends Savin's theorem to surfaces with non-negative sectional curvature.
Solves Dirichlet problem for elliptic equations on Hermitian manifolds.
problem Solving Dirichlet problem for fully non-linear elliptic equations on Hermitian manifolds.
method Establishing a quantitative boundary estimate under a subsolution assumption.
result Derives solvability and regularity of the Dirichlet problem.
Uniform linear bounds on volume changes in 3D hyperbolic spaces.
problem Volume variation in hyperbolic 3-manifolds.
method Uniform linear bounds proof for drilling and filling operations.
result Uniform linear bounds on volume variation proved.
Paper establishes estimates for solutions on compact manifolds.
problem Solving fully non-linear equations on compact almost Hermitian manifolds.
method Establishes a priori estimates for solutions.
result Solves complex Hessian and Monge-Ampère equations.
Linear inequality found for certain curved spaces.
problem Finding isoperimetric inequalities for curved spaces.
method Extending known result to homogeneous Hadamard manifolds.
result Linear isoperimetric inequality proven for higher-dimensional cycles.
Study shows long-term solutions for complex equations on curved spaces.
problem Long-term behavior of solutions to fully non-linear parabolic equations on Hermitian manifolds.
method Used general assumptions and derived a Harnack inequality for the linearized equation.
result Proved the long-time existence and convergence of solutions.
Sprays on Frechet manifolds connect connections and tangent structures.
problem Characterizing linear symmetric connections on Frechet manifolds.
method Constructing connection maps and linear symmetric connections on tangent and second-order tangent bundles using sprays.
result A bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.
This is a survey on the analytic theory of linear wave equations on globally hyperbolic Lorentzian manifolds. There is no claim of originality.
The study examines the stability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.
problem Linear instability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.
method Analysis of the second and third Betti numbers for Sasaki Einstein and nearly parallel G2 manifolds.
result Positive second and third Betti numbers lead to linear instability for the respective manifolds.
New proof for unique semi-symmetric compatible linear connection on Finsler manifolds.
problem Existence and uniqueness of semi-symmetric compatible linear connections on Finsler manifolds.
method New linear algebra proof without integration, using convex body properties and intrinsic equations.
result Uniqueness of semi-symmetric compatible linear connection proved.
The paper solves linearized Ricci curvature equations on compact manifolds.
problem Linear analysis of Ricci curvature equations on general compact Riemannian manifolds.
method Established solvability and uniqueness conditions using cohomology of a cochain complex.
result Vanishing theorems for cohomology under geometric assumptions on boundary and error term.
Paper generalizes sub-slope definition and solves complex equations on compact manifolds.
problem Solving complex equations on compact almost Hermitian manifolds.
method Generalized sub-slope definition and proved existence of solutions for a class of equations.
result Solved complex Hessian quotient and deformed Hermitian-Yang-Mills equations.
Proves torsion and curvature are unique for smooth manifolds.
problem Characterizing unique torsion-curvature pairs on smooth manifolds.
method Analyzes linear connections on smooth manifolds of dimension n≥4. result Torsion and curvature are the only pair satisfying Bianchi identities for linear connections.
Study on spectral asymptotics in elasticity on smooth manifolds.
problem Analyzing spectral asymptotics in linear elasticity on smooth manifolds.
method Established two-term spectral asymptotics for boundary value problems in linear elasticity.
result Corrected erroneous results in previous studies.
Most of existing manifold learning methods rely on Mean Squared Error (MSE) or ℓ2 norm. However, for the problem of image quality assessment, these are not promising measure. In this paper, we introduce the concept of an image structure manifold which captures image structure features and discriminates image dist…
Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integr…
This paper investigates how data augmentation improves linear separation of manifold data.
problem Understanding how data augmentation enhances linear separation of manifold data.
method Investigates the conditions under which self-supervised representations can linearly separate multi-manifold data.
result Self-supervised learning can linearly separate manifolds with a smaller distance than unsupervised learning.
Extends parabolic study to flat hyperkähler manifolds.
problem Study problems in hyperhermitian geometry.
method Extends elliptic approach to parabolic setting.
result Solves problems in hyperhermitian geometry.
Study the structure of linear bundle morphisms between vector bundles.
problem Understanding the structure of linear bundle morphisms between vector bundles.
method Analyzing the set of smooth linear bundle morphisms between fibre bundles.
result Detailed structure of linear bundle morphisms between vector bundles.
The paper investigates compatible linear connections on Randers spaces and finds a unique extremal connection.
problem Investigating compatible linear connections on Randers spaces.
method Transformed compatibility equations by taking torsion components as variables and determined when these equations have solutions.
result Characterized Randers spaces as non-Riemannian generalized Berwald spaces with a positive constant norm of perturbing term.
We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.
Diffusion models converge linearly to complex data manifolds.
problem Sampling from high-dimensional complex data distributions.
method Score-matching generative models with novel integration scheme.
result Linear convergence in KL divergence to intrinsic dimension d. Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.
problem Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.
method Comparison with auxiliary complex Monge-Ampère equations, Hölder-Young inequality, and De Giorgi iteration lemma.
result Improved L∞ estimates for fully non-linear elliptic equations on Kähler and Hermitian manifolds.
The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.
problem Boundary estimates for solutions to fully non-linear elliptic equations on Hermitian manifolds.
method Unified approach using quantitative boundary estimates, gradient estimates, and existence results.
result Established gradient estimates and unified approach to Dirichlet problem solutions.
We characterize the conjugate linearized Ricci flow and the associated backward heat kernel on closed three--manifolds of bounded geometry. We discuss their properties, and introduce the notion of Ricci flow conjugated constraint sets which characterizes a way of Ricci flow averaging metric dependent geometrical data. …
Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.
problem Proving uniqueness of asymptotic limits for noncollapsed Ricci flat manifolds with linear volume growth.
method Relating uniqueness to the existence of a harmonic function asymptotic to a Busemann function, proving uniqueness via a monotone quantity.
result Proves uniqueness of the asymptotic limit and establishes a polynomial convergence rate.
We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.
The paper examines scalar fourth-order linear differential operators and their invariants.
problem Equivalence problem of scalar fourth-order linear differential operators.
method Investigation of differential invariants.
result Application of differential invariants to the equivalence problem.
Families of linear connections are constructed on almost contact manifolds with Norden metric. An analogous connection to the symmetric Yano connection is obtained on a normal almost contact manifold with Norden metric and closed structural 1-form. The curvature properties of this connection are studied on two basic cl…
Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this resul…
We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.
Let G be a compact Lie group. (Compact) topological G-manifolds have the G-homotopy type of (finite-dimensional) countable G-CW complexes (2.5). This partly generalizes Elfving's theorem for locally linear G-manifolds [Elf96], wherein the Lie group G is linear (such as compact).
For compact CR manifolds of hypersurface type which embed in complex projective space, we show that for all k large enough there exist linear systems of O(k) which when restricted to the CR manifold are generic in a suitable sense. These systems are constructed using approximately holomorphic geometry.