Related papers: Nilpotency, almost nonnegative curvature and the g…

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at…

We consider the following properties of compact oriented irreducible graph-manifolds: to contain a $\pi_1$-injective surface (immersed, virtually embedded or embedded), be (virtually) fibered over $S^1$, and to carry a metric of nonpositive…

We prove that every homotopy class of almost contact structures on a closed 5-dimensional manifold admits a contact structure.

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold…

It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.

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…

In [Bre19], Simon Brendle showed that any compact manifold of dimension $n\geq12$ with positive isotropic curvature and contains no nontrivial incompressible $(n-1)-$dimensional space form is diffeomorphic to a connected sum of finitely…

We classify compact almost-K\"ahler four manifolds with nonnegative biorthogonal curvature.

Let M be a graph manifold. We show that \pi_1M is the fundamental group of a compact nonpositively curved cube complex if and only if M is chargeless. We also prove that in that case \pi_1M is virtually compact special.

We study non-positively curved closed manifolds $M$ and $n$-dimensional totally geodesic submanifolds of $M \times M$ which satisfy a transversality condition. We prove that, under some mild irreducibility requirements on $M$, if $M \times…

We establish positive mass type theorems for asymptotically locally flat (ALF) manifolds, which have asymptotic ends modeled on circle bundles over a Euclidean base with fibers of constant length. In particular for dimensions $n\leq 7$, the…

We show the existence of a family of manifolds on which all (pointwise or absolutely) partially hyperbolic systems are dynamically coherent. This family is the set of 3-manifolds with nilpotent, non-abelian fundamental group. We further…

A nilpotent quandle is a quandle whose inner automorphism group is nilpotent. Such quandles have been called reductive in previous works, but it turns out that their behaviour is in fact very close to nilpotency for groups. In particular,…

We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.

We give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups, namely, they are precisely the M\"obius band, and the products of a line with the total spaces of flat vector bundles over…

All the connections, pure toward the nilpotent structure, are found. Examples of manifolds, for which the curvature tensor is pure or hybrid, are given. For a manifold of B-type a necessary and sufficient condition for purity of the…

Let $M=V\setminus D$ be a smooth quasi-projective variety for some smooth projective variety $V$ and a divisor $D$ with normal crossings. Assume that $M$ is diffeomorphic to a non-compact nilmanifold $\Gamma\backslash N\times\mathbb{R}^m$.…

The class of the Riemannian almost product manifolds with nonintegrable structure is considered. Some identities for curvature tensor as certain invariant tensors and quantities are obtained.

We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of…

We classify $n$-dimensional geometric graph manifolds with nonnegative scalar curvature, and first show that if $n>3$, the universal cover splits off a codimension 3 Euclidean factor. We then proceed with the classification of the…