The paper reduces normal curvature and enhances homology recovery via embedded submanifolds.
problem Recovering the homology of submanifolds with narrow cycles.
method Embedding submanifolds into scaled oriented Grassmannian bundles to reduce normal curvature and stabilize Čech persistent homology.
result The Čech persistent homology is stable with respect to the interleaving distance and provides lower bounds on scales for homology recovery.
We study the global geometry of surfaces in Sasakian space forms whose mean curvature vector is parallel in the normal bundle (these include the Riemannian Heisenberg space of dimension 2n+1). We prove a codimension reduction theorem. We introduce two holomorphic quadratic differentials on anti-invariant such surface…
Closed Riemannian manifolds with positive mixed sectional curvature
problem Constructing closed Riemannian manifolds with positive mixed sectional curvature
method Explicit construction using totally geodesic foliations
result Positive mixed sectional curvature alone does not imply the Ferus--Adams estimate on closed manifolds
Theory of symplectic reduction in infinite dimensions developed.
problem Challenges in symplectic reduction in infinite dimensions.
method Normal form of momentum map for infinite-dimensional equivariant maps.
result Theory of singular symplectic reduction in infinite dimensions.
CurvSSL improves SSL by aligning local manifold curvature.
problem Improving self-supervised learning by capturing local manifold geometry.
method CurvSSL augments Barlow Twins with a curvature-based regularizer to align and decorrelate embeddings across augmentations.
result Curvature-regularized SSL yields competitive or improved linear evaluation performance.
New method computes affine normal directions efficiently for sparse polynomials.
problem Computing affine normal directions is computationally expensive in high dimensions.
method Reduces third-order tensor contraction to matrix-free formulation using log-determinant gradient.
result Scalable implementations with near-linear scaling in dimension and sparsity.
Normal distributions ensure asymptotic variance reduction in moment matching Monte Carlo.
problem Asymptotic variance reduction in general integration problems.
method Characterization of conditions for asymptotic variance reduction using normal distributions.
result Asymptotic variance reduction is guaranteed for normal distributions in moment matching Monte Carlo.
Study characterizes naturally reductive metrics on homogeneous manifolds.
problem Characterizing naturally reductive (α1,α2) metrics on homogeneous manifolds. method Characterization through local f-products and equivalence of properties. result Explicit flag curvature formula for naturally reductive metrics.
Study integrability of geodesic flow on specific Lie groups.
problem Integrability of geodesic flow on metabelian nilpotent groups.
method Symplectic reduction procedure applied to sub-Riemannian geodesic flow on metabelian nilpotent groups.
result Showed integrability of normal Hamiltonian flow in Engel-type groups.
Formula derived for Laplace-Beltrami spectrum on homogeneous spaces.
problem Calculating the spectrum of the Laplace-Beltrami operator on homogeneous spaces.
method Formula derivation based on eigenvalues of a generalized Casimir operator and spherical representations.
result First detailed computation and investigation of the spectrum for a family of metrics on the Aloff-Wallach manifold.
It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal ho…
Minimal normal curvature immersions in the unit ball studied.
problem Minimal normal curvature immersions in the unit ball.
method Gromov's problem, differentiable sphere theorem, existence result.
result Determined the minimal possible value of the normal curvature of SnimesS1. Optimization geometrodynamics simplifies adaptive optimizer dynamics.
problem Hidden states in adaptive optimizers complicate gradient-based learning.
method Develops a variational theory to eliminate hidden states and compose across hierarchies.
result Yields interaction curvature that integrates to finite contrasts.
We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kaehler manifold of a reductive group is of Vaisman type, if the normalizer of the isotropy group is compact. We also show that such a result does not hold in t…
New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
PSMM method optimizes matrix sufficient dimension reduction.
problem Feature matrices with row- and column-wise interpretations require efficient dimension reduction.
method PSMM method converts matrix problem into classification problems using rank-1 normal matrix.
result PSMM outperforms existing methods and provides strong interpretability.
Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) …
We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension n=2, we prove an optimal dimension reduction theorem for complete translating Kähler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reducti…
We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere Sn, under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property K+KN=1 for the Gauss curvature K and the normal curvature KN if the Gauss curvature i…
Investigates solving curvature equations on special Lie groups.
problem Solving curvature equations on non-compact simple Lie groups.
method Analyzes left-invariant naturally reductive metrics and conditions for solvability.
result Obtains conditions for the solvability of curvature equations.
A local normal form theorem for smooth equivariant maps between Fréchet manifolds is established. Moreover, an elliptic version of this theorem is obtained. The proof these normal form results is inspired by the Lyapunov-Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces, and uses a s…
This note analyzes the normal form of gradient Ricci 4-solitons.
problem Understanding the curvature operator of gradient Ricci 4-solitons.
method Analyzing the normal form of the operator R^+21H^ and curvature operator R^ of Koiso-Cao soliton. result The curvature operator of the Koiso-Cao soliton inherits a normal form relative to the space of algebraic Kähler curvature operators.
In this paper the geometry of normal metric contact pair manifolds is studied under the flatness of conformal, concircular and quasi-conformal curvature tensors. It is proved that a conformal flat normal metric contact pair manifold is an Einstein manifold with a negative scalar curvature and has positive sectional cur…
Classifies special submanifolds with specific curvature properties.
problem Classifying submanifolds with constant Moebius curvature and flat normal bundle.
method Analyzes isometric immersions with constant Moebius curvature and flat normal bundle.
result Classifies submanifolds with these curvature properties.
Formula for sectional curvatures on matrix groups.
problem Calculating curvatures on matrix groups.
method Simple formula derivation for sectional curvatures.
result Valid formula for general linear and reductive Lie groups.
Study flag curvature in homogeneous Finsler spaces with a specific metric.
problem Analyzing flag curvature in homogeneous Finsler spaces with a generalized m-Kropina metric. method Provided explicit formula for flag curvature, showed equivalence of definitions, and studied curvature of naturally reductive spaces.
result Equivalence of two definitions of naturally reductive homogeneous Finsler spaces for the generalized m-Kropina metric. We present a new and shorter proof of Stocking's result that any strongly irreducible Heegaard surface of a closed orientable triangulated 3-manifold is isotopic to an almost normal surface. We also re-prove a result of Jaco and Rubinstein on normal spheres. Both proofs are based on the "reduction" technique introduced…
Veronese minimizes normal curvatures to sphere.
problem Bounding normal curvatures of submanifolds.
method Veronese embeddings of projective planes.
result Optimal bound on normal curvatures guarantees sphere.
A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous sp…
This paper analyses the parabolic geometries generated by a free n-distribution in the tangent space of a manifold. It shows that certain holonomy reductions of the associated normal Tractor connections, imply preferred connections with special properties, along with Riemannian or sub-Riemannian structures on the man…
The study proves a rigidity theorem for compact manifolds with boundary.
problem Rigidity of compact manifolds with boundary in low dimensions.
method Dimension reduction argument for mean curvature, extending Schoen-Yau's for scalar curvature.
result Sharp spherical radius rigidity and best NNSC fill-in in terms of mean curvature.
We prove that codimension two surfaces satisfying a nonlinear curvature condition depending on normal curvature are smoothly deformed by mean curvature flow to round points.
The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale …
Normalization layers control deep neural network capacity, improving stability and generalization.
problem Excessive capacity in deep neural networks leads to overfitting and poor generalization.
method Developed a theoretical framework to explain normalization's role in capacity control.
result Normalization layers reduce the Lipschitz constant exponentially, smoothing the loss landscape and enhancing generalization.
We introduce in this paper normal twistor equations for differential forms and study their solutions, the so-called normal conformal Killing forms. The twistor equations arise naturally from the canonical normal Cartan connection of conformal geometry. Reductions of its holonomy are related to solutions of the normal t…
Lower bounds on average normal curvature for submanifolds in Riemannian domains.
problem Finding bounds on the average normal curvature of submanifolds in Riemannian domains.
method Using an invariant measuring optimal n-trace convexity under unit-gradient normalization. result Lower bounds for the average normal curvature expressed in terms of an invariant.
Proves a special type of submanifolds in a curved space.
problem Characterizing submanifolds with specific properties in a curved space.
method Uses the properties of flat normal bundle and parallel mean curvature to prove the submanifolds are warped products.
result Einstein submanifolds with flat normal bundle and parallel mean curvature are warped product of isometric immersions.
Study on generalized quasi-Einstein structures in contact geometry.
problem Characterizing and understanding generalized quasi-Einstein structures in contact geometry.
method Investigation of properties, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifolds.
result Normal metric contact pair manifolds with generalized quasi-constant curvature are generalized quasi-Einstein manifolds.
In this paper, we study normal complex contact metric manifolds and we get some general results on them. Moreover, we obtained the general expression of the curvature tensor field for arbitrary vector fields. Furthermore, we show that the necessary and succient conditions to be normal a complex contact metric manifold.…
This work examines consistency issues in Gaussian Mixture Model reduction algorithms.
problem Consistency issues in Gaussian Mixture Model reduction algorithms.
method Discussion of the importance of dissimilarity measure choice and consistency of GMR algorithms.
result Most existing GMR algorithms are not consistent with a unique measure, leading to suboptimal reduced GMs.
Develops an oblique projection technique to approximate a foliation for non-normal dynamics.
problem Modeling dynamics far from a primary Spectral Submanifold (SSM) in non-normal systems.
method Oblique projection technique based on experimental data.
result Approximates a stable invariant foliation for non-normal dynamics efficiently.
RC flow learns molecular kinetics in low dimensions.
problem Discovering interpretable low-dimensional models of molecular kinetics.
method Normalizing flow for coordinate transformation and Brownian dynamics for kinetics approximation.
result Tractable and trainable model of reduced kinetics in continuous time and space.
In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.
In this paper we form relations for the determination of the elements of the Eötvös matrix of the Earth's normal gravity field. In addition a relation between the Gauss curvature of the normal equipotential surface and the Gauss curvature of the actual equipotential surface both passing through the point P is presented…
Study on immersions with flat normal bundle in curved spaces.
problem Behavior of isometric immersions with negative curvature.
method Investigation of second fundamental form growth in space forms.
result Second fundamental form grows exponentially if normal bundle is flat.
We discuss Levi-Civita connections on Courant algebroids. We define an appropriate generalization of the curvature tensor and compute the corresponding scalar curvatures in the exact and heterotic case, leading to generalized (bosonic) Einstein-Hilbert type of actions known from supergravity. In particular, we carefull…
Introduces new algebraic structures for relational groupoids and proves a reduction theorem.
problem Developing algebraic tools for relational groupoids.
method Introduces relational groupoids and convolution algebras, provides examples, and proves a reduction theorem.
result Establishes a reduction theorem recovering the usual convolution of Lie groupoids.
Muon dynamics study uses spectral Wasserstein flow for optimization stability.
problem Optimizing deep learning models with gradient normalization.
method Introduces Spectral Wasserstein distances for matrix flows, proving equivalence with Benamou--Brenier formulation.
result Gradient-flow interpretation of mean-field normalized training dynamics.