Researchers introduce new invariants for a specific type of edge geometry.
problem Investigating geometric properties of 5/2-cuspidal edges. method Introducing secondary cuspidal curvature and bias, proving their properties and product's invariance.
result Real analytic 5/2-cuspidal edges with non-vanishing limiting normal curvature admit non-trivial isometric deformations. Study geometric inequalities for CR-submanifolds using curvature invariants.
problem Geometric inequalities for CR-submanifolds in almost Hermitian spaces.
method Comparing mutual curvature invariants with Chen-type invariants and proving geometric inequalities.
result Proved geometric inequalities with intermediate mean curvature squared for CR-submanifolds.
Study invariant hypersurfaces with linear mean curvature in Euclidean space.
problem Understanding hypersurfaces with linear mean curvature.
method Explicit parametrizations and classification of rotationally invariant hypersurfaces.
result Obtained explicit parametrizations of constant curvature hypersurfaces.
Defines and proves CR invariants on five-manifolds.
problem Defines and studies CR invariants on CR five-manifolds.
method Defines global secondary CR invariants and proves their linear combination.
result Any global secondary CR invariant is a linear combination of total Q′-curvature, total I′-curvature, and a local CR invariant. Constructs 7-manifolds with SO(3) invariant non-negative curvature.
problem Finding exotic spheres with non-negative curvature.
method Six-parameter family of highly connected 7-manifolds constructed with invariant metrics.
result All exotic spheres in 7 dimensions have non-negative curvature.
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.
New rational curvature measures for 2-complexes.
problem Measuring curvature in 2-dimensional cell complexes.
method Defined and proved rational curvature invariants.
result Computable rational curvature bounds for 2-complexes.
Study new Ricci flow invariant curvature conditions.
problem Topology of manifolds with pinched curvature.
method Provide quantitative evidence for an unpublished conjecture.
result Topology of manifolds with pinched curvature studied.
Study magnetic curvature on Lie groups, extending Milnor's work.
problem Exploring magnetic curvatures on Lie groups.
method Computing magnetic curvatures and analyzing algebraic properties.
result Extending results from Milnor's classic paper on left-invariant metrics.
New CVIs generalize scalar curvature stability results.
problem Stability and rigidity of Riemannian invariants.
method Establishing stability and rigidity results for CVIs.
result Generalizing scalar curvature stability results.
A surface in homogenous space Sol is said to be an invariant surface if it is invariant under some of the two 1-parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classi…
Study on curvature invariants near singularities of wavefronts.
problem Conditions for extendibility and boundedness of curvature invariants.
method Investigation of Gaussian curvature, Mean curvature, and principal curvatures near singularities.
result Relationship between convergence to infinity and uniform approximation of fronts.
The paper generalizes CR invariants using renormalized characteristic forms.
problem Defining new CR invariants via renormalized characteristic forms.
method Introducing new curvatures for each renormalized characteristic form.
result The new curvatures' integrals match CR invariants constructed by Marugame.
Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.
problem Understanding the relationship between Gaussian curvature and singularities of Gauss maps of cuspidal edges.
method Analyzes geometric invariants and types of singularities of Gauss maps to define and characterize positivity/negativity of cusps.
result Defines and characterizes positivity/negativity of cusps of Gauss maps by geometric invariants of cuspidal edges, and shows relation between sign of cusps and Gaussian curvature.
Classifies invariant hypersurfaces with singularities.
problem Classifying invariant hypersurfaces with singularities.
method Analyzing O(p)imesO(q)-invariant constant mean curvature hypersurfaces. result Solved Wu-yi Hsiang's conjecture.
I-non-degenerate spaces are spacetimes that can be characterized uniquely by their scalar curvature invariants. The ultimate goal of the current work is to construct a basis for the scalar polynomial curvature invariants in three dimensional Lorentzian spacetimes. In particular, we seek a minimal set of alg…
In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Ran…
The paper characterizes surfaces using invariants.
problem Determining surfaces in 3D space without umbilical points.
method Introducing canonical principal parameters and using invariants (principal curvatures or Gauss and mean curvature) to uniquely identify surfaces.
result Surfaces are uniquely determined by a pair of invariants satisfying a partial differential equation.
The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then, these invariants expressed by focal curvatures of the curve. And then, we give t…
Solves scalar curvature equations for rotation invariant Kähler metrics on complex space.
problem Finding rotation invariant Kähler metrics with constant scalar curvature on complex space.
method Reduces the scalar curvature equation to a system of ODEs and solves them.
result Obtains complete lists of rotation invariant metrics with zero or positive scalar curvature in lower dimensions.
We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p,q)-curvatures. They are a generalization of the p-curvat…
Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.
problem Relating Yamabe invariants on asymptotically Poincare-Einstein manifolds.
method Established a sharp inequality using lower Ricci curvature bounds.
result Sharp inequality relating type II Yamabe invariant of the interior to the Yamabe invariant of the conformal infinity.
Study finds metrics with positive intermediate Ricci curvature on specific low-dimensional manifolds.
problem Existence of invariant metrics with positive intermediate Ricci curvature on low-dimensional cohomogeneity one manifolds.
method Construction of invariant metrics with positive intermediate Ricci curvature on specific manifolds.
result Invariant metrics with positive 4th-intermediate Ricci curvature exist but not for 3rd-intermediate Ricci curvature on certain manifolds.
We derive a curvature-variation formula for a path of left-invariant metrics on a compact Lie group, beginning at a bi-invariant metric. We prove rigidity theorems for paths which remain nonnegatively curved, and we make progress towards a classification of the left-invariant metrics with nonnegative curvature on SO(4)…
In this article we review the recent results about the flag curvature of invariant Randers metrics on homogeneous manifolds and by using a counter example we show that the formula which obtained for the flag curvature of these metrics is incorrect. Then we give an explicit formula for the flag curvature of invariant Ra…
In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac …
Lower bound for L^2-norm of Hermitian scalar curvature derived from Futaki invariant.
problem Finding lower bounds for the L^2-norm of Hermitian scalar curvature.
method Using the symplectic Futaki invariant as an asymptotic obstruction to the existence of constant Hermitian scalar curvature almost-Kähler metrics.
result Deduced a lower bound for the L^2-norm of the Hermitian scalar curvature.
Develops a graphical calculus for stable curvature invariants.
problem Calculating stable curvature invariants of Riemannian manifolds.
method Graphical calculus based on trivalent graphs with colored edges.
result Derives a curvature identity for compact Einstein manifolds.
Study uses regularized mean curvature flow on invariant hypersurfaces in Hilbert space.
problem Analyzing invariant hypersurfaces in Hilbert space.
method Regularized mean curvature flow with specific conditions.
result Invariant hypersurfaces collapse to group orbits under flow.
Study left invariant spray structures on Lie groups, calculating curvature and geodesics.
problem Understanding curvature and geodesics in left invariant spray structures on Lie groups.
method Use invariant frames and canonical bi-invariant Berwald spray structure to analyze left invariant spray structures.
result Established correspondence between geodesics and inverse integral curves of spray vector fields.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
The second H. Weyl curvature invariant of a Riemannian manifold, denoted h4, is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of h4 is that it is nonnegative for Einstein manifolds, hence it p…
Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.
problem Bounding Laplacian eigenvalues on manifolds with non-negative scalar curvature.
method Investigation of invariant spectrum on compact Riemannian manifolds with large isometry groups.
result Upper bounds for eigenvalues of the invariant spectrum assuming non-negative scalar curvature.
Invariants from Seiberg-Witten equations link scalar curvature to diffeomorphisms.
problem Linking positive scalar curvature metrics to diffeomorphisms of 4-manifolds.
method Introducing an invariant using families of Seiberg-Witten equations.
result Invariant yields applications to homotopy groups of positive scalar curvature metrics.
The paper studies new curvature properties in Finsler geometry.
problem Properties of projectively equivalent Finsler metrics and their curvature structures.
method Introducing new characterizations of quadratic curvature properties in Finsler manifolds.
result Novel insights into curvature behavior under generalized projective sprays.
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.
It is known that the spectrum of the Laplace operator on functions of a closed Riemannian manifold does not determine the integrals of the individual fourth order curvature invariants scal2, ∣ric∣2, ∣R∣2, which appear as summands in the second heat invariant a2. We study the an…
Rationality proved for curvature invariants of 2-complexes.
problem Curvature invariants of 2-dimensional cell complexes.
method Proved rationality through explicit rational linear-programming problems.
result Curvature invariants are rational, computable, and algorithmically realisable.
Sharp inequalities for submersions from Sasakian space forms.
problem Understanding curvature properties in submersions from Sasakian space forms.
method Analyzing Ricci and scalar curvatures for anti-invariant submersions.
result Deriving precise inequalities involving curvature measures.
Researchers extend M-eigenvalues to Riemann curvature tensor, finding real values and invariants.
problem Extending M-eigenvalues to Riemann curvature tensor in general relativity.
method Introduced M-eigenproblem definition from minimization of associated function, proved existence and real values of M-eigenvalues.
result M-eigenvalues of Riemann curvature tensor are real and invariants, related to curvature invariants.
The paper studies invariant functions and their relation to Landsberg surfaces.
problem Investigating the geometry of invariant functions and their applications to Landsberg surfaces.
method Investigating the geometry of S-invariant functions and their associated vertical subdistribution, and relating the holonomy distribution to these subdistributions. result For Landsberg surfaces, if the flag curvature is S-invariant, it is constant, and the surface is Riemannian. Study constant mean curvature tubes in homogeneous spaces.
problem Global geometry of constant mean curvature tubes.
method Screw-motion invariants, foliation, numerical isoperimetric profile.
result Foliation result and embeddedness proof.
The paper explores Kähler-like metrics on generalized flag manifolds.
problem Finding invariant almost Hermitian structures with specific scalar curvature properties.
method Investigating invariant almost Hermitian geometry on generalized flag manifolds, focusing on Kähler-like metrics.
result Examples of Kähler-like metrics satisfying s=2smC are provided. 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.
Ricci flow on certain homogeneous spaces creates metrics with positive curvature.
problem Finding metrics with positive Ricci curvature on specific homogeneous spaces.
method Normalized Ricci flow on simply connected homogeneous spaces with two equivalent isotropy summands.
result Every G-invariant metric evolves to one with positive Ricci curvature under Ricci flow.
Classification of constant curvature surfaces in Berger spheres.
problem Identifying complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres.
method Complete classification through detailed analysis of Clifford tori and spheres.
result Rotationally invariant spheres with constant Gauss curvature are the only topological spheres in Berger spheres for K>KP. We extend the vanishing theorem for the Seiberg-Witten invariants of a manifold with positive scalar curvature to the case when the curvature is allowed to be negative on a set of small volume. (The precise curvature bounds are described in the paper.) The idea is to combine the method of `semigroup domination' with th…
The paper classifies invariant translators for a specific curvature flow.
problem Classifying invariant translators for a specific curvature flow.
method Classification of λ-translators invariant under translations and rotations. result All λ-translators are classified.