Study Brownian motion on Grassmann manifold using matrix stochastic calculus.
problem Understanding Brownian motion on non-compact Grassmann manifold.
method Realize Brownian motion as matrix diffusion process, use matrix stochastic calculus, and hyperbolic Stiefel fibration.
result Connection to generalized Maass Laplacian of complex hyperbolic space.
New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.
problem Defining Morse-Bott functions on non-linear Stiefel manifolds.
method Replacing linear height function with a quadratic one, proving it as a Morse-Bott function.
result Critical submanifolds are fibrations of products of Grassmannians, not Grassmannians themselves.
Study on determinants of unitary Brownian motion and their asymptotic laws.
problem Understanding determinants of unitary Brownian motion and their behavior over time.
method Using Stiefel fibration and skew-product decomposition of the Stiefel Brownian motion.
result Prove asymptotic laws for determinants of block entries of unitary Brownian motion.
Study shows non-vanishing Stiefel-Whitney classes and absence of spin^C structures in certain hyperbolic manifolds.
problem Existence of manifolds without spin^C structures and non-vanishing higher order Stiefel-Whitney classes.
method Analysis of cusped arithmetic hyperbolic manifolds of simplest type.
result Existence of manifolds with non-vanishing Stiefel-Whitney classes and absence of spin^C structures.
The study finds hyperbolic manifolds without spin^c structures in dimensions 5 and above.
problem Existence of closed hyperbolic manifolds without spin^c structures.
method Proof of existence and commensurability classes for manifolds with non-vanishing third Stiefel-Whitney class.
result Infinitely many commensurability classes of closed hyperbolic manifolds without spin^c structures.
The paper identifies 2^(k+1) distinct components of hyperbolic representations.
problem Understanding the structure of representations of non-orientable surfaces.
method Analysis of square map and Stiefel-Whitney classes.
result There are 2^(k+1) connected components of representations.
New hyperbolic manifolds with diverse features created.
problem Creating new types of hyperbolic manifolds.
method Using cubulations and hyperbolization procedures.
result First examples of hyperbolic manifolds with non-trivial Stiefel-Whitney and Pontryagin classes.
There are hyperbolic 3-manifolds that fiber over the circle but that do not admit fibrations by minimal surfaces. Furthermore these manifolds do not admit fibrations by surfaces that are even approximately minimal.
By a work of Thurston, it is known that if a hyperbolic fibred 3-manifold M has Betti number greater than 1, then M admits infinitely many distinct fibrations. For any fibration ω on a hyperbolic 3-manifold M, the number of fibrations on M that are commensurable in the sense of Calegari-Sun-Wang to ω is…
We discuss fibered commensurability of fibrations on a hyperbolic 3-manifold, a notion introduced by Calegari, Sun and Wang. We construct manifolds with non-symmetric but commensurable fibrations on the same fibered face. We also prove that if a given manifold M does not have any hidden symmetries, then M does not admi…
The study finds a limit on subgroup complexity in hyperbolic 3-manifold groups.
problem Understanding subgroups of bounded rank in hyperbolic 3-manifold groups.
method Proving a finiteness theorem for subgroups of bounded rank.
result Every bounded rank covering tower of closed hyperbolic 3-manifolds is a tower of finite covers associated to a fibration over a 1-orbifold.
A fibration of a Riemannian manifold is fiberwise homogeneous if there are isometries of the manifold onto itself, taking any given fiber to any other one, and preserving fibers. Examples are fibrations of Euclidean n-space by parallel n-planes, and the Hopf fibrations of the round n-sphere by great n-spheres. In this …
Unified method to compute Laplace spectra on homogeneous principal bundles.
problem Computing the Laplace-Beltrami spectrum on homogeneous principal bundles.
method Unified representation-theoretic approach using generalized canonical variations and spectral branching criterion.
result Explicit formulas for the full spectra of several geometric families.
Study stretch laminations in hyperbolic 3-manifolds via circle-valued maps.
problem Characterize stretch laminations in hyperbolic 3-manifolds.
method Use Thurston norm and Dehn filling slope length to determine stretch laminations as unions of core curves.
result Show existence of infinitely many examples with fibration and only closed leaves.
New groups algebraically fibre with high-dimensional hyperbolic groups.
problem Finding new quasi-isometry classes of hyperbolic groups.
method Constructing infinitely many hyperbolic groups as finite-index subgroups of right-angled Coxeter groups.
result Groups algebraically fibre with finitely presented kernels, expanding finiteness properties.
Stiefel-Whitney classes of moment-angle manifolds are trivial.
problem Analyzing the topological properties of moment-angle manifolds.
method Proving triviality of Stiefel-Whitney classes for moment-angle manifolds, including partial quotients.
result Stiefel-Whitney classes of moment-angle manifolds are trivial.
This paper studies algebraic fibrations in hyperbolic 4-manifolds.
problem Understanding algebraic fibrations in hyperbolic 4-manifolds.
method Analyzing exact sequences and properties of groups associated with hyperbolic 24-cell and 120-cell.
result The fiber-kernel groups are finitely generated but not finitely presented.
Proves non-hyperbolicity of symplectic varieties with specific properties.
problem Non-hyperbolicity of primitive symplectic varieties with certain conditions.
method Uses ergodicity, birational contractions, and cycle spaces.
result Vanishing Kobayashi pseudometric for symplectic varieties with Lagrangian fibrations.
We obtain infinitely many (non-conjugate) representations of 3-manifold fundamental groups into a lattice in the holomorphic isometry group of complex hyperbolic space. The lattice is an orbifold fundamental group of a branched covering of the projective plane along an arrangement of hyperplanes constructed by Hirzebru…
The paper calculates ranks and bounds for Stiefel manifolds over different fields.
problem Computing ranks and bounds for Stiefel manifolds over various fields.
method Computation of upper characteristic ranks and cup lengths, providing bounds and necessary conditions for maps.
result Bounds and necessary conditions for S3-maps between quaternionic Stiefel manifolds. Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
Paper proves convergence of Kalman filter on Stiefel manifolds with measurement errors.
problem Filtering constant particle with measurement errors on Stiefel manifolds.
method Extended Kalman filter applied to Stiefel manifold-valued observations.
result Convergence of the extended Kalman filter proved for constant system process.
A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with b2≥7 admits only finitely…
Researchers compute curvatures of Stiefel manifolds with new metrics.
problem Computing curvatures of Stiefel manifolds with specific metrics.
method Two approaches: global curvature formula and left-invariant metrics.
result Stiefel manifolds always carry an Einstein metric and have non-negative sectional curvature.
Improved Kalman filter for Stiefel manifold measurements.
problem Improving accuracy in measurements on Stiefel manifolds.
method Generalization of extended Kalman filter for Stiefel manifold-valued measurements.
result Significant improvement over raw measurements.
A new retraction on Stiefel manifold with a closed-form inverse.
problem Efficiency in Riemannian computing applications.
method Introduces a new retraction on the compact Stiefel manifold with a closed-form inverse.
result The retraction is second-order accurate and features a closed-form inverse.
ManifoldFlow relaxes fixed-spectrum Stiefel layers to learn a positive spectrum.
problem Fixed-spectrum Stiefel layers impose rigid spectral constraints.
method Introduces ManifoldFlow, a relaxation that learns a positive spectrum while keeping the basis on the Stiefel manifold.
result Learnable SPD spectrum improves performance in various settings.
Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.
problem Determining the shortest geodesic loops and injectivity radius on Stiefel manifold.
method Combining bounds on sectional curvature with existing metrics.
result Exact value of the injectivity radius for a wide range of metrics.
James's octonionic Stiefel spaces questions answered partially.
problem Two fundamental questions about octonionic Stiefel spaces.
method Partial answers to James's questions about octonionic Stiefel spaces.
result Partial answers to James's questions about octonionic Stiefel spaces.
Upper bound on Stiefel manifold's injectivity radius found.
problem Finding the maximum distance within which the Stiefel manifold remains injective.
method Exhibited conjugate points and calculated the minimum of geodesic lengths.
result Upper bound on Stiefel manifold's injectivity radius is conjectured to be equal to the injectivity radius.
A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to connected components in an explicit form. This is done by considering a natural n…
Study connects surface projections in fibered 3-manifolds.
problem Understanding surface projections in fibered 3-manifolds.
method Uniformly relates structure of surface projections across fibrations.
result Extends previous work to general cases.
Sedenions have geometric submanifolds with special properties.
problem Characterizing zero divisors in sedenion algebra.
method Analyzing zero divisors as submanifolds and proving isometries.
result Zero divisors form submanifolds isometric to Lie groups and Stiefel manifolds.
New theorem shows embedding restrictions for manifold skeletons.
problem Embedding restrictions for triangulated manifolds.
method Proves van Kampen-Flores theorem for manifolds with specific Stiefel-Whitney classes.
result Triangulated manifolds with non-trivial Stiefel-Whitney classes cannot embed into R2d. Study on rolling Stiefel manifolds with specific metrics.
problem Intrinsic and extrinsic rolling of Stiefel manifolds with α-metrics. method Investigation of intrinsic rolling of normal naturally reductive homogeneous spaces, derivation of ODEs for rolling, and explicit solutions.
result Explicit solutions for intrinsic and extrinsic rolling of Stiefel manifolds.
Helix surfaces in Anti-de Sitter space maintain constant Gaussian curvature.
problem Understanding helix surfaces in Anti-de Sitter space with Berger-like metrics.
method Proved helix surfaces have constant Gaussian curvature and described them explicitly.
result Explicit local description of helix surfaces in terms of isometries and curves.
We discuss conditions for the integrability of an almost complex structure defined on the total space of an induced Hopf S^3-bundle over a Sasakian manifold . As an application, we obtain an uncountable family of inequivalent complex structures on the Stiefel manifolds of orthonormal 2-frames in C^{n+1}, non compatible…
Injectivity radius on Stiefel manifold is π.
problem Determining the injectivity radius of the compact Stiefel manifold.
method Utilized the property of geodesics being space curves of constant Frenet curvatures.
result The injectivity radius on the Stiefel manifold under the Euclidean metric is π.
Formula derived for Laplace-Beltrami on Stiefel manifold.
problem Finding Laplace-Beltrami operator on Stiefel manifold.
method Using the general framework of Laplace operators on constraint manifolds, derived the explicit formula in terms of ambient Euclidean coordinates.
result Extended previously known formulas for sphere and special orthogonal group.
New example of hyperbolic 6-manifold with circle-valued Morse function.
problem Finding Morse functions on hyperbolic manifolds.
method Constructing a circle-valued Morse function on a hyperbolic 6-manifold.
result First example of a hyperbolic 6-manifold with a perfect circle-valued Morse function.
Efficiently optimizes CNN and RNN parameters on Stiefel manifold.
problem Computational expense in optimizing orthonormal matrices on Stiefel manifold.
method Cayley transform for efficient retraction and vector transport on Stiefel manifold.
result Cayley SGD and ADAM achieve faster convergence and less training time.
New retraction on symplectic Stiefel manifold with closed-form inverse.
problem Efficient mapping of manifold data to Euclidean domain.
method Introduces a new retraction map with a closed-form inverse.
result The new retraction has a closed-form inverse, unlike previous methods.
Researchers find examples of real Bott manifolds with nonzero dual Stiefel-Whitney class wbar_{n-ahat(n)} for all n nonzero mod 4.
problem Finding compact orientable manifolds with nonzero dual Stiefel-Whitney classes of largest possible grading.
method Constructing real Bott manifolds for all n nonzero mod 4.
result Examples of real Bott manifolds with the desired property are found for all n nonzero mod 4.
A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology cl…
A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety X is given by means of the conormal cycle of an embedding of X in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. W…
New method for identifying autoregressive systems on manifolds.
problem Identifying autoregressive systems on Stiefel and Grassmann manifolds.
method Defining parameters as orthogonal group elements, averaging over observations, conjugate gradient descent on manifolds.
result System parameters can be estimated efficiently using the proposed algorithm.
Study subelliptic heat kernel on octonionic anti-de Sitter space.
problem Heat kernel of octonionic anti-de Sitter space.
method Lift Laplacian of octonionic hyperbolic space and use sub-Laplacian.
result Two integral representations for subelliptic heat kernel.
Study geodesic orbit metrics in quaternionic Stiefel manifolds.
problem Characterize geodesic orbit spaces in quaternionic Stiefel manifolds.
method Analyze homogeneous Riemannian spaces (M=G/H,g) with geodesics as orbits of subgroups. result Identify conditions for $(\Sp(n)/\Sp(n_1) imes \cdots imes \Sp(n_s), g)$ to be a geodesic orbit space.