The symmetry-rank of a riemannian manifold is by definition the rank of its isometry group. We determine precisely which smooth closed manifolds admit a positively curved metric with maximal symmetry-rank.
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Proves a higher rank rigidity theorem for convex real projective manifolds.
problem No specific problem stated; focuses on proving a theorem.
method Analogue of Ballmann and Burns-Spatzier's higher rank rigidity theorem.
result Proves a higher rank rigidity theorem for convex real projective manifolds.
New compact ECS manifolds with rank 2 discovered, differing from previous rank 1 examples.
problem Finding new compact ECS manifolds with rank 2.
method Constructing new examples of compact pseudo-Riemannian manifolds with parallel Weyl tensor, rank 1 or 2.
result New compact ECS manifolds of rank 2, locally homogeneous, and geodesically incomplete.
We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.
Researchers describe the metric structure of compact ECS manifolds.
problem Understanding the metric structure of compact rank-one ECS manifolds.
method Analyzing pseudo-Riemannian manifolds with nonzero parallel Weyl tensor.
result Compact rank-one ECS manifolds are either translational or noncompact.
Sparse reduced-rank regression selects variables and ranks via manifold optimization.
problem Traditional rank selection fails when true rank is high.
method Sparse regularization and manifold optimization for rank and variable selection.
result Accurate estimation of coefficient parameter with high true rank.
The study establishes uncertainty principles on harmonic manifolds of rank one.
problem Developing uncertainty principles for harmonic manifolds of rank one.
method Derivation of various uncertainty principles including Heisenberg, Morgen, Schrödinger, and Hömanders principles.
result Generalization of Hausdorff-Young inequality to harmonic manifolds of rank one.
Study on rank-one ECS manifolds, focusing on dilational type.
problem Characterizing ECS manifolds with specific properties.
method Analyzing properties of pseudo-Riemannian manifolds with parallel Weyl tensor.
result Generic compact rank-one ECS manifolds are either translational or locally homogeneous.
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.
Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.
problem Spurious critical points on the boundary of low-rank matrix manifold.
method Riemannian gradient descent with dynamical low-rank approximation and rescaled gradient flow.
result Riemannian gradient descent escapes some spurious critical points on the boundary of the manifold.
We construct a counterexample to the Rank versus Genus Conjecture, i.e. a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus. Moreover, we show that the discrepancy between rank and Heegaard genus can be arbitrarily large for hyperbolic 3-manifolds. We also constr…
New insights into compact rank-one ECS manifolds, proving they are bundles over circles.
problem Understanding the structure of compact rank-one ECS manifolds.
method Analyzing the properties of pseudo-Riemannian manifolds with parallel Weyl tensor.
result Compact rank-one ECS manifolds are bundles over the circle with specific leaf structures.
Symmetry rank bound for manifolds with positive intermediate Ricci curvature.
problem Bounding the symmetry rank of manifolds with positive intermediate Ricci curvature.
method Local argument and effective isometric action on manifolds.
result Symmetry rank bound for positive intermediate Ricci curvature.
An LCK manifold with potential is a compact quotient of a Kahler manifold equipped with a positive Kahler potential , such that the monodromy group acts on by holomorphic homotheties and multiplies by a character. The LCK rank is the rank of the image of this character, considered as a function from the …
New rigidity results for manifolds with maximal symmetry rank and positive intermediate Ricci curvature.
problem Understanding the structure of manifolds with maximal symmetry rank and positive intermediate Ricci curvature.
method Recovering stronger topological rigidity results using higher intermediate Ricci curvatures and nontrivial fundamental groups.
result Stronger topological rigidity results for manifolds with maximal symmetry rank and positive intermediate Ricci curvature.
We complete the classification of rank two affine manifolds in the moduli space of translation surfaces in genus three. Combined with a recent result of Mirzakhani and Wright, this completes the classification of higher rank affine manifolds in genus three.
Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
problem Understanding the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
method Reduction of computations to solving PDEs, explicit solutions on specific manifolds, analysis of curve of almost complex structures, classification of Lie algebras.
result Classification of Lie algebras admitting almost complex structures with specific Nijenhuis tensor ranks.
Closed ray affine manifoldsmath.DG
Study closed manifolds with rank one ray structures, proving completeness or covering properties.
problem Characterize closed manifolds with specific affine structures.
method Analyze the developing map and automorphism group properties.
result Closed manifolds with rank one ray structures are either complete or cover the complement of an affine subspace.
Study finite group actions on 4-manifolds, finding rank bounds.
problem Understanding finite group actions on 4-manifolds.
method Investigate Borel spectral sequence for G-equivariant cohomology.
result Establish new bounds on the rank of G for homologically trivial actions.
Study manifolds with positive intermediate Ricci curvature and large symmetry rank.
problem Closed, simply connected manifolds with positive 2nd-intermediate Ricci curvature and large symmetry rank.
method New tools for studying isometric actions on closed manifolds with positive kth-intermediate Ricci curvature, including isotropy rank lemma, symmetry rank bound, and connectedness principle.
result Even dimensional manifolds with large symmetry rank must have trivial odd degree integral cohomology and are either spheres or complex projective spaces.
New method reduces computational cost for nonnegative low rank matrix approximation.
problem Efficiently compute nonnegative low rank matrix approximation for nonnegative matrices.
method Alternating projections onto tangent spaces of fixed rank matrices manifold and nonnegative matrix manifold.
result Sequence converges linearly to optimal solutions, showing better performance in terms of computational time and accuracy.
Derives smooth homogeneous structures for low-rank tensors.
problem Understanding the geometry of low-rank tensors.
method Analyzes sets of fixed CP, multilinear, and TT rank tensors to derive smooth homogeneous manifolds.
result Derives Riemannian metrics with complete geodesics.
We say that a Riemannian manifold M has rank at least k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns-Spatzier, later generalized by Eberlein-Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank at least 2 (the high…
Stochastic gradient descent on manifolds improves low-rank approximation.
problem Efficiently approximate large matrices with lower rank.
method Stochastic gradient descent on a manifold.
result Algorithm outperforms Euclidean space methods on Netflix Prize data.
Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and po…
The paper encourages Kleinian group thinking for higher rank Lie groups.
problem No specific problem stated; encouraging new thinking.
method Discussion of Kleinian group ideas applied to higher rank Lie groups.
result Encouragement to think about higher rank Lie groups using Kleinian group theory.
Study bounds on lattice actions on compact pseudo-Riemannian manifolds.
problem Bounding real-rank of lattices acting on compact pseudo-Riemannian manifolds.
method Investigates conformal actions of cocompact lattices in higher-rank Lie groups on compact pseudo-Riemannian manifolds.
result Proves a general bound on the real-rank of the lattice and shows manifold conformally flat when real-rank is maximal.
Local invertibility of higher rank tensor fields on curved manifolds proven.
problem Local invertibility of geodesic ray transform on tensor fields of rank four.
method Proved local invertibility up to potential fields on Riemannian manifolds with strictly convex boundary.
result Local invertibility of tensor fields of rank four on curved manifolds proven.
Finite groups with a hyperelliptic involution have a 2-rank of at most 4.
problem Finite groups acting on hyperelliptic 3-manifolds and their sectional 2-rank.
method Analyzing sectional 2-rank of finite groups containing hyperelliptic involutions.
result The sectional 2-rank of such groups is at most 4, with 4 being the best possible upper bound.
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 -maps between quaternionic Stiefel manifolds.
We show that any effective isometric torus action of maximal rank on a compact Riemannian manifold with positive (sectional) curvature and maximal symmetry rank, that is, on a positively curved sphere, lens space, complex or real projective space, is equivariantaly diffeomorphic to a linear action. We show that a compa…
Study on invariant anti-quasi-Sasakian structures on compact manifolds.
problem Existence and classification of invariant anti-quasi-Sasakian structures of maximal rank.
method Analysis of invariant structures on compact homogeneous Riemannian manifolds and nilpotent Lie groups.
result Classification of invariant anti-quasi-Sasakian structures on nilpotent Lie groups.
Proposes tensor Q-rank for better tensor rank recovery in complex data.
problem Improving tensor rank recovery for complex data with low sampling rate.
method Introduces tensor Q-rank and two selection methods for , proposing VMTQN and MOTQN models.
result Demonstrates superior performance in tensor completion problems compared to TNN-based methods.
The study proves stabilizing of ascending chains in specific groups.
problem Stabilization of ascending chains in bounded rank subgroups of 3-manifold groups.
method Reduction to hyperbolic 3-manifolds and use of geometrization.
result Ascending chains in toral relatively hyperbolic groups stabilize.
In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic …
We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group with principal isotropy group and cohomogeneity such that $k - (\rank G - \ran…
We classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.
Equivalent formulations for low-rank matrix optimization are proven.
problem Low-rank matrix optimization with rank constraints.
method Established geometric landscape connections between manifold and factorization formulations.
result Equivalence between manifold and factorization formulations at FOSPs, SOSPs, and strict saddles.
This paper studies knots in sutured manifolds achieving minimum instanton homology rank.
problem Properties of knots achieving minimum instanton homology rank in sutured manifolds.
method Analyzes sutured manifolds and instanton homology to find knots achieving minimum rank.
result Knots achieving minimum instanton homology rank in sutured manifolds are the unknot.
We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any embedded ball in the convex core of M.
Let M be a positive quaternionic Kaehler manifold of dimension 4m. If the isometry group Isom(M) has rank at least m/2 +3, then M is isometric to HP^m or Gr_2(C^{m+2}). The lower bound for the rank is optimal if m is even.
Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on ma…
We give a construction of hyperbolic 3-manifolds with rank two fundamental groups and report an experimental search to find such manifolds. Our manifolds are all surface bundles over the circle with genus two surface fiber. For the manifolds so obtained, we then examine whether they are of Heegaard genus two or not. As…
We classify closed, simply-connected non-negatively curved 5-manifolds admitting an (almost) effective, isometric or action. As a direct consequence, we show that for any manifold, of dimensions up to and including 9 under the same hypotheses, the maximal symmetry rank is equal to and the free rank…
This work learns low-rank hyperbolic embeddings for tasks with hierarchical structures.
problem Learning hyperbolic embeddings of tasks with hierarchical structures.
method Formulated as manifold optimization problems and proposed computationally efficient algorithms.
result Efficacy of the proposed approach demonstrated through empirical results.
Affirmative proof that rank 3 3-manifolds have filling links.
problem Existence of filling links in rank 3 3-manifolds.
method Introduced and used the concept of filling links, proved for rank 3 manifolds.
result Every rank 3 closed orientable 3-manifold contains a filling link.
A Riemannian manifold has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of lie in the interval , and is closed, we show that is a locally symmetric space of rank one. This…
Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.
problem Investigate critical exponents for vanishing L^p-cohomology in higher rank Lie groups and manifolds.
method Examine SL(R) and 5-dimensional solvable Lie groups, use spectral sequence arguments.
result Discover a continuum of quasi-isometry classes of rank 2 solvable Lie groups.