Proves a theorem for normal distributions on manifolds with boundary.
problem Normal distributions on manifolds with boundary require a new approach to integration.
method Introduces neat integral manifolds with boundary and conditions for integrability.
result Conditions for integrability expressed in terms of adapted collars and integrability on interior and boundary.
Equivariant flows learn symmetrical distributions on manifolds.
problem Learning symmetrical distributions on arbitrary manifolds.
method Equivariant manifold flows.
result Learned gauge invariant densities over SU(n) in quantum field theory.
Random matrix ensembles yield uniform distributions on manifolds.
problem Understanding distributions of vectors in random matrix ensembles.
method Analyzing eigenvalues, singular values, and Autonne-Takagi vectors of various random matrix ensembles.
result Uniform distributions on specific manifolds for different types of random matrix ensembles.
The paper establishes a correspondence between normal distributions and neat foliations on manifolds with boundary.
problem Understanding normal distributions on manifolds with boundary.
method Develops a theory analogous to Stefan and Sussmann's for integrable distributions, focusing on neat foliations.
result A one-to-one correspondence between neatly integrable normal distributions and neat foliations by manifolds with boundary.
The paper proves integral formulas for manifolds with multiple orthogonal distributions.
problem Understanding geometric properties of manifolds with multiple orthogonal distributions.
method Develops integral formulas for Riemannian manifolds with k>2 orthogonal complementary distributions. result Generalizes known formulas for k=2 and applies to manifold splitting and immersions. New HyperKahler structure found for 3-contact distributions on Sasakian manifolds.
problem Finding a HyperKahler structure for 3-contact distributions on Sasakian manifolds.
method Defined a special metric connection and proved curvature properties.
result 3-Sasakian manifolds with constant φα-sectional curvatures have constant holomorphic sectional curvatures in their HyperKahler contact distribution. The paper reinterprets Bayesian priors and posteriors using Riemannian manifolds.
problem The dependence of maximum a posteriori estimates on parametrization.
method Assuming a Riemannian manifold with Fisher metric, the paper reinterprets priors and posteriors as distributions over probability distributions, making estimates independent of parametrization.
result A maximum a posteriori estimate independent of parametrization is defined.
Simplified proofs and new distributions on anti-quasi-Sasakian manifolds.
problem Properties of anti-quasi-Sasakian manifolds.
method Simplified proofs and discussion of new invariant distributions.
result New invariant distributions exist on every anti-quasi-Sasakian manifold.
Characterizes connections on normal distributions manifold.
problem Geometric characterization of connections on normal distributions.
method Homogeneous statistical manifold structure and Lie group analysis.
result Geometric characterization of α-connections on Lie group. In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. A…
The distributional category bounds manifold invariants and imposes constraints.
problem Bounding manifold invariants and understanding constraints.
method Using geometric conditions like non-negative Ricci curvature, the distributional category bounds invariants such as the first Betti number and macroscopic dimension.
result Equality of bounds imposes specific constraints on the manifold.
Maps to manifolds transverse to certain distributions satisfy an h-principle.
problem Maps to manifolds transverse to certain distributions.
method Proving the complete h-principle for transverse maps. result Maps to manifolds transverse to certain distributions satisfy the h-principle. Study h-principles for non-integrable distributions on manifolds.
problem Existence and classification of maximally non-integrable distributions of derived length one.
method Introduced formal structures and used h-principles to discuss existence and classification.
result Discussed existence and classification of maximally non-integrable distributions of derived length one.
Paper classifies structures on 5D manifolds with specific rank and conditions.
problem Classifying tangent distributions on 5D manifolds.
method Established necessary and sufficient topological condition for existence.
result Classification of structures up to homotopy as formal Cartan distributions.
Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.
problem Characterizing metabelian distributions and geodesics in sub-Riemannian manifolds.
method Characterization of metabelian distributions in terms of principal bundle structures. Proof of geodesic properties for rank-2 distributions.
result For rank-2 metabelian distributions, geodesics are of class C1. New distributions on manifolds for better sampling.
problem Creating flexible distributions on Riemannian manifolds.
method Area-preserving maps and isometries for constructing distributions.
result Flexibility and straightforward sampling of distributions.
New method upsamples sparse, non-uniform point clouds more accurately.
problem Suboptimal results from existing point cloud upsampling methods.
method Imposes manifold distribution constraints using Gaussian functions.
result Generates higher-quality, more uniformly distributed dense point clouds.
Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prov…
This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroid and Banach Lie-Poisson manifold. T…
Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions. Svensson [7] has shown that this formula simplifies to a Bochner type formula when we are dealing with Kähler manifolds and holomorphic (integrable) distributions. Here, …
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.
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.
This paper tackles learning functions on manifolds using parallel distributed learning.
problem Learning real-valued functions on manifolds from input-output data pairs.
method Filtered hyperinterpolation and parallel distributed learning.
result Optimal approximation order for non-distributed case, and quantitative relations for distributed case.
This paper constructs Brownian motion on complex flag manifolds and finds joint distribution of stochastic areas.
problem Modeling stochastic areas on complex partial flag manifolds.
method Constructs Brownian motion on complex partial flag manifolds and uses it to find joint distribution of stochastic areas.
result Limit law of stochastic areas is a multivariate Cauchy distribution.
The paper defines a new metric space invariant and computes it for various manifolds.
problem Computing the distributional LS-category of manifolds.
method Defining and analyzing the distributional LS-category of metric spaces and applying it to manifolds.
result Several sufficient conditions for the distributional LS-category of a closed manifold to be maximum are derived.
The study generalizes a specific geometric correspondence to higher dimensions.
problem Understanding nondegenerate lines on holomorphic contact manifolds.
method Analyzing nondegenerate lines and corresponding distributions on higher-dimensional manifolds.
result A generalization of the (2,3,5)-distributions to higher dimensions. New flows model distributions on Riemannian manifolds without domain knowledge.
problem Limited modeling of distributions on Riemannian manifolds.
method Riemannian convex potential maps using optimal transport.
result These flows can model standard distributions on spheres and tori.
The paper introduces exponential-wrapped distributions on symmetric spaces for better data modeling.
problem Challenges in statistical modeling due to curvature of data spaces.
method Construction and use of exponential-wrapped distributions on affine locally symmetric spaces.
result Exponential-wrapped distributions on symmetric spaces have useful properties for practical use.
Proposes QQE for transforming and embedding data distributions.
problem Transforming and embedding data distributions for better representation or visualization.
method Quantile-Quantile Embedding (QQE) using quantile-quantile plot concept.
result QQE allows for better discrimination of classes in some cases.
Proves rigidity of boundaries with constant mean curvature in warped product manifolds.
problem Rigidity and compactness of boundaries with constant mean curvature in warped product manifolds.
method Distributional CMC-rigidity proof for rectifiable boundaries.
result Characterizes limits of boundaries with converging mean curvatures.
Defines conformal submersion with horizontal distribution and provides necessary conditions for its existence.
problem Existence and conditions for conformal submersion with horizontal distribution.
method Definition and analysis of conformal submersion with horizontal distribution, dual connections, and necessary conditions.
result Necessary and sufficient conditions for conformal submersion with horizontal distribution and geodesics.
The article proves the existence of horizontal immersions into fat distributions and contact structures.
problem Proving the existence of horizontal immersions in fat distributions and contact structures.
method Gromov's sheaf theoretic and analytic techniques of h-principle. result Existence of horizontal immersions of an arbitrary manifold into degree 2 fat distributions and quaternionic contact structures.
Study Hochschild cohomology of dg manifolds linked to integrable distributions.
problem Understanding Hochschild cohomology of dg manifolds associated with integrable distributions.
method Analyzing the Hochschild cohomology of (F[1],dF) and relating it to the algebra of functions on leaf space. result Established a canonical isomorphism between the Hochschild cohomology of (F[1],dF) and the algebra of functions on leaf space. The study reveals a persistent bias in the distribution of holonomy on compact hyperbolic 3-manifolds.
problem The distribution of holonomy on compact hyperbolic 3-manifolds is not uniformly distributed.
method An asymptotic count of closed geodesics by their length and holonomy, and analysis of spectral parameters.
result A normalized, smoothed bias count of holonomy is distributed according to a probability distribution, controlled by the number of zero spectral parameters.
We show that, for an affine submersion π:M⟶B with horizontal distribution, B is a statistical manifold with the metric and connection induced from the statistical manifold M. The concept of conformal submersion with horizontal distribution is introduced, which i…
The paper extends Bochner's technique to singular distributions on manifolds.
problem Analyzing the curvature and null space of Hodge Laplacian on singular distributions.
method Defining modified statistical connection, exterior derivative, and Weitzenbock type curvature operator.
result Derivation of Bochner-Weitzenbock type formula leading to vanishing theorems.
This paper introduces a new method to compare collections of distributions on manifolds and graphs.
problem Comparing collections of probability distributions over diverse domains.
method Intrinsic slicing construction for Wasserstein distances, Hilbert embedding, resampling, p-value combination.
result Powerful and well-calibrated p-values for comparing distributions on manifolds and graphs.
In this paper, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection ∇ˉ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler conta…
Study variational problem on manifold with special distributions.
problem Generalize Einstein metrics on manifold with multiple distributions.
method Define functional of pseudo-Riemannian metric and contorsion tensor, prove critical pairs make distributions totally umbilical.
result Metrics in critical pairs make all distributions totally umbilical.
Study curvature of orthogonal distributions on manifolds.
problem Understanding curvature of orthogonal distributions on manifolds.
method Derived Euler-Lagrange equations for a functional of Riemannian metrics.
result Examples of critical metrics for specific distributions.
Study curvature invariants in sub-Riemannian manifolds.
problem Understand curvature invariants in sub-Riemannian geometry.
method Prove geometrical inequalities for submanifolds with orthogonal distributions.
result Inequalities for submanifolds with orthogonal distributions are derived.
I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on ma…
We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular hol…
Study properties of pointwise k-slant submanifolds in Kähler manifolds.
problem Characterize the integrability of component distributions in Kähler manifolds.
method Characterization through integrability and totally geodesic cases.
result Characterize the integrability of component distributions in Kähler manifolds.
Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.
problem Integrability of projective limits of involutive bundles on Banach manifolds.
method An integrability criterion for a projective limit of Banach distributions.
result Result of integrability of projective limit of involutive bundles on a projective sequence of Banach manifolds.
We are interested in comparing probability distributions defined on Riemannian manifold. The traditional approach to study a distribution relies on locating its mean point and finding the dispersion about that point. On a general manifold however, even if two distributions are sufficiently concentrated and have unique …
We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold M there is a metric, such that th…
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.