Manifolds with almost nonnegative curvature operator and principal bundles

We study closed manifolds with almost nonnegative curvature operator (ANCO) and derive necessary and/or sufficient conditions for the total spaces of principal bundles over (A)NCO manifolds to admit ANCO connection metrics. In particular, we provide first examples of closed simply connected ANCO manifolds which do not admit metrics with nonnegative curvature operator.     

Manifolds of nonnegative Ricci curvature and infinite topological type

Geometriae Dedicata - We construct a family of complete n-dimensional ( $$nge 4$$ ) manifolds with nonnegative Ricci curvature and infinite topological type. Moreover, the curvature decay,...     

A splitting theorem for manifolds of almost nonnegative Ricci curvature

Cheeger and Gromoll proved that a closed Riemannian manifold of nonnegative Ricci curvature is, up to a finite cover, diffeomorphic to a direct product of a simply connected manifold and a torus. In this paper, we extend this theorem to manifolds of almost nonnegative Ricci curvature.     

Manifolds of positive Ricci curvature,quadratically asymptotically nonnegative curvature,and infinite Betti numbers

Science China Mathematics - In a previous paper (Jiang and Yang (2021)), we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and...     

The gradient of certain harmonic functions on manifolds of almost nonnegative Ricci curvature

In this paper we prove that when the Ricci curvature of a Riemannian manifoldM ⁿ is almost nonnegative, and a ballB _L (p)⊂M ⁿ is close in Gromov-Hausdorff distance to a Euclidean ball, then the gradient of the harmonic functionb defined in [ChCo1] does not vanish. In particular, these functions can serve as harmonic coordinates on balls sufficiently close to an Euclidean ball. The proof, is based on a monotonicity theorem that generalizes monotonicity of the frequency for harmonic functions onR ⁿ .     

Collapsing sequences of solutions to the Ricci flow on 3-manifolds with almost nonnegative curvature

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilating about infinite time singularities. In this case we classify the possible Gromov-Hausdorff limits and construct 2-dimensional virtual limits. The virtual limits are constructed using Fukaya theory of the limits of local covers. We then show that the virtual limit arising from appropriate dilations of a Type IIb singularity is always Hamilton's cigar soliton solution. Partially supported by NSF grant DMS-0203926.     

Open manifolds of nonnegative curvature

A survey of results on the geometry and topology of open manifolds of nonnegative sectional curvature obtained to 1989 is given.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 21, pp.67–91, 1989.     

Topological obstructions to nonnegative curvature

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 metrics. Received February 11, 2000 / Published online February 5, 2001     

Complete open manifolds of nonnegative curvature

Siberian Mathematical Journal -     

Twisted submersions in nonnegative sectional curvature

In [16], 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 show that the same phenomenon often occurs for Riemannian submersions from nonnegatively curved spaces even without the strict positive curvature assumption and irrespective of the particular metric.     

Noncompact manifolds with nonnegative Ricci curvature

Let (M, d) be a metric space. For 0 < r < R, let G(p, r, R) be the group obtained by considering all loops based at a point p ∈ M whose image is contained in the closed ball of radius r and identifying two loops if there is a homotopy between them that is contained in the open ball of radius R. In this article we study the asymptotic behavior of the G(p, r, R) groups of complete open manifolds of nonnegative Ricci curvature. We also find relationships between the G(p, r, R) groups and tangent cones at infinity of a metric space and show that any tangent cone at infinity of a complete open manifold of nonnegative Ricci curvature and small linear diameter growth is its own universal cover.     

3-Manifolds with nonnegative Ricci curvature

For a complete noncompact 3-manifold with nonnegative Ricci curvature, we prove that either it is diffeomorphic to ?³ or the universal cover splits. This confirms Milnor’s conjecture in dimension 3.     

Manifolds with wells of negative curvature

Oblatum 25-XI-1989 & 15-II-1990     

Three-dimensional open Riemannian space of nonnegative curvature

Let V³ be a connected three-dimensional open complete Riemannian manifold with nonnegative sectional curvature. It is proved that if at some point all the sectional curvatures are positive, then V³ is diffeomorphic to a Euclidean space R³.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 103–113, 1976.     

Manifolds with pure non-negative curvature operator

We prove that if a simply connected compact Riemannian manifold has pure non negative curvature operator then its irreducible components (in the de Rham decomposition) are homeomorphic to spheres.     

Manifolds with invariant semi-Riemannian almost product structure

A manifold M with semi-Riemannian almost product structure invariant relative to a transformation group G is considered. A connection with special G-invariance property is constructed in the corresponding bundle of frames.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 926–931, July, 1992.     

Manifolds of nonpositive curvature and their buildings

Let M be a complete Riemannian manifold of bounded nonpositive sectional curvature and finite volume. We construct a topological Tits building Δ associated to the universal cover of M. If M is irreducible and rank (M)≥2, we show that Δ is a building canonically associated with a Lie group and hence that M is locally symmetric. Supported in part by NSF Grant MCS-82-04024 and M.S.R.I., Berkeley. Supported in part by NSF Grant DMS-84-01760 and M.S.R.I., Berkeley.