Related papers: An exotic sphere with positive sectional curvature
The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the…
We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan-Baas singularities. The manifolds we consider are Spin and simply connected. We prove an analogue of the…
This paper extends widely the work in \cite{GT13}. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy $n$-sphere ($n>4$) carries…
In this paper, we consider the problem when a differential equation y"(z)=Q(z)y(z) is Fuchsian on H* and apparent on H, where Q(z) is a meromorphic modular form of weight 4 on SL(2,Z) and H denotes the complex upper half-plane. Such a…
We use certain Morse functions to construct conformal metrics such that the eigenvalue vector of modified Schouten tensor belongs to a given cone. As a result, we prove that any Riemannian metric on compact 3-manifolds with boundary is…
We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved $2$-spheres.
In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and…
We prove a formula involving the scalar curvature of a Riemannian manifold endowed with a distribution in terms of an adapted orthonormal frame for its tangent bundle. Using the formula, we then investigate the effect of collapsing the…
In this article, we investigate the geometry of a central extension $\widehat{\mathcal D}_{\mu}(S^{2})$ of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with the $L^{2}$-metric, whose geodesics correspond solutions…
We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ such that the quotient space can be…
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…
In a 2013 paper, Gromov proves that if smooth Riemannian metrics $g_i$ converge to a smooth Riemannian metric $g$ uniformly, and $g_i$ have scalar curvature uniformly bounded below, then $g$ shares the same scalar curvature lower bound. In…
We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements of infinite order in higher homotopy and homology groups of these spaces, which,…
We generalize the famous result of Gromov and Lawson on the nonexistence of metric of positive scalar curvature on enlargeable manifolds to the case of foliations, without using index theorems on noncompact manifolds.
We prove a Geroch type result for isolated conical singularity. Namely, we show that there is no Riemannian metric $g$ on $ X \# T^n $ with an isolated conical singularity which has nonnegative scalar curvature on the regular part, and is…
In this paper we study the behaviour of the holomorphic sectional curvature (or Gaussian curvature) of the Bergman metric of planar annuli. The results are then utilized to construct a domain for which the curvature is divergent at one of…
Consider a compact Riemannian manifold M of dimension n whose boundary \partial M is totally geodesic and is isometric to the standard sphere S^{n-1}. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least…
The present paper shows that for a given integer k greater than 2 it is possible to construct an at least k-differentiable Riemannian metric on the sphere of a certain dimension such that the cut locus of a point of it becomes a fractal.…
We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products. Using this theory we construct positive scalar…
Inspired by Goette-Semmelmann \cite{GSSU2002}, we derive an estimate for the scalar curvature without a nonnegativity assumption on curvature operator. As an application, we show that, on an even dimensional closed manifold with nonzero…