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.
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. 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.
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.
We introduce two invariants called the secondary cuspidal curvature and the bias on 5/2-cuspidal edges, and investigate their basic properties. While the secondary cuspidal curvature is an analog of the cuspidal curvature of (ordinary) cuspidal edges, there are no invariants corresponding to the bias. We prove that t…
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method Provide quantitative evidence for an unpublished conjecture.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.
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.
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.
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.
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 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…
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.
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)…
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.
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 …
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.
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.
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.
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…
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 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.
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.
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…
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.
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. 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. New invariant for 4D hypersurfaces ensures smooth critical points.
problem Understanding smoothness of curvature energies on 4D hypersurfaces.
method Developed a new conformally invariant energy.
result Critical points of new energy are smooth.
We classify left invariant metrics with nonnegative curvature on SO(3) and U(2).
Study curvature and torsion from cross-ratios in discrete curves.
problem Define curvature and torsion for discrete curves using cross-ratios.
method Use Möbius invariant point-insertion-rule to construct circles and express torsion using cross-ratio.
result Discrete curvature and torsion defined using cross-ratios converge to smooth curvature and torsion as sampling density increases.
This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions g…
For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely its sectional curvature and scalar curvature on one side; and its main extrinsic invariant, namely squared mean curvature on the other side…
In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-neg…
We prove that every quasitoric manifold admits an invariant metric of positive scalar curvature.
In this letter we provide an invariant characterization for all spacetimes with all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives vanishing except those zeroth order curvature invariants expressed as polynomials in Λ, the cosmological constant. Using this invariant des…
Classifies solitons on invariant surfaces in solvable Lie group.
problem Classifying solitons on invariant surfaces in a specific Lie group.
method Analyzes solitons associated with Killing vector fields and proves rigidity results.
result Identifies the only solitons for specific invariant surfaces.
The paper studies conformal invariants of Riemannian manifolds and proves vanishing theorems and inequalities.
problem Analyzing conformal invariants of Riemannian manifolds and their implications.
method Defining new conformal invariants and proving vanishing theorems and inequalities.
result Established inequalities relating conformal invariants to other geometric invariants.
Study on Berwald-Weyl curvature with projective invariance and vanishing results.
problem Characterizing Berwald-Weyl curvature for spray/Finsler metrics.
method Analyzing expressions and proving vanishing conditions for curvature.
result Berwald-Weyl curvature vanishes for certain spray/Finsler metrics.