The paper examines Einstein warped product manifolds and finds a flat Base-manifold condition.
problem Investigating conditions for (2+2)-Einstein warped product manifolds. method Analyzing the (2+2)-Einstein warped product manifolds with specific curvature conditions. result The f-curvature-Base condition is equivalent to a flat Base-manifold. This paper extends curvature results to higher fiber dimensions.
problem Investigating curvature constraints on base for (2+m)-Einstein warped product manifolds. method Extending previous work to higher fiber dimensions, considering (2,m)-PNDP manifolds. result The dimension of the fiber does not affect curvature results.
A new metric-based principal curve method learns 1D manifolds from spatial data.
problem Learning 1D manifolds from spatial data.
method Metric-based Principal Curve (MPC) approach.
result The method effectively learns the shape of 1D manifolds from synthetic and real datasets.
Study Einstein warped products with Einstein base and fiber.
problem Characterize Einstein warped products with Einstein base and fiber.
method Investigate necessary and sufficient conditions for a warped product to be Einstein.
result Explicitly determine the warping function when the base is hyperbolic space.
The paper studies Riemannian maps with Ricci soliton base manifolds.
problem Analyzing Riemannian maps with specific properties of base manifolds.
method Analyzing Riemannian curvature tensor, Ricci tensor, scalar curvature, and necessary conditions for Ricci soliton leaves.
result Necessary and sufficient conditions for harmonicity and biharmonicity of Riemannian maps.
The based loop space homology of a special family of homogeneous spaces, flag manifolds of connected compact Lie groups is studied. First, the rational homology of the based loop space on a complete flag manifold is calculated together with its Pontrjagin structure. Second, it is shown that the integral homology of the…
An almost Clifford and an almost Cliffordian manifold is a G--structure based on the definition of Clifford algebras. An almost Clifford manifold based on $\mathcal O:= \cc l (s,t)$ is given by a reduction of the structure group GL(km,R) to GL(m,O), where k=2s+t and m∈N. An…
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.
Extends geometrical description of tensor manifolds in tree-based formats.
problem Geometrical description of tensor manifolds in tree-based formats.
method Provided a new geometrical description of manifolds of tensors in tree-based format.
result Geometrical description compatible with Tucker format.
RSGMs extend SGMs to Riemannian manifolds for better data modeling.
problem Current SGMs are limited to Euclidean spaces; RSGMs handle Riemannian manifolds.
method RSGMs use a noising stage with a diffusion process and a denoising model approximating the time-reversal of the diffusion on Riemannian manifolds.
result RSGMs improve generative modeling for data on Riemannian manifolds.
Develop intrinsic consensus-based optimization framework on Riemannian manifolds with bounded curvature.
problem Nonconvex optimization on manifolds
method Intrinsic consensus-based optimization on Riemannian manifolds with bounded curvature
result Global convergence of the mean-field equation toward a global minimizer of the objective function.
In this paper we give a method to construct Heegaard splittings of oriented graph manifolds with orientable bases. A graph manifold is a closed 3-manifold admitting only Seifert-fibered pieces in its Jaco-Shalen decomposition; for technical reasons, we restrict our attention to the fully oriented case, i.e. both the …
We derive a formula for the Dijkgraaf-Witten invariants of orientable Seifert 3-manifolds with orientable bases.
Topology-aware generative models improve manifold-based defenses against adversarial examples.
problem Adversarial examples compromise the reliability of ML models, especially DNNs.
method Investigate if generative models used in manifold-based defenses need to be topology-aware.
result Topology-aware generative models enhance the robustness of manifold-based defenses.
The paper studies Clairaut maps on Sasakian manifolds.
problem Investigating Clairaut maps on Sasakian manifolds.
method Analyzing necessary and sufficient conditions for geodesics and biharmonicity.
result Conditions for Clairaut anti-invariant Riemannian maps on Sasakian manifolds.
We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.
Simplifies denoising score matching for manifold learning.
problem Learning distributions on manifolds is computationally intensive.
method Modifies denoising score matching to implicitly account for the manifold.
result Reduces computational burden while maintaining efficiency.
Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
problem Determining when cosymplectic manifolds are diffeomorphic based on their cosymplectomorphism groups.
method Characterized Reeb flow, used to descend isomorphism to symplectic base manifolds, preserved monodromy class ensuring bundle equivalence.
result Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
We construct new examples of manifolds with cyclic-parallel Ricci tensor, so called A-manifolds, on a r-torus bundle over a product of almost Hodge A-manifolds.
The paper studies maps between Riemannian and Kähler manifolds, focusing on Clairaut semi-invariant Riemannian maps.
problem Analyzing maps between Riemannian and Kähler manifolds, particularly Clairaut semi-invariant Riemannian maps.
method Recalled and defined Clairaut semi-invariant Riemannian maps, derived necessary and sufficient conditions for geodesic curves and maps, and explored foliations and product manifolds.
result Necessary and sufficient conditions for various properties of Clairaut semi-invariant Riemannian maps were derived.
We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is η-E…
Study graph-based algorithms for multi-manifold clustering with sufficient conditions.
problem Clustering data from a union of manifolds with different dimensions and intersections.
method Investigate sufficient conditions for similarity graphs to capture geometric information.
result High probability error bounds for spectral approximation of tensorized Laplacian.
Paper proposes methods to learn sub-manifolds and estimate densities in normalizing flows.
problem Normalizing flows struggle with finding sub-manifolds in high-dimensional data.
method Introduces per-pixel penalized log-likelihood and hierarchical training approaches.
result Validated superior performance in manifold learning and density estimation.
The paper investigates geometrical aspects of static spacetime with almost gradient Ricci solitons.
problem Geometrical properties of static spacetime with almost gradient Ricci solitons.
method Analyzing conditions and properties of static spacetime with almost gradient Ricci solitons.
result Conditions and properties of static spacetime with almost gradient Ricci solitons are determined.
Existence of hypersurfaces in warped product manifolds proven.
problem Existence of closed hypersurfaces in warped product manifolds.
method Standard degree theory based on a priori estimates.
result Existence of solutions to prescribed Weingarten curvature equations.
Deep neural networks (DNNs) have been successfully applied to a wide variety of acoustic modeling tasks in recent years. These include the applications of DNNs either in a discriminative feature extraction or in a hybrid acoustic modeling scenario. Despite the rapid progress in this area, a number of challenges remain …
We use Heath's, Moriah's and Schultens's results to prove that irreducible Heegaard splittings of orientable Seifert manifolds with nonorientable base space are either vertical or horizontal.
The tangent bundle of an almost Norden manifold and the complete lift of the Norden metric is considered as a 4n-manifold. It is equipped with an almost hypercomplex Hermitian-Norden structure. It is characterized geometrically. The case when the base manifold is an h-sphere is considered.
New examples of manifolds with large symmetry found.
problem Finding manifolds with significant continuous symmetry.
method Analyzing the solvable radicals of isometry group actions on universal covers.
result Examples of aspherical manifolds with large symmetry that do not support locally homogeneous metrics.
2-knot manifolds have Seifert fibered base orbifolds.
problem Characterizing Seifert fibered 2-knot manifolds.
method Analyzing aspherical 2-orbifolds with orbifold fundamental groups of weight 1.
result Aspherical 2-orbifolds with orbifold fundamental groups of weight 1 are Seifert fibered bases of 2-knot manifolds.
It is proven that a local Lie algebra in the sense of A. A. Kirillov determines the base manifold up to a diffeomorphism provided the anchor map is nowhere-vanishing. In particular, the Lie algebras of nowhere-vanishing Poisson or Jacobi brackets determine manifolds. This result has been proven for different types of d…
SBMs learn manifold-like structures by mixing samples with a non-conservative field.
problem How SBMs learn data distributions on low-dimensional manifolds.
method Investigating linear approximations and subspaces of local feature vectors during diffusion.
result SBMs mix samples by a non-conservative field within the manifold, maintaining manifold-like structure.
Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
problem Characterize fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
method Construct a class of involutions, extend product involutions, and use double covering.
result Any fiber-preserving, orientation-reversing involution factors as a product of an orientation-preserving and a specific class of involutions.
Establishes a link between heat diffusion and manifold distances in data.
problem No theoretical link between diffusion-based manifold learning and geodesic distances.
method Formulates heat geodesic embeddings based on Riemannian geometry.
result Method outperforms state-of-the-art in preserving manifold distances and cluster structure.
Paper extends group actions on Seifert manifolds with non-orientable base space.
problem Finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds with non-orientable base space.
method Constructing a double cover and isomorphism between fiber-preserving diffeomorphisms and actions on the cover.
result Construction of all actions satisfying a condition on the obstruction term and structure of finite groups.
In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.
Study gradient Yamabe solitons on warped product manifolds.
problem Characterize and provide examples of gradient Yamabe solitons on warped product manifolds.
method Prove triviality results and provide nontrivial examples using a conformal base and translation group action.
result Provide infinitely many explicit examples of complete steady gradient Yamabe solitons.
We present a simple combinatorial method to encode 3-dimensional manifolds, based on their Heegaard diagrams. The notion of a Gauss diagram of a 3-manifold is introduced. We check the conditions for a Gauss diagram to represent a closed manifold and a manifold with boundary.
This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the bas…
The paper calculates curvature formulas for stable Higgs bundles.
problem Investigating the geometry of stable Higgs bundles over complex manifolds.
method Applying curvature formulas from Schumacher-Toma and Biswas-Schumacher to calculate and analyze the curvature of the base manifold.
result Explicit calculation of holomorphic sectional curvature for a complex Finsler metric.
A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifo…
MLDL preserves manifold geometry in vector transformations.
problem Geometric deterioration in neural network transformations.
method Locally isometric smoothness (LIS) and Markov random field (MRF) encoding.
result Enhanced vector transformations into well-behaved metric homeomorphisms.
Open manifolds can be covered by Rm with finite or infinite degree.
problem Covering open manifolds with Rm. method Proving simple branched covers for open manifolds of dimensions 2 and 3, and investigating universal bases.
result Open manifolds can be covered by Rm with finite or infinite degree. Introduces a new manifold from a graph subgraph.
problem None explicitly stated in the abstract.
method Defined geometrically from a combinatorial subgraph of the Hoffman-Singleton graph.
result Geometric properties of the new manifold presented.
By a classical theorem of Gallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is dec…
A new graph-based approach for estimating complex data with manifold structure.
problem Regression of large-scale, complex data with underlying geometric structure and noises.
method Constructing a skeleton graph to capture geometric structure, defining metrics, and applying nonparametric regression.
result Statistical guarantees and effectiveness demonstrated through simulations and real data examples.
A method to fix radius distortion in generative models on curved spaces.
problem Distortion in geodesic radius measurements across different charts on Riemannian manifolds.
method Radial Compensation (RC) adjusts the tangent-space base distribution to match the geodesic radius law, improving model stability and interpretability.
result RC ensures that the model's geodesic radius matches the intended distribution, improving numerical stability and curvature interpretation.
We propose a new framework for manifold denoising based on processing in the graph Fourier frequency domain, derived from the spectral decomposition of the discrete graph Laplacian. Our approach uses the Spectral Graph Wavelet transform in order to per- form non-iterative denoising directly in the graph frequency domai…