The paper establishes new Casorati inequalities for various Riemannian maps and submersions.
problem Developing new inequalities for Riemannian maps and submersions.
method Using general forms of Casorati inequalities, the paper derives inequalities for specific Riemannian spaces.
result The paper provides new Casorati inequalities for Riemannian maps and submersions.
Establish generalized Chen inequalities for Riemannian submersions and Riemannian maps with applications.
problem Generalized Chen inequalities for Riemannian submersions and Riemannian maps.
method Employing generalized δ-invariants introduced by Chen.
result Optimal inequalities involving δ-invariants and extrinsic invariants.
Classification of specific pseudo-Riemannian manifolds.
problem Classifying conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds.
method Complete classification through mathematical analysis.
result Conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds are either de Sitter or anti-de Sitter spacetimes.
Paper establishes new inequality for Riemannian maps and applies it to various space forms.
problem Developing a new inequality for Riemannian maps and its applications.
method Proposed and utilized a general Chen's first inequality for Riemannian maps and applied it to various space forms.
result Validated the new inequality and compared results with existing approaches.
We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equa…
The paper derives Chen-Ricci inequalities for Riemannian submersions and maps.
problem Chen-Ricci inequalities for Riemannian submersions and maps.
method General forms of Chen-Ricci inequalities for Riemannian submersions and maps are derived, involving curvatures of subspaces.
result New, easy, and elegant techniques for Chen-Ricci inequalities are established.
Generative models use Riemannian manifolds to improve latent space interpretation.
problem Generative models often bias latent space interpretations.
method Use Riemannian manifolds to define latent space paths that respect ambient geometry.
result Improves interpretability of learned representations for both stochastic and deterministic generators.
Two-root Riemannian manifolds have no odd-dimensional examples.
problem Characterizing Riemannian manifolds with specific eigenvalues of the Jacobi operator.
method Investigation of k-root manifolds, focusing on one-root and two-root cases. result There are no two-root Riemannian manifolds of odd dimension.
We completely classify the algebraic Ricci solitons of four-dimensional pseudo-Riemannian generalized symmetric spaces.
The paper examines geometric curvatures in generalized Riemannian spaces.
problem Understanding the physical meaning of scalar curvatures in generalized Riemannian spaces.
method Developed Madsen's formulae for pressures and energy-densities, analyzed with different concepts of generalized Riemannian spaces.
result Linearities of energy-momentum tensor, pressure, energy-density, and state-parameter are examined.
The paper sets new bounds for Ricci-curvature in submersions and maps.
problem Establishing bounds for Ricci-curvature in submersions and maps.
method Developed upper and lower bounds for Ricci-curvature in submersions and maps, providing geometric characterizations.
result New bounds for Ricci-curvature in submersions and maps have been derived.
We classify the space-like biharmonic surfaces in 3-dimension pseudo-Riemannian space form, and construct explicit examples of proper biharmonic hypersurfaces in general ADS space.
The paper proves unique properties of Riemannian twistor spaces under specific curvature conditions.
problem Characterizing Riemannian twistor spaces under vanishing curvature conditions.
method Moving frame method, classification, and nonexistence proofs.
result The only twistor space with parallel Bochner tensor is CP3. The paper introduces a differentially private method for optimization on Riemannian manifolds.
problem Differential privacy in optimization constrained to Riemannian manifolds.
method Adding Gaussian noise to the Riemannian gradient on the tangent space, with privacy and utility guarantees.
result Privacy and utility guarantees for differentially private Riemannian optimization.
This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics …
Generalized Blaschke rolling theorem for curved spaces.
problem Extending classical theorem to curved spaces.
method Generalization to Riemannian manifolds with bounded curvature.
result Sharp results in arbitrary dimensions, new even in constant curvature spaces.
The paper classifies submanifolds in pseudo-Riemannian space forms.
problem Characterizing and classifying totally umbilical submanifolds.
method Classification of congruent classes of totally umbilical submanifolds in non-flat pseudo-Riemannian space forms.
result Some moduli spaces of isometric immersions between space forms are non-Hausdorff.
The paper derives inequalities for Riemannian submersions and applies them to specific space forms.
problem Deriving inequalities for Riemannian submersions.
method Introducing and deriving inequalities for vertical, horizontal, and mixed distributions of Riemannian submersions.
result Established relationships between intrinsic and extrinsic invariants of Riemannian submersions.
Constructs stochastic processes on sub-Riemannian manifolds using Cartan connections.
problem Developing stochastic processes on sub-Riemannian manifolds.
method Introduces stochastic development using Cartan connections, derives generator, and provides conditions for existence.
result Derives a general expression for the generator of the stochastic process and provides conditions for the existence of a Cartan connection.
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.
In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. A…
Analogously to the concept of a curvature of curve and surface, in the differential geometry, in the main part of this paper the concept of the curvature of the hyper-dimensional vector spaces of Riemannian metric is generally defined. The defined concept of the curvature of Riemannian spaces of higher dimensions M: M>…
The paper derives optimal inequalities for bi-slant submanifolds in metallic Riemannian space forms.
problem Understanding geometric properties of bi-slant submanifolds in metallic Riemannian product space forms.
method Deriving generalized Wintgen inequality, optimal inequalities involving δ-invariants, Ricci curvature, shape operator invariants, and generalized normalized δ-Casorati curvatures.
result Established optimal inequalities for bi-slant submanifolds in metallic Riemannian product space forms.
The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is p…
The paper proves a new inequality for submanifolds in Riemannian space forms and applies it to find minimal submanifolds.
problem Finding necessary conditions for minimal submanifolds in Riemannian space forms.
method Proving a first Chen inequality for general warped product submanifolds and applying it to derive conditions for minimality.
result A necessary condition for submanifolds to be minimal in Riemannian space forms.
Researchers study solitons on homogeneous spaces, finding useful geometric structures.
problem Finding solitons on homogeneous spaces.
method Investigating solitons with a focus on equivalence relations and optimal tangent directions.
result Solitons on homogeneous spaces have been identified as a useful tool in various geometric contexts.
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geome…
Riemannian metrics on orbifolds are equivalent to diffeological ones.
problem Equivalence of Riemannian and diffeological orbifolds.
method Framework of Riemannian diffeology and analysis of 2-metrics.
result Riemannian metrics on orbifolds are equivalent to diffeological ones.
We construct compactifications of Riemannian locally symmetric spaces arising as quotients by Anosov representations. These compactifications are modeled on generalized Satake compactifications and, in certain cases, on maximal Satake compactifications. We deduce that these Riemannian locally symmetric spaces are topol…
No isometric immersion of hyperbolic space into Euclidean space.
problem Isometric immersions of hyperbolic space into Euclidean space.
method Generalized Gauss-Bonnet formula for Riemannian polyhedra.
result Hilbert's Theorem generalized to higher dimensions.
Total curvatures of certain hypersurfaces are continuous.
problem Continuity of curvatures in geometric settings.
method Hausdorff distance for hypersurfaces and convex bodies in Riemannian manifolds and Cartan-Hadamard spaces.
result Total generalized mean curvatures are continuous.
Unified framework for human motion generation on Riemannian manifolds.
problem Learning valid human motion in Euclidean spaces.
method Riemannian Motion Generation (RMG) on product manifolds, Riemannian flow matching.
result Achieves state-of-the-art FID (0.043) on HumanML3D and surpasses strong baselines on MotionMillion.
The study of spectral-tightness in Riemannian manifolds and its topological implications.
problem Understanding the spectral properties of Riemannian manifolds and their coverings.
method Analyzing the fundamental group and the Euclidean local de Rham factor to characterize spectral-tightness.
result Spectral-tightness is a topological property of the fundamental group, and it can be characterized by the dimension of the Euclidean local de Rham factor.
We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry between the horizontal tangent spaces is realized by a global isometry. We will show that these spaces have a canonical choice of partial connection on their horizontal bundle, which is determined by isom…
Generalizes Connes's tangent groupoid for sub-Riemannian geometry.
problem Calculating tangent cones in sub-Riemannian geometry.
method Constructs a completion of MimesMimesR+imes using sub-Riemannian metric. result Calculates all tangent cones in Gromov-Hausdorff distance.
Completes the space of vector-valued one-forms on manifolds.
problem Metric incompleteness of the space of full-ranked one-forms.
method Distance equality and quotient structures.
result Concrete description of the metric completion of the space of full-ranked one-forms.
This paper generalizes optimization techniques to diffeological spaces.
problem Challenges in applying optimization techniques to diffeological spaces due to various tangent space definitions.
method Suitable definition of tangent space, diffeological Riemannian space, diffeological gradient, and diffeological retraction.
result Formulation of an optimization algorithm on diffeological spaces.
We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in R2,1 to other Lorentzian space forms. We also characterize immersions of Riemannia…
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.
Smooth submetries between curved spaces are smooth.
problem Smoothness of submetries between curved spaces.
method Proving smoothness of submetries in a general setting, including Riemannian submersions and isometric actions.
result Smoothness of the base manifold is implied by the smoothness of the total manifold without curvature assumptions.
We extend Gaussian Differential Privacy to curved Riemannian manifolds.
problem Extending Gaussian Differential Privacy to curved spaces.
method Developed a Riemannian Gaussian distribution using the Bishop-Gromov theorem and a MCMC-based algorithm.
result Achieved Gaussian Differential Privacy on general Riemannian manifolds with bounded Ricci curvature.
New method synthesizes data on curved spaces for better interpolation.
problem Synthesizing data on curved spaces for better interpolation.
method Riemannian Diffusion Schrödinger Bridge
result Generalizes Diffusion Schrödinger Bridge to curved spaces for better interpolation.
In this work we study riemannian metrics on flag manifolds adapted to the symmetries of these homogeneous nonsymmetric spaces. We first introduce the notion of riemannian Γ-symmetric space when Γ is a general abelian finite group, the symmetric case corresponding to Γ=Z2. We describe and study all the riemannia…
We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures Ω∈P(P(M)) on Wasserstein space, extending on one hand, results in the Euclidean case (for ba…
Consider a singular Riemannian foliation (s.r.f for short) on a compact manifold. By successive blow-ups along the strata, we construct a regular Riemannian foliation on another compact Riemannian manifold and a desingularization map that projects leaves of the regular Riemannian foliation into leaves of the s.r.f. Thi…
Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of …
Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky g…
Introduces new metric for Riemannian metrics, extending unbalanced optimal transport.
problem Extending unbalanced optimal transport to Riemannian metrics.
method Dynamic and static formulations of unbalanced optimal transport on Riemannian metrics.
result Wasserstein--Ebin metric provides a new Riemannian structure on the space of Riemannian metrics.