The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in R^n with negative flag curvature and constant S-curvature. In th…
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48 results for “Riemannian curvatures”
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.
Classification of biharmonic Riemannian submersions from manifolds with constant sectional curvaturemath.DG
This paper generalizes biharmonic Riemannian submersions to higher dimensions.
problem Classifying biharmonic Riemannian submersions from manifolds with constant sectional curvature.
method Constructing an adapted orthonormal frame to simplify the biharmonic equation and analyzing curvature properties.
result A Riemannian submersion is biharmonic if and only if it is harmonic from an -dimensional manifold with constant sectional curvature to an -dimensional manifold.
If is a Riemannian Submersion and has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of…
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma calculus and --Bochner's formula.
result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
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.
Study adds scalar curvatures of mapped manifolds to Riemannian products.
problem Additivity of scalar curvatures in Riemannian products.
method Stabilized scalar curvatures of mapped manifolds.
result Proves additivity in some cases.
The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Ri…
The paper studies geometric structures of curvature radii on Riemannian manifolds.
problem Exploring geometric structures of curvature radii on Riemannian manifolds.
method The main method involves constructing a pair of global vector fields that encode intrinsic geometry.
result Existence of sub-Riemannian manifolds associated with curvature radii and investigation of their properties.
Study curvature invariants in sub-Riemannian manifolds.
problem Understand curvature invariants in sub-Riemannian geometry.
method Prove geometrical inequalities for submanifolds with orthogonal distributions.
result Inequalities for submanifolds with orthogonal distributions are derived.
The paper derives curvature inequalities for submersions from quaternionic space forms.
problem Deriving curvature inequalities for submersions from quaternionic space forms.
method Analyzing Ricci and scalar curvatures of horizontal and vertical distributions in anti-invariant submersions.
result Established Ricci curvature inequality for anti-invariant submersions.
Survey shows deformations of homogeneous metrics in curvature homogeneous manifolds.
problem Understanding curvature homogeneous manifolds with nontrivial κ-nullity.
method Analyzes deformations of homogeneous metrics along submersions.
result Identifies two new examples of curvature homogeneous manifolds.
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…
The paper explores geometric properties of Riemannian warped product maps and their curvature.
problem Investigating the geometric properties of Riemannian warped product maps.
method The approach involves establishing conditions for geodesics, deriving curvature tensors, and examining various types of maps.
result Derivation of integral formula for scalar curvature of conformal Riemannian warped product maps.
Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature fo…
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.
The paper proves uniformization for specific curvature types on manifolds.
problem Uniformizing fourth order conformal curvature on Riemannian manifolds.
method Proving existence of conformal deformations for specific curvature conditions.
result Existence of conformal deformations for positive Yamabe invariant and total Q-curvature.
The paper studies variations of -curvature for submanifolds in Riemannian manifolds.
problem Understanding the behavior of -curvature under variations of submanifolds.
method Analyzes the functional of -curvature for submanifolds of arbitrary codimension in Riemannian manifolds.
result Provides insights into the variational properties of -curvature.
We show that any universal curvature identity which holds in the Riemannian setting extends naturally to the pseudo-Riemannian setting. Thus the Euh-Park-Sekigawa identity also holds for pseudo-Riemannian manifolds. We study the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that as in t…
New Einstein metrics found from para-Sasaki-like Riemannian manifolds.
problem Finding new Einstein metrics in Riemannian geometry.
method Cone construction and hyperbolic extension of paracontact paracomplex Riemannian manifolds.
result New complete Einstein para-Sasaki-like Riemannian manifolds with negative scalar curvature.
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.
Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
Study on curvatures of surfaces in specific Lie groups.
problem Analyzing curvatures of surfaces in 3D contact sub-Riemannian Lie groups.
method Riemannian approximation scheme to derive formulas for curvatures.
result Classification of surfaces with constant horizontal curvatures.
We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation theory to prove the existence of a infinite number of conformal classes with at leas…
Curvature measures uniquely determined by invariance under embeddings.
problem Characterizing curvature measures uniquely.
method Applied Weyl principle and Künneth-type formula.
result Curvature measures uniquely characterized by invariance under isometric embeddings.
The paper studies bi-slant Riemannian maps to Kenmotsu manifolds and derives inequalities.
problem Investigating bi-slant Riemannian maps and their properties.
method Introducing and studying bi-slant Riemannian maps, deriving curvature relations and inequalities.
result Construction of Chen-Ricci inequalities, DDVV inequalities, and optimal inequalities involving Casorati curvatures.
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.
Riemannian submersions can preserve positive intermediate Ricci curvature, but not necessarily.
problem Understanding the conditions under which Riemannian submersions preserve positive intermediate Ricci curvature.
method Analyzing the Gray--O'Neill Horizontal curvature equation and constructing perturbations of metrics.
result Riemannian submersions that do not preserve positive Ricci curvature are dense in the -topology.
The paper bends Riemannian manifolds to achieve metrics with negative Ricci curvature.
problem Achieving metrics with negative Ricci curvature on closed Riemannian manifolds.
method Solving a fully nonlinear equation to conformally bend the manifold.
result Metrics of quasi-negative Ricci curvature are conformal to metrics with negative Ricci curvature.
The paper estimates curvature for a specific flow on manifolds.
problem Estimating curvature for Ricci-harmonic flow on manifolds.
method Local estimate and De Giorgi-Nash-Moser iteration method.
result Local boundedness of Riemannian curvature proved.
Study shows limits of metrics with positive scalar curvature on spheres.
problem Non-negativity of scalar curvature is not preserved under certain limits.
method Examined metrics conformal to the round metric on for .
result Any conformal metric to the round metric on for can be a limit of metrics with positive scalar curvature.
Proves curvature comparison for Riemannian bands in low dimensions.
problem Curvature comparison in Riemannian bands with lower bounds.
method Uses warped products over scalar-flat manifolds with log-concave warping.
result Scalar and mean curvature comparison results proven.
The study classifies Riemannian manifolds with curvature nullity.
problem Classifying Riemannian manifolds with nontrivial curvature nullity.
method Classification theorems based on curvature nullity, scalar curvature, and quotient existence.
result New classification theorems and revisited previous results.
This paper describes cyclic Riemannian Lie groups and their curvatures.
problem Understanding cyclic Riemannian Lie groups and their properties.
method Analyzing left-invariant vector fields and Riemannian metrics.
result Complete description and detailed analysis of cyclic Riemannian Lie groups and their curvatures.
Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.
problem Find a relationship between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds isometrically immersed in another Riemannian manifold.
method Establish an optimal inequality involving mixed scalar curvature and square of mean curvature.
result Optimal inequality that includes mixed scalar curvature and square of mean curvature.
Paper classifies fibers of fat Riemannian submersions with non-negative curvature.
problem Understanding fat bundles in Riemannian submersions.
method Structural analysis of fat Riemannian submersions with non-negative sectional curvature.
result Classification of fibers as symmetric spaces, isometric correspondence of foliations, rigidity of dual foliations.
Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvaturemath.DG
Every biharmonic Wintgen ideal submanifold in a Riemannian manifold of constant sectional curvature is either minimal or has constant mean curvature.
problem Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
method Show that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of nonpositive constant sectional curvature is minimal and that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature.
result Partial affirmative answers to Chen's conjecture, generalized Chen's conjecture in hyperbolic spaces, and Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.
3D metrics get scalar curvature bounds via IMCF.
problem Bounding scalar curvature for metrics.
method Inverse Mean Curvature Flow (IMCF) and Hawking mass monotonicity.
result Stability theorem for nonnegative scalar curvature.
Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.
problem Integral of scalar curvature on manifolds with a pole.
method Asymptotic scaling invariant integral of scalar curvature equals a term determined by asymptotic volume ratio.
result Integral of scalar curvature equals a volume ratio term.
Super twisted productsmath.DG
The paper defines and analyzes curvature tensors on super twisted product spaces.
problem Investigating curvature tensors on super twisted product spaces.
method Defined -curvature tensor, computed curvature tensors and Ricci tensors, and studied curvature flatness.
result Mixed Ricci-flat super twisted product semi-Riemannian manifolds can be expressed as super warped product manifolds.
The study examines symmetries in spaces with positive or non-negative curvature.
problem Understanding symmetries in spaces with curvature constraints.
method Survey of existing results for Riemannian manifolds with specified curvature properties and symmetries.
result Results on symmetries in spaces with curvature bounds.
Novel coarse extrinsic curvature for Riemannian submanifolds.
problem Understanding extrinsic curvature of submanifolds.
method Derived from Wasserstein 1-distance between probability measures.
result New insights and approximation of mean curvature from data.
We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space …
Sharp bounds on scalar curvature spectrum and rigidity theorems.
problem Understanding scalar curvature bounds and rigidity on manifolds.
method Sharp upper bounds for the bottom spectrum of the Beltrami Laplacian, scalar curvature rigidity theorem.
result Sharp upper bound for the bottom spectrum of the Beltrami Laplacian and scalar curvature rigidity theorem.
Formula for scalar curvature under metric collapse.
problem Finding positive scalar curvature metrics.
method Formula involving scalar curvature and adapted orthonormal frame.
result Effect of metric collapse on scalar curvature.
The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.
problem Understanding the combinatorial types of 3D polytopes in specific Riemannian manifolds.
method Analysis of mean curvature convex Riemannian polyhedra with non-obtuse dihedral angles in positive scalar curvature 3-manifolds.
result Determination of combinatorial types of 3D simple convex polytopes.
Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.
problem Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.
method Perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor, with general multiplicity results via Lusternik-Schnirelman theory.
result Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.