Let $M^4$ be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, $S^2 \times S^2$. We prove that $M$ is a direct metric product of two spheres endowed with some polyhedral metrics. In other words, $M$ is a direct metric product of the surfaces of two convex polyhedra in $\mathbb{R}^3$.
The background for the question is the following. The classical H.Hopf's hypothesis states: for any Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature the curvature cannot be strictly positive at all points. There is no quick answer to this question: it is known that a Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature need not be a product metric. However, M.Gromov has pointed out that the condition of nonnegative curvature in the PL-case appears to be stronger than nonnegative sectional curvature of Riemannian manifolds and analogous to some condition on the curvature operator. So the motivation for the question addressed in this text is to settle the PL-version of the Hopf's hypothesis.
We show that in each dimension $n\ge 10$ there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian $n$-manifolds with $0\le \sec\le 1$, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non-homeomorphic. Finally, we construct examples of noncompact manifolds whose moduli spaces of complete metrics with $\sec\ge 0$ have infinitely many co…
We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on surfaces of positive Euler characteristic equipped with the topology of $C^γ$ uniform convergence on compact sets, when $γ$ is infinite or is not an integer. If $γ=\infty$, the space of metrics is homeomorphic to the separable Hilbert space. If $γ$ is finite and not an integer, the space of metrics is homeomorphic to the countable power of the linear span of the Hilbert cube. We also prove similar results for some other spaces of metrics including the space of complete smooth Riemannian metrics …
We determine the structure of the fundamental group of the regular leaves of a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold. We also study closed singular Riemannian foliations whose leaves are homeomorphic to aspherical or to Bieberbach manifolds. These foliations, which we call A-foliations and B-foliations, respectively, generalize isometric torus actions on Riemannian manifolds. We apply our results to the classification problem of compact, simply connected Riemannian 4- and 5-manifolds with positive or nonnegative sectional curvature.
A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S^4 or diffeomorphic to CP^2. This conclusion stills holds true if the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant. In particular, the Hopf conjecture on S^2 x S^2 holds in this c…
We study the space of complete Riemannian metrics of nonnegative curvature on the plane equipped with the C^k topology. If k is infinite, we show that the space is homeomorphic to the separable Hilbert space. For any k we prove that the space cannot be made disconnected by removing a finite dimensional subset. A similar result holds for the associated moduli space. The proof combines properties of subharmonic functions with results of infinite dimensional topology and dimension theory. A key step is a characterization of the conformal factors that make the standard Euclidean metric on the plan…
For each odd integer r greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise non-homeomorphic closed homogeneous spaces with fundamental group isomorphic to Z/r. As an application we construct the first examples of manifolds which possess infinitely many metrics of nonnegative sectional curvature with pairwise non-homeomorphic homogeneous souls of codimension three with trivial normal bundle, such that their curvatures and the diameters of the souls are uniformly bounded. These manifolds are the fi…
We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $\mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $\mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of $\mathbb{R}P^2$. We then classify closed three-dimensional $\mathsf{CD}^*(0,3)$-Alexandrov spaces.
We show that the moduli space of nonnegatively curved metrics on each member of a large class of 2-connected 7-manifolds, including each smooth manifold homeomorphic to $S^7$, has infinitely many connected components. The components are distinguished using the Kreck-Stolz $s$-invariant computed for metrics constructed by Goette, Kerin and Shankar. The invariant is computed by extending each metric to the total space of an orbifold disc bundle and applying generalizations of the Atiyah-Patodi-Singer index theorem for orbifolds with boundary. We develop methods for computing characteristic class…
We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of nonnegative sectional curvature.
Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $Σ$ be a compact connected and orientable surface immersed in $M$ which is a stable constant mean curvature (CMC) surface or an index one minimal surface. We prove that $Σ$ is homeomorphic either to a sphere or to a torus. Moreover, in case $Σ$ is homeomorphic to a torus, then it is embedded, minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unit …
For each nonnegative integer we find an open (4m+9)-dimensional simply-connected manifold admitting complete nonnegatively curved metrics whose souls are non-diffeomorphic, homeomorphic, and have codimension 2. We give a diffeomorphism classification of the pairs (N, soul) when N is a nontrivial complex line bundle over the product of 7-sphere and complex projective plane: up to diffeomorphism there are precisely three such pairs, distinguished by their non-diffeomorphic souls.
In this paper, we construct local and global solutions to the Kähler-Ricci flow from a non-collapsed Kähler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed Kähler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete Kähler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and m…
We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both $\mathbb{CP}^2$ and $\mathbb{S}^3 \times \mathbb{S}^1$ have $4\frac{1}{2}$-nonnegative curvature operator of the second kind. In higher dimensions $n\geq 5$, we show that closed Riemannian manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that $4\frac{1}{2}$-positive curvature operator of the second kind…
Let T(γ) be the total space of the canonical line bundle γover CP^1 and r an integer which is greater than one and coprime to six. We prove that L_r^3\times T(γ) admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where L_r^3 is the standard 3-dimensional lens space with fundamental group isomorphic to Z/r. We classify the total spaces of S^1-fibre bundles over S^2\times S^2 with fundamental group isomorphic to Z/r up to diffeomorphism and use these results to give examples of manifolds N which admit two complete metrics of nonnegativ…
We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspension of the real projective plane; we use this to classify, up to homeomorphism, closed, positively curved Alexandrov spaces of dimension three. Second, we classify closed three-dimensional Alexandrov spaces of nonnegative curvature. Third, we study the well-known Poincaré Conjectur…
