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.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
Study on Riemannian Poisson warped product spaces and their properties.
problem Characterizing and understanding Riemannian Poisson warped product spaces.
method Formal treatment of Killing and 2-Killing 1-forms on Riemannian Poisson manifolds, including Bochner type results.
result Characterization of 2-Killing 1-form on (R2,g,Π) and Bochner type results on compact spaces. Paper studies quaternionic space forms and Riemannian maps inequalities.
problem Investigate DDVV-type inequality for quaternionic space forms.
method Analyze Riemannian maps from quaternionic space forms to manifolds.
result Derived inequality with equality conditions discussed.
Rigidity of Wasserstein spaces over Riemannian manifolds
problem Isometric rigidity of L2 Wasserstein spaces over Riemannian manifolds
method Showing L2 Wasserstein spaces are isometrically rigid if and only if their underlying manifolds do not admit a Euclidean de Rham factor
result Isometry of L2 Wasserstein spaces over non-Euclidean manifolds
Introduces a new framework for Riemannian diffeology.
problem No specific problem stated; focuses on a new framework.
method Uses tangent functor and metric from Iglesias-Zemmour to establish weak Riemannian diffeological spaces.
result Establishes a category of weak Riemannian diffeological spaces and shows induced pseudodistance is a distance under technical conditions.
Develops calculus on Wasserstein spaces for Riemannian manifolds.
problem Characterizing and understanding the geometry of Wasserstein spaces.
method Intrinsic formalism for topology, smooth structure, and Riemannian geometry of Wasserstein spaces.
result Wasserstein spaces of closed manifolds are geodesically convex.
We prove that Berwald spaces whose flag curvature is nowhere vanishing are in fact Riemannian spaces. This means that any Berwald space with flag curvature bounded below by a positive number must be also Riemannian. This rigidity result shows the importance of non-Riemannian examples when imposing flag curvature bounds…
Paper defines minimal hypersurfaces in Euclidean and Riemannian spaces.
problem Characterizing minimal hypersurfaces in different spaces.
method Analyzes conditions for hypersurfaces to be minimal or stable.
result Minimal and stable hypersurfaces are hyperplanes in Euclidean spaces and totally geodesic submanifolds in Riemannian manifolds.
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.
Minimal polynomial found for Riemannian C_0-spaces.
problem Understanding the structure of Riemannian C_0-spaces.
method Constructing polynomial functions on tangent spaces and gluing them globally.
result The degree of the polynomial provides an upper bound for the Singer invariant.
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.
In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Rie…
We classify pseudo-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a pseudo-Riemannian manifold. Also, we obtain the classification of the pseudo-Riemannian submersions with (para-)complex connected totally geodesic fibres from a (para-)complex pseudo-hyperbolic sp…
The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
problem Understanding the structure of leaf spaces of Riemannian foliations.
method Analyzing the holonomy groupoid and using diffeological spaces.
result The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal sectional curvatures are locally equivalent to canal hypersurfaces in Euclidean spa…
We classify the semi-Riemannian submersions from a pseudo-hyperbolic space onto a Riemannian manifold under the assumption that the fibres are connected and totally geodesic. Also we obtain the classification of the semi-Riemannian submersions from a complex pseudo-hyperbolic space onto a Riemannian manifold under the …
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 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.
New submersion proves complex-valued harmonic map existence.
problem Existence of non-constant harmonic morphisms.
method Constructing harmonic Riemannian submersions from symmetric spaces.
result Existence of non-constant, globally defined complex-valued harmonic morphism.
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.
Paper derives sub-Riemannian versions of Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for twisted BCV spaces.
problem Deriving sub-Riemannian versions of the Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for twisted BCV spaces.
method Derives sub-Riemannian versions of the Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki theorems for the twisted BCV spaces.
result Computes Connes conformal invariants for the twisted product and sub-Riemannian limits of these invariants for the twisted BCV spaces.
Study on sets with positive reach in Euclidean and Riemannian spaces.
problem Understanding sets with positive reach in various spaces.
method Structural results on subsets of positive reach.
result New insights into sets with positive reach in Euclidean and Riemannian spaces.
We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.
The paper studies Finsler spaces with semi-concurrent vector fields and their equivalence to Riemannian spaces.
problem Characterizing Finsler spaces with semi-concurrent vector fields.
method Analyzing various Finsler spaces and proving conditions for equivalence to Riemannian spaces.
result Various Finsler spaces (quasi-C-reducible, C3-like, Ch-recurrent, P2-like) are equivalent to Riemannian spaces if they admit a semi-concurrent vector field. Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Understanding Riemannian metrics on lens spaces and their geometric properties.
method Geometric control theory methods applied to axisymmetric metrics.
result Cut loci and cut times converge to sub-Riemannian structure's values.
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Analyzing Riemannian metrics on lens spaces.
method Geometric control theory methods.
result Cut loci and cut times converge to sub-Riemannian structure's cut locus and time.
Survey on smooth function and form density in Riemannian Sobolev spaces.
problem Density of smooth functions and forms in Sobolev spaces on Riemannian manifolds.
method Careful examination of weak covariant derivatives and partial derivatives.
result Equivalence of weak covariant derivatives to weak partial derivatives.
The paper derives inequalities for Riemannian submersions and their applications.
problem Characterizing Casorati inequalities for Riemannian submersions.
method Algebraic and geometric analysis of Casorati inequalities for normalised scalar and Casorati curvatures.
result Characterization of equality cases for Casorati inequalities in Riemannian submersions.
The paper derives inequalities for Riemannian maps and submersions involving quaternionic space forms.
problem Establishing optimal inequalities for Riemannian maps and submersions involving quaternionic space forms.
method Deriving Casorati inequalities for Riemannian maps and submersions involving quaternionic space forms.
result Geometric characterizations of equality cases for Riemannian maps and submersions involving quaternionic space forms.
The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from the Satake diagram, in a way that is suited for the use with comput…
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.
Developed a sub-Riemannian version of Minkowski problem in Heisenberg groups.
problem Minkowski type problem in Heisenberg groups.
method Variational method.
result Positive answer to sub-Riemannian Minkowski type problem.
New sub-Riemannian spaces with boundary meet curvature-dimension condition.
problem Finding sub-Riemannian manifolds with boundary satisfying curvature-dimension condition.
method Constructing specific sub-Riemannian structures on half-spaces and hemispheres.
result Provided new examples of sub-Riemannian manifolds with boundary that meet RCD(K,N) condition. 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.
Consider the sum of the first N eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for N sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree N to be thos…
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
Classifies homogeneous Riemannian structures on 3D Lie groups.
problem Classifying homogeneous Riemannian structures on 3D Lie groups.
method Classification based on left invariant metrics and previous classifications.
result Complete classification of homogeneous Riemannian structures on 3D Lie groups.
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.
It is proved that the isometry classes of pointed connected complete Riemannian n-manifolds form a Polish space, M∗∞(n), with the topology described by the C∞ convergence of manifolds. This space has a canonical partition into sets defined by varying the distinguished point into each manifo…
Geodesic orbit property studied for Lorentz manifolds.
problem Geodesic orbit property for Lorentz manifolds.
method Defined naturally reductive for pseudo-Riemannian manifolds and proved theorems for Lorentz nilmanifolds.
result Geodesic orbit property holds for Lorentz nilmanifolds under specific conditions.
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…
In this article we show how holomorphic Riemannian geometry can be used to relate certain submanifolds in one pseudo-Riemannian space to submanifolds with corresponding geometric properties in other spaces. In order to do so, we shall first rephrase and extend some background theory on holomorphic Riemannian manifolds,…
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.
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.
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 obtain sharp inequalities involving the Ricci curvature and the scalar curvature for anti-invariant Riemannian submersions from Sasakian space forms onto Riemannian manifolds.
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.