Study infinitesimal properties of idempotent sets in Banach algebras.
problem Investigate infinitesimal aspects of idempotent sets in Banach algebras.
method Introduce and study Stiefel bundles on flag manifolds, using connections on infinite-dimensional bundles.
result Developed new connections on infinite-dimensional bundles.
Non-trivial Clifford bundle from loop space tangent bundle.
problem Triviality obstruction of Clifford bundle on loop space.
method Constructing Clifford algebra bundle from loop space tangent bundle, showing non-triviality through Stiefel-Whitney and Pontrjagin classes.
result Clifford bundle is non-trivial, obstructed by manifold's Stiefel-Whitney and Pontrjagin classes.
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.
Proves Massey's theorems on complex structure obstructions.
problem Finding complex structures on real vector bundles.
method Fractional Stiefel-Whitney classes and combinations of Pontryagin, Chern, and Euler classes.
result Determines the second obstruction for rank six bundles.
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…
A 4-manifold is parallelizable if its Stiefel-Whitney and Pontryagin classes vanish.
problem Characterizing parallelizable 4-manifolds.
method Classification of SO(4)-bundles over the 4-sphere using Euler and first Pontryagin classes. result A closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class, and Euler characteristic vanish.
Non-orientable 4-manifolds are simple branched coverings of RP^4 and twisted S^3-bundles.
problem Characterizing non-orientable 4-manifolds as branched coverings.
method Showing that every closed connected non-orientable PL 4-manifold is a simple branched covering of $\RP^4$ and a twisted S3-bundle. result Non-orientable 4-manifolds are simple branched coverings of $\RP^4$ and twisted S3-bundles, with specific conditions on the degree and branch set. The paper develops strong lower bounds for projective Stiefel manifolds, resolving special cases with the Browder-Dupont invariant.
problem Developing strong lower bounds for the span of projective Stiefel manifolds.
method Elementary stability properties of vector bundles, with special attention to the Browder-Dupont invariant for odd dimensions.
result Characterization of n for which the Browder-Dupont invariant is well-defined and use of this invariant to obtain lower bounds for the span. Proof of orientable 3-manifolds parallelizability using knot theory.
problem Proving all orientable 3-manifolds are parallelizable.
method Using knot theory and relationships between tangent and normal bundles.
result Completion and modification of a proof of Stiefel's theorem.
In the paper we consider the Stiefel manifold Vn;k as a principal U(k)- bundle over the Grassmann manifold and study the cut locus from the unit element. We gave the complete description of this cut locus on Vn;1 and presented the sufficient condition on the general case. At the end, we study the complement…
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 of instantons on Stiefel manifold with G2 and Sasakian structures.
problem Characterizing and classifying instantons on Stiefel manifold.
method Reductive decomposition, spinorial approach, Weitzenböck-type formula with torsion.
result Classification of invariant connections and rigidity results for G2-instantons. 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.
Study geodesic orbit metrics on specific homogeneous spaces.
problem Characterize geodesic orbit spaces in a class of homogeneous bundles.
method Analyze geodesic orbit spaces of compact Lie groups with semisimple subgroups.
result Identify conditions for a metric to be geodesic orbit.
Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid b…
The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description …
A canonically defined mod 2 linear dependency current is associated to each collection of m sections of a real rank n vector bundle. This current is supported on the linear dependency set of the collection of sections. It is defined whenever the collection satisfies a weak measure theoretic condition called "atomicity"…
Let h be a Real bundle, in the sense of Atiyah, over a space X. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that BU is equipped with a structure of conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern cla…
Computes conditions for real spinor bundles on pseudo-Riemannian manifolds.
problem Obtaining Dirac operators on real spinor bundles of complex type.
method Using Lipschitz structures and Karoubi Stiefel-Whitney classes, reformulating the problem in terms of Spin^o_α structures.
result Explicit construction and characterization of real spinor bundles of irreducible complex type.
A proper etale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel-Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of th…
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.
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. In this paper we define a Poincaré-Reidemeister scalar product on the determinant line of the cohomology of any flat vector bundle over a closed orientable odd-dimensional manifold. It is a combinatorial "torsion-type" invariant which refines the PR-metric, introduced earlier by the first author, and contains an additi…
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.
We describe a diagram containing the zero sets of the moment maps associated to the diagonal U(1) and Sp(1) actions on the quaternionic projective space HP^n. These sets are related both to focal sets of submanifolds and to Sasakian-Einstein structures on induced Hopf bundles. As an application, we construct a complex …
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.
The article computes the isometry groups of generalized Stiefel manifolds.
problem Computing the isometry groups of generalized Stiefel manifolds.
method The approach involves a non-associative algebra associated with the affine structure of the Stiefel manifold, computing the automorphism group and then inducing this to the isometry group.
result Explicit computation of the isometry groups of generalized Stiefel manifolds.
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.
This is a slightly expanded version of the talk given by Ch.O. at the conference "Instantons in complex geometry", at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces". In this paper w…
We prove a Theorem on homotheties between two given tangent sphere bundles SrM of a Riemannian manifold M,g of dim≥3, assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constan…
The paper reformulates clustering as matrix factorization on the Stiefel manifold.
problem Clustering high-dimensional data like images and gene expression.
method Reformulates clustering as low-rank matrix estimation, using Burer-Monteiro factorization on the Stiefel manifold.
result Proves novel prediction bounds for clustering and proposes a componentwise Langevin sampler.
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 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.
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.
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.
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.
Paper computes Stiefel-Whitney classes on real Bott manifolds.
problem Computing Stiefel-Whitney classes on real Bott manifolds.
method Analyzes real Bott manifolds with holonomy group Z2k and diagonal type. result Extends results to compute even and odd Stiefel-Whitney classes.
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.
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.
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.
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.