相关论文: Maximal diameter sphere theorem for manifolds with…

We prove that if a complete connected $n$-dimensional Riemannian manifold $M$ has radial sectional curvature at a base point $p\in M$ bounded from below by the radial curvature function of a two-sphere of revolution $\widetilde M$ belonging…

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold $M$ with a base point $p \in M$ whose radial curvature at $p$ is bounded from below by that of a non-compact model surface of…

The survey is devoted to Toponogov's conjecture, that {\it if a complete simply connected Riemannian manifold with sectional curvature $\le 4$ and injectivity radius $\ge \pi/2$ has extremal diameter $\pi/2$, then it is isometric to CROSS}.…

In this paper, we prove that, for any integer $n\ge 2,$ there exists an $\epsilon_{n} \ge 0$ so that if $M$ is an n-dimensional complete manifold with sectional curvature $ K_{M}\ge 1$ and if $M$ has conjugate radius bigger than…

We extend the classical theory of sphere theorems to the transverse geometry of Riemannian foliations. In this setting, we establish transverse analogues of the Grove-Shiohama diameter sphere theorem and of the Berger-Klingenberg…

The smallest $r$ so that a metric $r$-ball covers a metric space $M$ is called the radius of $M$. The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with…

We prove the following rigidity theorem: For an n-dimensional compact Riemannian manifold with boundary whose Ricci curvature is bounded by n-1 from below, if its boundary is isometric to the standard sphere of dimension n-1 and totally…

We will construct surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we…

We show two sphere theorems for the Riemannian manifolds with scalar curvature bounded below and the non-collapsed $\mathrm{RCD}(n-1,n)$ spaces with mean distance close to $\frac{\pi}{2}$.

In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says that if a complete $n$-manifold $M$ of Ricci curvature, $\operatorname{Ric}_M\ge (n-1)$, has the maximal diameter $\pi$, then $M$ is isometric to the unit sphere…

A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with…

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that…

For three dimensional complete Riemannian manifolds with scalar curvature no less than one, we obtain the sharp upper bound of complete stable minimal surfaces' diameter.

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is…

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 this paper, we successfully set up a generalized sphere theorem for compact Riemannian manifolds with radial Ricci curvature bounded.

We prove that, for a Finsler space, if the weighted Ricci curvature is bounded below by a positive number and the diam attains its maximal value, then it is isometric to a standard Finsler sphere. As an application, we show that the first…

For a Riemannian manifold $M^{n+1}$ and a compact domain $\Omega \subset M^{n+1}$ bounded by a hypersurface $\partial \Omega$ with normal curvature bounded below, estimates are obtained in terms of the distance from $O$ to $\partial \Omega$…

In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend…

We show that spheres in all dimensions $\geq3$ can be deformed to have diameter larger than the distance between any pair of antipodal points. This answers a question of Yurii Nikonorov.