Study of harmonic Riemannian submersions from 3D geometries.
problem Characterizing harmonic Riemannian submersions from specific 3D geometries.
method Using generalized integrability data and classifications of Thurston's 3D geometries, 3D BCV spaces, and Berger sphere.
result Complete classifications and explicit constructions of harmonic Riemannian submersions.
Review of metallic Riemannian geometry advances.
problem No specific problem stated; focuses on advancements.
method No specific method mentioned; focuses on review of advances.
result Rich potential and diverse applications of metallic Riemannian geometry.
Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.
problem Unified framework for Riemannian and sub-Riemannian geometries.
method Study of gauge metric measure spaces.
result Unified synthetic Ricci curvature lower bounds for both Riemannian and sub-Riemannian structures.
Course notes on Lie groups and Riemannian geometry, focusing on applications and low-dimensional examples.
problem Exploring Lie groups and their representations in Riemannian geometry.
method Review of well-known topics and recent advances in Riemannian geometry with symmetries.
result First construction of exceptional holonomy metrics.
Introduces systolic inequalities in Riemannian and symplectic geometry.
problem Exploring systolic inequalities in different geometric settings.
method Comparing classical Riemannian metrics to recent symplectic measurements.
result Illustrates connections between Riemannian and symplectic geometry.
Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.
problem Understanding the geometry induced by slice Riemannian metric.
method Developed Lie theoretic study, computed isometry group, compared with quaternionic Poincaré geometry.
result Isometry group of slice Riemannian metric is built from symmetries of Sp(1,1) group.
Quantum field theory connects Riemannian geometry to quantum fluctuations.
problem Generating Riemannian structures from quantum fluctuations.
method QFT approach to Riemannian Geometry, focusing on Ricci curvature.
result Ricci curvature is crucial in generating Riemannian structures.
Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description …
Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalisation of the la…
New Riemannian geometry for Compound Gaussian distributions applied to efficient change detection.
problem Change detection in multivariate image times series.
method Developed a recursive approach based on Riemannian optimization.
result Optimal performance achieved with computational efficiency.
Proposes a scalable framework for extracting data manifold geometry.
problem Efficiently mapping and learning data manifold geometry.
method Score-based pullback Riemannian geometry integrating pullback Riemannian geometry and generative models.
result High-quality geodesics and reliable intrinsic dimension estimation.
The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable li…
Mathematical analysis of Prytz planimeter using sub-Riemannian geometry.
problem Historical use of Prytz planimeter to approximate areas.
method Sub-Riemannian geometry and connections/horizontal lifts.
result Mathematical description and analysis of Prytz planimeter.
Reconstructs Riemannian geometry from diffusion properties.
problem Recovering Riemannian geometry from diffusion data.
method Intrinsic reconstruction from diffusion semigroup and calculus.
result Reveals Riemannian structure from diffusion properties.
The paper examines curvature properties of twistor spaces.
problem None explicitly stated in the abstract.
method Review of Riemannian and almost Hermitian geometry of twistor spaces.
result Curvature properties of twistor spaces are explored.
Sub-Riemannian geometry connects bike paths to mathematical curves.
problem Understanding bike paths and their mathematical properties.
method Relating sub-Riemannian geometry to bicycle motion and curve shapes.
result Geodesics in sub-Riemannian geometry correspond to specific bike paths.
Revisits geometric PDE uniqueness in Riemannian and CR geometry.
problem Uniqueness of solutions to geometric PDEs in Riemannian and CR geometry.
method New proofs and reconstruction of Jerison-Lee identity.
result Stronger uniqueness result in CR geometry.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
The study extends inscription problems to non-Euclidean geometries.
problem Generalizing inscription problems to non-Euclidean geometries.
method Symplectic and Riemannian geometry techniques.
result Proved generalized inscription theorems for hyperbolic and spherical surfaces.
We consider two Riemannian geometries for the manifold M(p,m×n) of all m×n matrices of rank p. The geometries are induced on M(p,m×n) by viewing it as the base manifold of the submersion π:(M,N)↦MNT, selecting an adequate Riemannian metric on the total space, and …
We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids.
We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.
Survey on four-dimensional Thurston geometries with Riemannian metrics.
problem Understanding homogeneous manifolds and their Riemannian metrics.
method Description and analysis of specific Thurston geometries in four dimensions.
result Exhibition of all Riemannian metrics invariant under group actions.
Mass in relativity linked to polyhedra geometry.
problem Mass in general relativity.
method Riemannian polyhedra geometry.
result Mass connected to polyhedra geometry.
Extends Riemannian geometry inequalities with sharper estimates.
problem Deriving new inequalities on Riemannian manifolds.
method Investigates advanced Hardy and Rellich-type inequalities on complete noncompact manifolds with weight functions.
result Provides sharper estimates conforming to the geometry and structure of the manifold.
Lecture notes for the minicourse "Holonomy Groups in Riemannian geometry", a part of the XVII Brazilian School of Geometry, to be held at UFAM (Amazonas, Brazil), in July of 2012.
We discuss some basic concepts of semi-Riemannian geometry in low-regularity situations. In particular, we compare the settings of (linear) distributional geometry in the sense of L. Schwartz and nonlinear distributional geometry in the sense of J.F. Colombeau.
A new Lagrangian formulation of the Raychaudhuri equation in non-Riemannian geometry.
problem Formulating the Raychaudhuri equation in non-Riemannian geometries.
method Established a formal connection between the expansion scalar and the cross-sectional volume of the congruence. Derived a Lagrangian and Hamiltonian formulation.
result The expansion scalar equals the fractional rate of change of volume, weighted by a scalar factor.
Develops Riemannian geometry for optimization on manifolds with detailed derivations.
problem Abstract high-level optimization on nonlinear spaces like matrix manifolds.
method Systematic derivation of geometric structures and constructions in coordinates and matrix form.
result Unified treatment of Riemannian geometry for optimization on manifolds.
Researchers solve a Riemannian geometry problem using warped products.
problem Solving a Moser-Bernstein problem in warped Riemannian manifolds.
method Study entire solutions to the minimal hypersurface equation in warped products.
result Solves the Moser-Bernstein problem in a broader class of Riemannian manifolds.
The paper proves complex geometry results for manifolds of the form X × R², answering a 1994 conjecture.
problem Proving complex geometry results for manifolds of the form X × R².
method Using Riemannian and complex geometry techniques, the authors show the existence of metrics with positive scalar curvature.
result The paper answers a 1994 Rosenberg-Stolz conjecture for X × R², extending results to noncompact manifolds.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
We construct natural Riemannian metrics on Seiberg-Witten moduli spaces and study their geometry.
Lecture notes on geodesics in differential geometry.
problem Understanding geodesics in differential geometry.
method Expository lecture notes with exercises.
result Explains the geometry of geodesics.
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwe…
After defining generalizations of the notions of covariant derivatives and geodesics from Riemannian geometry for reductive Cartan geometries in general, various results for reductive Cartan geometries analogous to important elementary results from Riemannian geometry are proven using these generalizations. In particul…
This review explores Ricci soliton inequalities in Riemannian geometry.
problem Understanding geometric and analytic characteristics of Riemannian manifolds.
method Comprehensive study of Ricci soliton inequalities, summarizing historical evolution and current developments.
result Complex interactions between curvature conditions and geometric inequalities.
We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)Kähler geometry.
We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.
A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extend…
Riemannian metric matching learns the geometry of high-dimensional datasets using neural networks.
problem Estimating the geometry of high-dimensional datasets from samples
method Riemannian metric matching using neural networks
result Riemannian metric matching rivals or improves k-NN-based diffusion geometry estimators Study on the geometry of spacelike hypersurfaces in spacetime.
problem Understanding the geometry of compact spacelike Cauchy hypersurfaces.
method Analysis of a weak Riemannian metric on the manifold of hypersurfaces.
result Positive geodesic distance and non-positive sectional curvature.
Embedding theorem for tractor bundles applied to conformal geometry.
problem Embedding theorem for tractor bundles in Cartan geometries.
method Extension of Gromov-Zimmer embedding theorem to tractor bundles.
result Rigidity result for conformal actions of special pseudo-unitary groups.
The Fisher-Rao geometry is applied to elliptical distributions for optimization and classification.
problem Optimizing and classifying covariance matrices using geometric tools.
method Riemannian optimization and intrinsic Cramér-Rao bounds.
result Geometric tools enhance covariance matrix estimation and classification.
Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian met…
We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.
Riemannian geometry improves protein dynamics analysis.
problem Efficient analysis of protein dynamics data in non-linear spaces.
method Developed a local approximation technique for geodesics and a smooth manifold of protein conformations.
result Geodesics approximate molecular dynamics trajectories and provide realistic summary statistics.
Develops Riemannian geometry for noncommutative super surfaces.
problem No specific problem stated; focuses on mathematical development.
method Introduces metric and connections on noncommutative super surfaces, showing compatibility and zero torsion under certain conditions.
result Noncommutative super surfaces have a well-defined Riemannian geometry with properties analogous to classical Riemannian geometry.