Related papers: The Degree Theorem in higher rank

The Hopf theorem states that homotopy classes of continuous maps from a closed connected oriented smooth $n$-manifold $M$ to the $n$-sphere are classified by their degree. Such a map is equivalent to a section of the trivial $n$-sphere…

Let $X$ be a locally symmetric space $\Gamma\backslash G/K$ where $G$ is a connected non-compact semisimple real Lie group with trivial centre, $K$ is a maximal compact subgroup of $G$, and $\Gamma\subset G$ is a torsion-free irreducible…

Let $f:G_{n,k}\longrightarrow G_{m,l}$ be any continuous map between any two distinct complex Grassmann manifolds of the same dimension where the target is not the complex projective space. We show that, for any given $k,l$, the degree of…

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we…

In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold $(M^{n},g)$ with scalar curvature $R_{g}\geq 6$ admits a non-zero degree and $1$-Lipschitz map to…

We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function $\Delta_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than…

We address two fundamental and well-known problems of Gromov and Lyndon: \demo{Problem A} (Gromov, see [5]). Consider a category $M_n$ of closed manifolds of dimension $n$ with nonzero-degree ways as morphisms. Study a partial order $M \ge…

A natural problem in the theory of 3-manifolds is the question of whether two 3-manifolds are homeomorphic or not. The aim of this paper is to study this problem for the class of closed Haken manifolds using degree one maps. To this purpose…

For oriented connected closed manifolds of the same dimension, there is a transitive relation: $M$ dominates $N$, or $M \ge N$, if there exists a continuous map of non-zero degree from $M$ onto $N$. Section 1 is a reminder on the notion of…

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the identity map on…

Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map $f: M\to S^n$ from a closed connected Riemannian spin manifold $M$ with scalar curvature $\mathrm{scal}\ge n(n-1)$ to the standard sphere $S^n$ is an isometry if the…

Let $M^n$ be a closed manifold of almost nonnegative sectional curvature and nonzero first de Rham cohomology group. For any $[\theta] \in H^1_{dR}(M^n), [\theta] \neq 0$, we show that the Morse- Novikov cohomology group $H^p(M^n, \theta)$…

In this note, we prove that if a compact even dimensional manifold $M^{n}$ with negative sectional curvature is homotopic to some compact space-like manifold $N^{n}$, then the Euler characteristic number of $M^{n}$ satisfies…

Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\tilde M^n$ contains no totally…

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p in M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact…

In the paper we investigate the degree and the homotopy theory of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ between manifolds, where the Young function $P$ satisfies a divergence condition and forms a slightly larger space than $W^{1,n}$,…

Let $(M,g^{TM})$ be a noncompact complete Riemannian manifold of dimension $n$, and let $F\subseteq TM$ be an integrable subbundle of $TM$. Let $g^F=g^{TM}|_{F}$ be the restricted metric on $F$ and let $k^F$ be the associated leafwise…

We study complete, finite volume $n$-manifolds $M$ of bounded nonpositive sectional curvature. A classical theorem of Gromov says that if such $M$ has negative curvature then it is homeomorphic to the interior of a compact…

We show there exists a closed locally symmetric manifold $M$ modeled on $SL_n(\mathbb R)/SO(n)$, and a non-trivial homology class in degree $dim(M)-rank(M)$ represented by a totally geodesic submanifold that contains a circle factor. As a…

The energy of any $C^1$ representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant…