Discussing Riemannian Hilbert manifolds, their properties, and extending a theorem.
problem Properties and extensions of Riemannian Hilbert manifolds.
method Analyzing singularities, completeness, and homogeneity; extending a theorem.
result Extended a theorem due to Nomizu and Ozeki to infinite dimensional Riemannian Hilbert manifolds.
The paper defines and studies isoparametric submanifolds in Riemannian Hilbert manifolds.
problem Defining and studying isoparametric submanifolds in Riemannian Hilbert manifolds.
method Introducing curvature-invariant submanifolds, regularizable submanifolds, and isoparametric submanifolds; proving the constancy of mean curvatures and independence of shape operators and normal Jacobi operators.
result Proving that certain submanifolds are isoparametric under specific conditions.
Extends Einstein-Hilbert functional definition for stable manifolds.
problem Stability of Einstein manifolds on Riemannian manifolds.
method Second variation of generalized Einstein-Hilbert functional.
result Properties of stable Einstein manifolds presented.
We prove the Focal Index Lemma and the Rauch and Berger comparison theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop spaces of a compact manifold with the L^2 metrics
Study RKHS on manifolds, linking Sobolev and diffusion spaces.
problem Characterizing RKHS on manifolds and their properties.
method Analyzing Sobolev and diffusion spaces on Riemannian manifolds.
result Sobolev spaces are RKHS under certain conditions, and diffusion spaces are introduced.
Extends cohomology to incomplete Riemannian manifolds.
problem Cohomology of harmonic forms on incomplete Riemannian manifolds.
method Abstract setting of Hilbert complexes.
result Geometric applications to Thom-Mather spaces.
Vanishing geodesic distances in infinite dimensions can be created.
problem Vanishing geodesic distances in infinite-dimensional spaces.
method Constructing a weak Riemannian metric in a Hilbert manifold.
result Vanishing geodesic distances can be engineered.
We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponentia…
This study describes the Fisher-Rao metric on Gaussian measures in infinite-dimensional spaces.
problem Understanding the Fisher-Rao metric in infinite-dimensional Gaussian settings.
method Explicit description and generalization of finite-dimensional quantities to infinite-dimensional Hilbert spaces.
result The Fisher-Rao metric and related geometric quantities generalize from finite to infinite dimensions.
The paper studies Einstein-Hilbert action on complex manifolds.
problem Deriving equations for the Einstein-Hilbert action on almost k-product manifolds. method Adapted variations of metric, deriving Euler-Lagrange equations.
result Presented a nice form of Einstein equation.
We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that are twisted with Hilbert C*-module bundles. Along the way we give a complete and s…
Infinite-dimensional geometry: completeness and geodesics in Hilbert manifolds.
problem Failure of Hopf--Rinow theorem in Hilbert manifolds.
method Investigates conformal flexibility and completeness properties in infinite-dimensional settings.
result Conformal class of metrics on Hilbert manifolds contains complete representatives.
New method interpolates training data and is consistent for various data distributions.
problem Establishing generalization guarantees for ensemble methods in the interpolating regime.
method Developed manifold-Hilbert kernel for Riemannian manifolds and used it in ensemble classification.
result Consistent ensemble classification method for broad data distributions.
We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.
Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.
The paper calculates variations of Einstein-Hilbert action on CR manifolds.
problem Variation of the Einstein-Hilbert action in pseudohermitian geometry.
method Computed first and second variations on CR manifolds, characterized critical points as pseudo-Einstein structures, and analyzed second variation on standard spheres.
result In three dimensions, the second variation of the Einstein-Hilbert action on CR structures differs from the Riemannian case due to embeddability.
For a closed Riemannian manifold we extend the definition of analytic and Reidemeister torsion associated to an orthogonal representation of fundamental group on a Hilbert module of finite type over a finite von Neumann algebra. If the representation is of determinant class we prove, generalizing the Cheeger-Müller the…
The paper introduces austere and arid submanifolds in Hilbert spaces.
problem Classifying minimal orbits in hyperpolar PF actions on Hilbert spaces.
method Introducing austere and arid submanifolds into PF submanifolds in Hilbert spaces.
result Examples of infinite dimensional austere and arid PF submanifolds in Hilbert spaces.
We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a dis…
This paper solves Hilbert's fourth problem for constant curvature metrics.
problem Classifying metric geometries with shortest straight lines in constant curvature settings.
method Analyzing Finsler manifolds with constant flag curvature, deriving distance formulas, and proving global geometry theorems.
result Complete characterization of global geometry for constant flag curvature metrics.
Study infinite circle patterns in the Weil-Petersson class using discrete harmonic functions.
problem Characterize infinite circle patterns in the Weil-Petersson class.
method Investigate circle patterns parameterized by discrete harmonic functions of finite Dirichlet energy, equipped with a Riemannian metric.
result Induced quasiconformal homeomorphisms from the unit disk to itself belong to the Weil-Petersson class.
The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
The study finds geometric obstructions for Einstein-Hilbert-Palatini theories.
problem Geometric obstructions for Einstein-Hilbert-Palatini theories.
method Generalization of Einstein-Hilbert-Palatini functional over n-manifolds, analysis of algebraic conditions for non-null functionals.
result Geometric obstructions for Einstein manifolds in various geometries.
New geometric variant of factorization homology for conformally flat manifolds.
problem Defining invariants of conformally flat manifolds.
method Introducing a metric-dependent geometric variant of factorization homology.
result Left Kan extensions of conformally flat d-disk algebras define invariants of conformally flat manifolds. This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the L2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the thesis is a description of the completion manifold of metrics with respect to the $L^…
New construction of isoparametric submanifolds in Hilbert spaces.
problem Constructing isoparametric submanifolds in Hilbert spaces.
method Analyzing holonomy maps and connections in Hilbert spaces.
result Each component of the inverse image of an equifocal submanifold by the holonomy map is an isoparametric submanifold.
The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…
The paper finds many infinite-dimensional weakly reflective PF submanifolds in Hilbert spaces.
problem Minimal submanifolds in Hilbert spaces with reflective properties.
method Introduced weakly reflective PF submanifolds into Hilbert spaces and showed their existence.
result Existence of infinite-dimensional weakly reflective PF submanifolds in Hilbert spaces.
Develops Hilbert geometries and characterizes their isometries.
problem Characterizing isometries in Hilbert geometries.
method Defining rank one isometries and using geometric group theory.
result Discrete subgroups containing rank one isometries are either virtually cyclic or acylindrically hyperbolic.
The paper proves compactness for Dirac-Einstein spin manifolds.
problem Compactness of Dirac-Einstein spin manifolds under specific conditions.
method Study of the Hilbert-Einstein-Dirac functional, proving compactness for critical points.
result Compactness result for Dirac-Einstein spin manifolds in dimensions three and four.
Research shows that certain metric spaces cannot contain rigid structures and provides evidence for loose embeddings into Euclidean spaces.
problem The inability of certain metric spaces to contain rigid structures like regular simplices or equidistant sequences.
method Proof of non-embeddability of certain metric spaces into finite-dimensional Euclidean spaces and a local-to-global principle for loose embeddability.
result Compact Riemannian manifolds cannot contain arbitrarily large regular simplices or long equidistant sequences, suggesting loose embeddings into Euclidean spaces.
Curvature defined for Hilbert modules and Kasparov modules.
problem Defining and studying curvature in Hilbert modules and Kasparov modules.
method Introduced curvature for densely defined universal connections on Hilbert C∗-modules relative to spectral triples. result Curvature only depends on the represented form of the universal connection modulo junk forms.
Stochastic Gradient Descent improved for various Hilbert scales and misspecified models.
problem Understanding and optimizing SGD in Hilbert scales for machine learning.
method Extending SGD analysis to Hilbert scales, including Sobolev and Diffusion spaces, and showing the effects of smoothness and preconditioning.
result Violation of smoothness assumption affects learning rate; preconditioning in Hilbert scales reduces the number of iterations for misspecified models.
Sub-Riemannian spectral distance defined using eigenfunctions of sub-Laplacian
problem Sub-Riemannian geometry and eigenvalues of sub-Laplacian
method Embedding manifold into Hilbert space using eigenfunctions
result Defined spectral distance between sub-Riemannian manifolds
In this paper we will investigate the global properties of complete Hilbert manifolds with upper and lower bounded sectional curvature. We shall prove the Focal Index Lemma that we will allow us to extend some classical results of finite dimensional Riemannian geometry such as Rauch and Berger Theorems and the Topogono…
Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…
Paper proves weighted Riemannian manifolds are Hilbertian, embedding tangent modules.
problem Infinitesimal Hilbertianity of weighted Riemannian manifolds.
method Proves infinitesimal Hilbertianity through Sobolev space and tangent module embedding.
result Weighted Riemannian manifolds are infinitesimally Hilbertian.
Proves sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary measures.
problem Infinitesimal Hilbertianity of sub-Riemannian manifolds with general measures.
method Embedding metric derivations into square-integrable sections, approximating sub-Finsler distances.
result Sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary Radon measures.
No exceptional orbits found in Hilbert spaces actions.
problem Proving the non-existence of exceptional orbits in Hilbert spaces.
method Analyzing polar actions on separable Hilbert spaces by connected Lie groups.
result Proved non-existence of exceptional orbits in Hilbert spaces.
Study on estimating distances between covariance operators and Gaussian processes.
problem Estimating distances between covariance operators and Gaussian processes.
method Riemannian distances, concentration results for Hilbert space-valued random variables, RKHS covariance and cross-covariance operators.
result Both distances converge in the Hilbert-Schmidt norm and can be consistently and efficiently estimated.
New theorems show non-embeddability of certain Lie groups and sub-Riemannian manifolds.
problem Non-embeddability of Lie groups and sub-Riemannian manifolds in specific metric spaces.
method Proving non-existence of quasi-isometric embeddings and biLipschitz embeddings into certain metric measure spaces.
result Connected nonabelian nilpotent Lie groups and sub-Riemannian manifolds cannot be embedded into specified metric spaces.
The Hilbert manifold Σ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits Ω⊂Σ is studied from the topological and metric viewpoints: we seek for conditions that ensure the existence of a smooth local structure for the set $…
The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.
problem Integral and variation formulas for mixed scalar curvature in multi-product manifolds.
method Generalizes results from pseudo-Riemannian almost product manifolds to multi-product structures.
result Generalizes formulas for mixed scalar curvature in multi-product manifolds.
The paper explores isometric models and Busemann functions for Funk and Hilbert discs.
problem Exploring isometric models and Busemann functions for Funk and Hilbert discs.
method Finding and describing isometric models and computing Busemann functions.
result Proving asymptotic harmonicity of the Funk disc and showing its dependence on measure.
Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.
problem Finding optimal bounds between Riemannian and Hilbert metrics in convex projective domains.
method Optimal control techniques applied to Riemannian metrics induced by centro-affine hypersurface immersions.
result Optimal inequalities between Riemannian and Hilbert metrics for a class of convex projective domains.
Introduces a new G2-Hilbert functional in G2-geometry.
problem None explicitly stated; focuses on introducing a new functional.
method Inspired by the Einstein-Hilbert functional, defines a new G2-Hilbert functional on G2-structures. result Torsion-free and nearly G2-structures are saddle critical points of the volume-normalized G2-Hilbert functional. A new Riemannian metric on curve spaces is complete and smooth.
problem Defining a complete metric on the space of embedded curves.
method Proposed a new Riemannian metric and proved its completeness.
result The proposed metric is complete in multiple senses.
Study of sphere bundles over Grassmann manifold with geometric properties.
problem Geometric properties of sphere bundles over Grassmann manifold.
method Analyzes the differentiable structure and metric properties of sphere bundles over a Grassmann manifold.
result Established the smooth action of unitary operators and studied geodesics.