We provide techniques for studying the nonnegatively curved left-invariant metrics on a compact Lie group. For "straight" paths of left-invariant metrics starting at bi-invariant metrics and ending at nonnegatively curved metrics, we deduce a nonnegativity property of the initial derivative of curvature. We apply this …
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48 results for “Nonnegatively curved”
Proves nonnegatively curved hypersurfaces in hyperbolic space are usually properly embedded.
problem Proving nonnegatively curved hypersurfaces in hyperbolic space are usually properly embedded.
method Analyzes Alexander and Currier's conjecture and proves it for most cases.
result Proves a conjecture about nonnegatively curved hypersurfaces in hyperbolic space.
Totally geodesic dual leaves on curved manifolds are also curved.
problem Characterizing dual leaves of nonnegatively curved polar manifolds.
method Proving dual leaves are totally geodesic and closed, and inducing a Riemannian submersion.
result Dual leaves of nonnegatively curved polar manifolds are themselves nonnegatively curved and totally geodesic.
Let V be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on V are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many V the s…
Sharp estimate shown to be rigid on curved surfaces.
problem Sharp Bezout estimate on nonnegatively curved Riemann surfaces.
method General three circle theorem applied.
result Rigidity of the sharp Bezout estimate.
Proves nonnegatively curved hypersurfaces on a sphere are convex disks.
problem Characterizing nonnegatively curved hypersurfaces with free boundary on a sphere.
method Analyzes hypersurfaces in Euclidean space with constant mean curvature.
result Compact hypersurfaces are embedded convex disks.
Study shows unbounded Pontryagin numbers on curved manifolds.
problem Understanding unbounded Pontryagin numbers on curved manifolds.
method Analyzing rational linear combinations of Pontryagin numbers and their relation to the universal elliptic genus.
result Proves existence of unbounded Pontryagin numbers on nonnegatively curved spin manifolds.
Study on curved spheres' topology.
problem Topology of curved spheres.
method Gromov-Hausdorff metric on isometry classes.
result Properties of curved spheres' metric space.
Study shows nonnegatively curved 7D manifolds with specific properties.
problem Classifying nonnegatively curved 7D manifolds.
method Examined cohomogeneity one manifolds, proving nonexistence of certain metrics.
result Reduced classification of 7D nonnegatively curved manifolds to one family.
New examples show nonnegatively curved metrics with same soul but different moduli space components.
problem Finding nonnegatively curved metrics with same soul but different moduli space components.
method Examples of open manifolds with infinitely many metrics of nonnegative sectional curvature.
result Found open manifolds with souls of codimensions 3 and 2, showing different moduli space components.
Study shows infinitely many nonnegatively curved metric components on Milnor spheres.
problem Understanding the moduli space of nonnegatively curved metrics on Milnor spheres.
method Analyzes the moduli space of metrics with nonnegative sectional curvature and positive Ricci curvature on Milnor spheres.
result The moduli space of nonnegatively curved metrics on Milnor spheres has infinitely many path components.
Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a …
We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimensi…
The paper shows Morse-Novikov cohomology vanishes for certain curved manifolds.
problem Analyzing cohomology groups of curved manifolds.
method Examining Morse-Novikov cohomology groups for manifolds with specific curvature properties.
result Morse-Novikov cohomology groups vanish for almost nonnegatively curved manifolds with nonzero first de Rham cohomology.
We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanif…
We prove that a class of asymptotically nonnegatively curved manifolds (in the sense of Abresch) satisfying some uniform Euclidean type volume growth conditions contains only finitely many homeomorphism types.
A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetrymath.DG
We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S^3 which can be realized as an extremal set with respect to an …
Study involutions on spaces to compute nonnegatively curved metrics dimensions.
problem Computing dimensions of spaces of nonnegatively curved metrics.
method Involution on pseudoisotopy spaces and free loop spaces.
result Explicit dimensions of manifolds with nontrivial rational homotopy groups.
Minimal graph level sets are strictly convex in curved spaces.
problem Regularity and convexity of minimal graph level sets in curved spaces.
method Continuity method to prove strict convexity.
result Minimal graph level sets are strictly convex.
Sobolev inequalities involving 2-tensor fields in manifolds with nonnegative sectional curvaturemath.DG
Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.
problem Establishing inequalities for tensor fields on curved manifolds.
method Applying the ABP method to symmetric tensor fields on manifolds with nonnegative sectional curvature.
result Log Sobolev and Michael Simon inequalities for tensor fields.
Study on spaces of nonnegatively curved surfaces and their topological properties.
problem Characterizing spaces of nonnegatively curved metrics on surfaces.
method Analysis of homeomorphism types of spaces of smooth complete nonnegatively curved metrics on surfaces.
result Spaces of metrics on surfaces have specific topological properties, including being homeomorphic to Hilbert spaces or countable powers of Hilbert cubes.
Study proves inequalities for mass-capacity on curved spaces.
problem Proving nonnegativity and positive lower bounds of mass on curved spaces.
method Applying mass-capacity inequalities from \cite{M22} to manifolds with nonnegative scalar curvature.
result Sufficient conditions for nonnegativity and positive lower bounds of mass.
We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of ``bounded geometry''.
We study nonnegatively curved metrics on S^2xR^4. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking's almost-positively curved metric on S2xS3 extends to a nonnegatively curved metr…
This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric circle-action admits an approximation in Gromov-Hausdorff topology by smooth, positively cur…
Study shows infinitely many nonnegatively curved metric spaces on exotic 7-manifolds.
problem Investigating nonnegatively curved metrics on exotic 7-manifolds.
method Using Kreck-Stolz invariant and Atiyah-Patodi-Singer index theorem for orbifolds with boundary.
result Moduli space of nonnegatively curved metrics has infinitely many connected components.
The study classifies singularities in curved 3D shapes.
problem Understanding nonnegatively curved polyhedral 3-manifolds.
method Classification of singularities with nonnegative curvature and integral monodromy.
result 32 isometry types of codimension 3 singularities are classified.
Study shows infinitely many nonnegatively curved metrics on quotient spaces.
problem Investigating nonnegatively curved metrics on specific quotient spaces.
method Used Lefschetz fixed point theorem and relative η-invariant to distinguish components.
result Found infinitely many path components of nonnegatively curved metrics.
Sharp inequality found for hypersurfaces in curved spaces.
problem Establishing geometric inequalities for hypersurfaces in curved spaces.
method Standard comparison methods in Riemannian Geometry.
result Sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature.
Proves conjecture about foliations on curved spaces.
problem Completeness of dual foliations on curved spaces.
method Analyzes Riemannian foliations on nonnegatively curved symmetric spaces.
result Foliations split into trivial and single dual leaf foliations.
We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metric.
We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension , there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to non…
B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We…
Study shows fundamental groups of certain curved spaces are simple.
problem Understanding fundamental groups of curved spaces with specific growth conditions.
method Proved using nonnegative Ricci curvature and volume growth conditions.
result Fundamental groups are trivial or finite for these spaces.
In this paper, we study a complete noncompact nonnegatively curved Alexandrov space with a soul of codimension two. We establish some structural results under additional regularity assumptions. As an application, we conclude that in this case Sharafutdinov retraction, , is a submetry.
We classify the complete metrics of nonnegative sectional curvature on M^2xR^2, where M^2 is any compact 2-manifold.
In curved spaces, isoperimetric sets don't exist for small volumes.
problem Nonexistence of isoperimetric sets in spaces of positive curvature.
method Constructing specific noncompact smooth Riemannian manifolds with positive curvature.
result Nonexistence of isoperimetric sets for small volumes in spaces of positive curvature.
We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnegative sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.
We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.
Note: Flatness proven for certain curved solitons.
problem Characterizing asymptotic behavior of curved solitons.
method Analyzing -noncollapsed steady Kähler-Ricci solitons with nonnegative bisectional curvature.
result Flatness proven for these solitons.
In this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact nonnegative curved manifolds admit (complete) metrics with nonnegative curvature.
New examples found of 4D spaces with curvature that flow to less curved spaces.
problem Finding 4D spaces with nonnegative curvature that change under Ricci flow.
method Exhibited new examples of 4-manifolds.
result First examples of 4D spaces losing nonnegative curvature under Ricci flow.
We construct several infinite families of nonnegatively curved manifolds of low cohomogeneity and small dimension which can be distinguished by their cohomology rings. In particular, we exhibit an infinite family of eight-dimensional cohomogeneity one manifolds of nonnegative curvature with pairwise non-isomorphic comp…
New vanishing theorems for genera derived under almost nonnegative Ricci curvature.
problem Vanishing theorems for genera under specific curvature conditions.
method Almost nonnegative Ricci curvature and infinite fundamental group.
result Vanishing theorems for Todd genus, -genus, elliptic genera, Witten genus, and Euler characteristic number for Alexandrov spaces.
Proves inequalities on curved spaces with positive curvature.
problem Proving inequalities on manifolds with nonnegative Ricci curvature.
method Analyzes manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Proves Heisenberg-Pauli-Weyl, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities.
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 shows nonnegative curvature metrics have complex homotopy structure.
problem Understanding metrics with nonnegative sectional curvature on open manifolds.
method Index theory and homotopy density of cylindrical ends.
result Space of complete metrics has nontrivial homotopy groups.
Sharp Sobolev and Michael-Simon inequalities on curved manifolds.
problem Proving inequalities on manifolds with nonnegative curvature.
method Analyzing asymptotic volume ratio and curvature properties.
result Sharp Sobolev and Michael-Simon inequalities established.