New findings on Kähler manifolds restrict orthogonal coordinates existence.
problem Existence of orthogonal coordinates on Kähler manifolds.
method Algebraic and geometric techniques applied to Kähler manifolds.
result No nontrivial self-dual Kähler 4-manifolds or Ricci-flat Kähler 4-manifolds support orthogonal coordinates.
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. Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.
problem Determine diameter bounds for Kähler and quaternionic Kähler manifolds under curvature positivity.
method Define orthogonal Bakry-Émery tensor, study diameter theorems, and derive Bonnet-Myers type bounds.
result Sharper diameter bounds than in Riemannian case under specific curvature assumptions.
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
The article proves Kähler 4-manifolds can only be products of 2 surfaces.
problem Finding orthogonal coordinates on Kähler manifolds.
method Using geometric and algebraic methods to construct orthogonal coordinates.
result Only Kähler 4-manifolds can be products of 2 surfaces.
The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.
problem Existence and multiplicity of orthogonal Finsler geodesic chords in a disk-like manifold.
method Study of Finsler geodesic chords under reversibility assumption.
result At least N orthogonal Finsler geodesic chords found in a disk-like manifold.
A parametric manifold can be viewed as the manifold of orbits of a (regular) foliation of a manifold by means of a family of curves. If the foliation is hypersurface orthogonal, the parametric manifold is equivalent to the 1-parameter family of hypersurfaces orthogonal to the curves, each of which inherits a metric and…
New complex orthogonal structures found on a 3D manifold.
problem Finding new geometric structures on a 3D manifold.
method Constructing uniformizable complex orthogonal and projective structures.
result A manifold has multiple uniformizable complex orthogonal structures.
New metrics improve landing algorithms for orthogonality constraints.
problem Optimizing landing algorithms with orthogonality constraints.
method Proposed a family of metrics over full-rank matrices to enhance landing algorithms.
result Natural extension of β-metric improves landing performance.
New algorithms reduce orthogonality constraint enforcement time in machine learning.
problem Efficiently solving orthogonality constraints in machine learning.
method Extending the landing algorithm to Stiefel manifold, incorporating stochastic and variance reduction techniques.
result All proposed methods achieve the same convergence rate as Riemannian counterparts enforcing constraints.
Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.
problem Computing Laplace-Beltrami on constrained submanifolds.
method Embedded gradient vector field method, explicit formula derivation.
result Explicit formula for Laplace-Beltrami on orthogonal group.
New characterization of Osserman tensors using Jacobi-orthogonality.
problem Characterizing Osserman tensors.
method Introducing Jacobi-orthogonality as a new potential characterization.
result Jacobi-orthogonal tensors are Osserman, and all known Osserman tensors are Jacobi-orthogonal.
In every dimension n≥3 we introduce a class of orthogonal graph-manifolds and prove that the fundamental group of any orthogonal graph-manifold quasi-isometrically embeds into a product of n trees. As a consequence, we obtain that asymptotic and linearly-controlled asymptotic dimensions of such group are equal t…
Introduces new curvature concept for Kähler manifolds.
problem Optimizing curvature constraints for projective Kähler manifolds.
method Introduces weighted orthogonal Ricci curvature and proves vanishing theorems.
result Proves optimal curvature constraints for projective Kähler manifolds.
Complex and quaternionic projective spaces lack local orthogonal coordinates.
problem Lack of local orthogonal coordinates in complex and quaternionic projective spaces.
method Analysis of Riemannian manifolds and canonical metrics.
result Complex and quaternionic projective spaces do not have local systems of orthogonal coordinates.
We construct a decomposition of the identity operator on a Riemannian manifold M as a sum of smooth orthogonal projections subordinate to an open cover of M. This extends a decomposition of the real line by smooth orthogonal projection due to Coifman, Meyer and Auscher, Weiss, Wickerhauser, and a similar decomposit…
In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on Kähler manifolds of nonnegative orthogonal bisectional curvature.
A new quasi-Newton method tackles NMF with transform learning on orthogonal manifolds.
problem Efficiently learning transforms for NMF in non-convex optimization on orthogonal manifolds.
method Derives a quasi-Newton method on the orthogonal matrix manifold using sparse approximations of the Hessian.
result Outperforms state-of-the-art methods by orders of magnitude in experiments on synthetic and real audio data.
The study explores globally defined eigenfamilies on closed manifolds, providing existence and orthogonality results.
problem Existence and properties of globally defined eigenfamilies on closed Riemannian manifolds.
method Analyzes topological properties, provides non-/existence results, and uses combinatorial identities.
result Highly rigid orthogonality relations for eigenfunction powers in L2(M). The paper explores geometric decompositions for Ricci tensors and their applications.
problem Understanding Ricci tensors on compact Riemannian manifolds.
method Utilizes Berger-Ebin and York L2-orthogonal decompositions. result New insights into Ricci almost solitons and harmonic maps.
New Einstein metrics found on orthogonal groups without natural reductivity.
problem Finding non-naturally reductive Einstein metrics on orthogonal groups.
method Using real flag manifolds and symmetry assumptions on left-invariant metrics.
result Obtained new invariant Einstein metrics on $\SO(n)$.
In this short note, using Siu-Yau's method [14], we give a new proof that any n-dimensional compact Kahler manifold with positive orthogonal bisectional curvature must be biholomorphic to Pn.
A new algorithm avoids retractions to optimize orthogonal matrices efficiently.
problem Optimizing functions over the manifold of orthogonal matrices efficiently.
method Landing algorithm that avoids retractions using potential energy.
result The landing algorithm is faster and less prone to numerical errors than retraction-based methods.
The paper explores conditions for the existence of orthogonal almost complex structures on manifolds.
problem Conditions for the existence of orthogonal almost complex structures on manifolds.
method Analyzes the Nijenhuis tensor and its squared norm to determine the existence of orthogonal almost complex structures.
result There exists no orthogonal almost complex structure on the standard sphere \(S^6\) with \(|N|^2 < \frac{64}{5}\) everywhere.
SOFARI improves inference on multi-task learning latent factors.
problem Challenges in precise inference on multi-task learning latent factor matrices.
method High-dimensional manifold-based Neyman near-orthogonality inference on Stiefel manifold structure.
result Easy-to-use bias-corrected estimators for latent factor vectors and singular values with asymptotic normal distributions.
We introduce a topological invariant, it a type of a graph-manifold, which takes natural values. For a 4-dimensional graph-manifold, whose type does not exceed two, it is proved that its universal cover is bi-Lipschitz equivalent to a universal cover of an orthogonal graph-manifold (for any Riemannian metrics on graph-…
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.
We develop matrix models for Grassmann, flag, and Stiefel manifolds.
problem Creating efficient models for Grassmann, flag, and Stiefel manifolds.
method Orthogonally-equivariant matrix submanifold models derived for each manifold.
result Exhaustive list of orthogonally-equivariant submanifold models for the lowest dimensions.
In this paper, we study any Kähler manifold where the positive orthogonal bisectional curvature is preserved on the Kähler Ricci flow. Naturally, we always assume that the first Chern class C1 is positive. In particular, we prove that any irreducible Kähler manifold with such property must be biholomorphic to $\math…
In this short note we show the following result: Let (M2n+1,g) (n≥2) be a compact Sasaki manifold with positive transverse orthogonal bisectional curvature. Then π1(M) is finite, and the universal cover of (M2n+1,g) is isomorphic to a weighted Sasaki sphere. We also get some results in the case of n…
A new optimizer preserves orthogonality constraints on matrices efficiently.
problem Optimization on Stiefel manifold with orthogonality constraints.
method Interplay between continuous and discrete dynamics leading to a gradient-based optimizer with momentum.
result The method optimizes matrices on Stiefel manifold efficiently and accurately.
In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author. We also prove a Liouville theorem for plurisubharmonic functions on such a manifolds, which generalizes a previous result …
Lower bound found for Kähler manifold eigenvalues.
problem Finding bounds for eigenvalues on Kähler manifolds.
method Comparison results of Li and Wang applied to Kähler manifolds.
result Explicit lower bound of the first eigenvalue determined.
Study of zonal spherical functions on partial flag manifolds using Jacobi polynomials.
problem Understanding zonal spherical functions on partial flag manifolds.
method Matrix variate version of Koornwinder's method for constructing orthogonal polynomials, using Hermitian Jacobi polynomials and multivariate Schur polynomials.
result Conjecture that Hermitian Jacobi polynomials are elementary zonal spherical functions.
We investigate submanifolds in space forms such that every geodesic orthogonal to the submanifold intersects a fixed totally geodesic submanifold. We obtain an application to horospheres in Hadamard manifolds.
We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provi…
Modified Laplacian connects to Yang-Mills instantons on manifolds.
problem Understanding instantons on 4D manifolds.
method Infinite dimensional Lévy Laplacian defined on manifolds, parameterized by curves in orthogonal rotations.
result Instantons on 4D manifolds are related to the modified Lévy Laplacian under specific curve conditions.
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 on Kähler Finsler manifolds with curvature bounds, proving theorems.
problem Understanding Kähler Finsler manifolds with curvature constraints.
method Analyzing partial parallelism of complex structure, proving theorems.
result Generalized comparison theorem for positively curved Kähler Finsler manifolds.
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.
The paper finds an upper limit for the length of geodesic chords on Riemannian manifolds.
problem Finding the maximum length of geodesic chords on Riemannian manifolds.
method Establishing an upper bound for geodesic chord length using geometric bounds on the manifold.
result An upper bound for the length of geodesic chords is derived, with a specific example for 2-dimensional spheres.
We consider four-dimensional Riemannian manifolds with commuting higher order Jacobi operators defined on two-dimensional orthogonal subspaces (polygons) and on their orthogonal subspaces. More precisely, we discuss higher order Jacobi operator J(X) and its commuting associated operator $\mathcal{J}(X^{\per…
Geodesic flows with diagonalisable integrals are orthogonal.
problem Understanding geodesic flows with specific integrals.
method Analyzing quadratic integrals for geodesic flows.
result Diagonalisable integrals imply orthogonal separation of variables.
Efficiently optimizes orthogonal and Stiefel matrices on parallel units.
problem Optimization over orthogonal groups on parallel units.
method CWY and T-CWY transforms for parametrization and optimization.
result CWY and T-CWY methods lead to convergence on parallel units.
Improved estimation of multiple principal components using manifold optimization and iterative deflation techniques.
problem Estimating multiple principal components efficiently and orthogonally.
method Extended SFPCA using manifold optimization and iterative deflation techniques.
result Alternative deflation schemes improve signal extraction and component estimation.
The paper studies geometric properties of Grassman manifolds within Euclidean spaces.
problem Understanding the geometric structure of Grassman manifolds.
method Analyzing Grassman manifold G(E) as a subset of Euclidean space E and orthogonal projections. result Explicit formulas for differential geometry of G(E) as a submanifold. LEGO estimates tangent spaces more robustly than LPCA in noisy data.
problem Estimating tangent spaces in high-noise settings.
method Spectral method using graph Laplacian eigenvectors and gradient orthogonization.
result LEGO yields more robust tangent space estimates than LPCA.
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.