Sampling random points can reveal submanifold topology.
problem Estimating the topology of submanifolds in Riemannian manifolds.
method Sampling random points in a neighborhood of the submanifold.
result Topology of the submanifold can be recovered with high confidence.
Develops a method for random manifolds and submanifolds, focusing on 3-ball knots.
problem Understanding random submanifolds in triangulated manifolds.
method Coloring vertices of triangulated manifolds to generate random submanifolds.
result Probability of generating an unknot decays exponentially in 3-ball with 3 colors.
Bayesian methods estimate regression functions on submanifolds using graph Laplacian eigenbasis.
problem Estimating regression functions on unknown smooth submanifolds.
method Random geometric graph structure, Bayesian priors based on random basis expansion in graph Laplacian eigenbasis.
result Posterior contraction rates are minimax optimal for any positive smoothness index.
Develops calculus for random submanifolds using zonoids.
problem Calculating properties of random submanifolds defined by zero sets of vector fields.
method Defines zonoid sections and uses them to compute expected volumes and currents.
result Establishes new inequalities and formulas for random submanifolds.
We extend Kac-Rice formula to compute expected intersections of random submanifolds.
problem Computing expected intersections of random submanifolds.
method Generalized Kac-Rice formula using measure theory and integration.
result Formula computes expected cardinality of preimages of submanifolds via random maps.
New method uses manifold learning to infer latent positions of 1D submanifolds in random dot product graphs.
problem Inference on latent positions of unknown 1D submanifolds in RDPGs.
method Apply Isomap for manifold learning to estimate arc lengths on the unknown submanifold.
result Test statistics based on Isomap converge to known submanifold power as auxiliary vertices increase.
Paper analyzes convergence of Laplacian eigenmaps on submanifolds with singularities.
problem Analyzing convergence of Laplacian eigenmaps on submanifolds with singularities.
method Using ε-neighborhood graphs constructed from random points on the submanifold, the paper provides a spectral approximation result for the Laplacian.
result The convergence rate for the eigenvalue of the Laplacian is \( O\left(\left(\log n/n
ight)^{1/(m+2)}
ight) \), where \( m \) and \( n \) are the dimension of the manifold and the sample size, respectively.
Random 3-manifolds have no totally geodesic submanifolds.
problem Existence of totally geodesic submanifolds in random 3-manifolds.
method Analysis of metrics on compact 3-manifolds in Cq-topology. result The set of such metrics contains an open and dense set in the Cq-topology for any q≥3. The aim of this paper is to establish two fundamental measure-metric properties of particular random geometric graphs. We consider ε-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of RK. We show t…
A new method reduces complexity for optimizing large-scale problems with orthogonality constraints.
problem Optimizing large-scale problems with orthogonality constraints.
method Randomized Riemannian submanifold method that restricts updates to random submanifolds.
result Significantly reduces per-iteration complexity for large-scale problems.
Estimates manifold dimension from random samples.
problem Estimating the dimension of a manifold from random samples.
method Explicit theoretical and heuristic bounds for data set size.
result Data set needs to be sufficiently large for accurate dimension estimation.
We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m-dimensional submanifold M in Rd as the sample size n increases and the neighborhood size h tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of…
For n≥4 we show that generic closed Riemannian n-manifolds have no nontrivial totally geodesic submanifolds, answering a question of Spivak. An immediate consequence is a severe restriction on the isometry group of a generic Riemannian metric. Both results are widely believed to be true, but we are not aware of…
The Freund family of distributions becomes a Riemannian 4-manifold with Fisher information as metric; we derive the induced α-geometry, i.e., the α-curvature, α-Ricci curvature with its eigenvales and eigenvectors, the α-scalar curvature etc. We show that the Freund manifold has a positive constant 0-scalar cur…
The paper describes fitting submanifolds to data using Sussmann's orbit theorem.
problem Fitting an immersed submanifold to random samples.
method Uses Sussmann's orbit theorem to ensure submanifold fitting. Reconstruction involves encoding times and decoding via flows of vector fields.
result A high-probability bound on excess risk for the reconstruction error.
For a non-uniform lattice in SL(2,R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. I…
Polynomial chaos expansions on Grassmannian submanifolds for high-dimensional stochastic systems.
problem Uncertainty quantification in high-dimensional stochastic systems.
method Principal Geodesic Analysis on the Grassmann manifold, adaptive algorithm for local submanifolds, polynomial chaos expansion.
result Efficient surrogate modeling of system behavior across different parameter spaces.
We show that gamma distributions provide models for departures from randomness since every neighbourhood of an exponential distribution contains a neighbourhood of gamma distributions, using an information theoretic metric topology. We derive also the information geometry of the 3-manifold of McKay bivariate gamma dist…
A new model clusters networks with community-specific submanifold structures.
problem Clustering networks with community-specific submanifold structures.
method Latent Structure Block Models (LSBM) for Bayesian spectral graph clustering.
result LSBM correctly recovers underlying communities in one-dimensional manifold structures.
The fields of compressed sensing (CS) and matrix completion have shown that high-dimensional signals with sparse or low-rank structure can be effectively projected into a low-dimensional space (for efficient acquisition or processing) when the projection operator achieves a stable embedding of the data by satisfying th…
A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden-Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifold…
Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from a random sample of a d-dimensional compact submanifold M in RD, we establish the spectral convergence rate…
The study characterizes submanifolds in product spaces.
problem Limited studies on pseudo-umbilical submanifolds.
method Using projections from product structure, conditions for submanifolds to be invariant, anti-invariant, or semi-invariant are derived.
result Necessary and sufficient conditions for pseudo-umbilical submanifolds in locally product Riemannian manifolds.
Researchers describe a specific type of submanifolds in Euclidean space.
problem Understanding inhomogeneous almost symmetric submanifolds.
method Completely describing submanifolds as unions of parallel symmetric submanifolds.
result Described inhomogeneous properly embedded almost symmetric submanifolds as unions of symmetric submanifolds.
We introduce the notions of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and pointwise semi-slant submanifolds. We have characterizations and investigate the integrability of distrib…
Develops methods for computing conformal invariants of submanifolds.
problem Computing conformal invariants of submanifolds.
method Direct construction of extrinsic ambient space, global invariants of conformally compact minimal submanifolds, introduction of conformal submanifold scalars.
result Derives an explicit Gauss--Bonnet--Chern-type formula and proves a rigidity result.
We introduce the {\it diffusion K-means} clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion K-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as …
We study lightlike submanifolds of indefinite statistical manifolds. Contrary to the classical theory of submanifolds of statistical manifolds, lightlike submanifolds of indefinite statistical manifolds need not to be statistical submanifold. Therefore we obtain some conditions for a lightlike submanifold of indefinite…
Survey of recent austere submanifold results.
problem Understanding austere submanifolds.
method Review of existing papers [24,25].
result Recent progress in the field of austere submanifolds.
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.
The paper derives Chen inequalities for statistical submanifolds in cosymplectic manifolds.
problem Deriving Chen inequalities for statistical submanifolds in cosymplectic manifolds.
method Analyzing statistical cosymplectic manifolds and Legendrian submanifolds to derive Chen inequalities.
result Chen inequalities for statistical submanifolds in cosymplectic manifolds and Legendrian submanifolds are derived.
Study attached submanifolds in solvmanifolds, generalizing symmetric space results.
problem Exploring submanifolds in non-symmetric spaces with unusual curvature properties.
method Generalizing Tamaru's construction to pseudo-Riemannian scalar products and root spaces.
result Ricci curvature restriction holds for attached submanifolds under specific algebraic conditions.
Study on null submanifolds in indefinite complex contact geometry.
problem Geometry of null submanifolds in indefinite complex contact manifolds.
method Analysis of quaternion null submanifolds and distributions on screen submanifolds.
result Quaternion null submanifolds are always totally geodesic.
Study various submanifolds in quaternionic skew-Hermitian spaces.
problem Characterize submanifolds in almost quaternionic skew-Hermitian manifolds.
method Construct explicit examples of submanifolds in semisimple quaternionic skew-Hermitian symmetric spaces.
result Explicit examples of submanifolds for each type considered.
Generative model for joint discrete distributions using randomized assignment flows.
problem Efficiently representing and sampling from complex joint distributions of discrete variables.
method Randomized assignment flows on the statistical submanifold of factorizing distributions.
result Our model can efficiently represent and sample from any target distribution and assess likelihood of unseen data points.
An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space PN(C). By means of the foca…
Proves Frankel theorem for generic submanifolds in Sasakian manifolds.
problem Intersection properties of generic submanifolds in Sasakian manifolds.
method Introduces a weaker notion of generic submanifolds and proves Frankel theorem under specific conditions.
result Derives topological information about generic submanifolds in Sasakian space forms.
The paper normalizes Poisson saturation of coregular submanifolds.
problem Normalizing the Poisson saturation of coregular submanifolds.
method Normal form construction and Poisson geometry analysis.
result Local Poisson saturation of coregular submanifolds is an embedded Poisson submanifold with a normal form.
The paper studies λ-submanifolds in Gauss spaces and proves theorems for complete proper ones.
problem Understanding λ-submanifolds in Gauss spaces and their properties. method Using divergence type theorems and Simons' identities, the authors prove theorems for complete proper λ-submanifolds. result Proves halfspace and gap theorems for complete proper λ-submanifolds, generalizing previous results. Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce these two notions into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infin…
In this paper, invariant submanifolds of a generalized Kenmotsu manifold are studied. Necessary and sufficient conditions are given on a submanifold of a generalized Kenmotsu manifold to be an invariant submanifold.In this case, we investigate further properties of invariant submanifolds of \ a generalized Kenmotsu man…
Study on submanifolds of Euclidean space, classifying their symmetry types.
problem Classifying symmetry types of submanifolds in Euclidean space.
method Analyzing properties of full irreducible almost symmetric submanifolds and their cohomogeneity.
result Classification of almost symmetric submanifolds into specific types.
Constructs biharmonic and r-harmonic submanifolds in cohomogeneity one manifolds.
problem Constructing biharmonic and r-harmonic submanifolds. method Using cohomogeneity one manifolds, the normal index of submanifolds is studied, and new examples are provided.
result Constructs metrics on the sphere with biharmonic non-minimal hypersurfaces.
The paper studies a new type of submanifolds in product spaces.
problem Characterizing and understanding warped product pointwise bi-slant submanifolds.
method Introduced and studied warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds.
result Characterization results and non-trivial examples of these submanifolds.
Recent developments on biconservative submanifolds in Riemannian geometry.
problem Characterizing and understanding biconservative submanifolds.
method Analyzing stress-energy tensors and bitension fields.
result Biconservative submanifolds and H-submanifolds coincide in Euclidean spaces. Diffeological submanifolds are a new type of submanifold in manifold theory.
problem Defining and understanding different types of submanifolds in manifold theory.
method Introducing diffeological submanifolds and comparing them with other types of submanifolds.
result A diffeological submanifold can be included in a manifold without being an immersion.
In this paper we study slant submanifolds of Lorentzian almost contact manifolds. We have taken the submanifold as a space like and then defined the slant angle on a submanifold and thus we extended the results of A. Lotta (Slant submanifolds in contact geometry [8]) and M. A. Khan et. al. (Slant submanifolds of Lorent…
The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being A-manifolds in the sense of A.Gray but rarely Ricci-parallel (\cite{QTY},\cite{LY},\cite{TY3}). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that …