Paper introduces method to estimate animal motion on unknown submanifolds using Koopman operator.
problem Estimating animal motion on unknown submanifolds in high-dimensional space.
method Data-dependent approximation of Koopman operator in RKHS over ambient space.
result Strong rates of convergence derived for estimates in terms of fill distance.
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.
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.
The paper projects unknown manifolds onto hyperspheres for efficient function approximation.
problem Function approximation from data on unknown manifolds with added errors.
method Projects unknown manifold onto hypersphere and uses localized spherical polynomial kernels.
result Optimal rates of approximation for rough functions are given.
By the Fox's re-embedding theorem, any compact submanifold of the 3-sphere can be re-embedded in the 3-sphere so that it is unknotted. It is unknown whether the Fox's re-embedding can be replaced with twistings. In this paper, we will show that any closed 2-manifold embedded in the 3-sphere can be unknotted by twisting…
Connected sums defined for codimension two locally flat submanifolds in higher dimensions.
problem Defining connected sums for codimension two locally flat submanifolds in various dimensions.
method Using results from higher dimensional topological manifolds and four-manifolds, defining connected sums for codimension two locally flat submanifolds.
result A well-defined connected sum exists up to orientation preserving homeomorphism.
This paper develops a method for estimating animal motion submanifolds using distribution-free learning.
problem Estimating submanifold models of animal motion supported on a configuration manifold.
method Formulates a novel method based on distribution-free learning over the manifold of measurements.
result Derives approximations of the unknown mapping that converge with rates dependent on sample size and dimensionality.
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.
New proof shows diffusion models implicitly estimate intrinsic dimensionality.
problem Estimating intrinsic dimensionality of data from diffusion models.
method Formal proof of FLIPD under realistic assumptions.
result FLIPD's correctness proven under realistic conditions.
Suppose M_t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E^3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals over M_t are constant. It is unknown whether there are nontrivial such bendings. The estimate…
An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The manifolds constructed here are all Haken manifolds in the sense of B. Foozwell and H. Rubinstein and can be systematically reduced to balls by suitably cutting them open alo…
New algorithms improve robustness in metric learning.
problem Improving robustness in metric learning for better performance.
method Robust Geometric Metric Learning (RGML) approach, shrinking covariance matrices towards their barycenter.
result Strong performance and robustness to mislabeled data demonstrated.
Paper proves all Lagrangians unobstructed if one is, using non-archimedean analytic structure.
problem Proving the existence of bounding cochains for unobstructed Lagrangians.
method Introducing non-archimedean analytic structure and using family Floer techniques.
result All Lagrangians in a connected family are unobstructed if one is.
Timely detection of abrupt anomalies is crucial for real-time monitoring and security of modern systems producing high-dimensional data. With this goal, we propose effective and scalable algorithms. Proposed algorithms are nonparametric as both the nominal and anomalous multivariate data distributions are assumed unkno…
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…
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 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.
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.
Given a sample Y from an unknown manifold X embedded in Euclidean space, it is possible to recover the homology groups of X by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set Y. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular…
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 …
Study submanifolds in hyperbolic space, focusing on their boundary and Laplace operator.
problem Understanding the geometry and regularity of submanifolds in hyperbolic space.
method Analyzing asymptotic geometry and regularity properties near the ideal boundary, computing essential spectra.
result Computed essential spectra of the Laplace operator on certain submanifolds.
Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
The paper examines the geometry of specific submanifolds in flag manifolds.
problem Understanding the geometry of invariant almost semi Kähler submanifolds.
method Analyzing homogeneous spaces as almost Hermitian submanifolds of flag manifolds.
result Minimal and totally geodesic properties of certain submanifolds.
Classifies minimal submanifolds in complex hyperbolic spaces.
problem Identifying minimal submanifolds in complex hyperbolic spaces.
method Classification based on extrinsic homogeneity.
result Classification of minimal extrinsically homogeneous submanifolds.
Study on submanifolds with specific types of factors in Kaehler manifolds.
problem Characterizing submanifolds with holomorphic, totally real, and slant factors.
method Introduced sequential warped product submanifolds, established Chen's and pinching inequalities.
result Found geometric results under equality cases of pinching inequalities.
We introduce a new class of lightlike submanifolds, namely, Screen Transversal Cauchy Riemann (STCR)-lightlike submanifolds, of indefinite Kaehler manifolds. We show that this new class is an umbrella of screen transversal lightlike, screen transversal totally real lightlike and CR-lightlike submanifolds. We give a few…
The notion of Lagrangian H-umbilical submanifolds was introduced by B. Y. Chen in 1997, and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanif…
Study finds rigid submanifolds in spacelike waves under specific conditions.
problem Understanding rigidity of submanifolds in spacelike waves.
method Proves submanifolds are contained in characteristic lightlike hypersurfaces under certain conditions.
result Complete codimension two submanifolds are wavefronts under specific conditions.