arXiv subjects
Publications and source records attributed to Curtis Pro.
We prove that for any $k\in \mathbb{R},$ $v>0,$ and $D>0$ there are only finitely many diffeomorphism types of closed Riemannian $4$-manifolds with sectional curvature $\geq k,$ volume $\geq v,$ and diameter $\leq D.$
Let $Y$ be a subset of a metric space $X.$ We say that $Y$ is $\eta $-Gromov provided $Y$ is $\eta $-separated and not properly contained in any other $\eta $-separated subset of $X.$ In this paper, we review a result of Chew which says that any $\eta $-Gromov subset of $\mathbb{R}^{2} $ admits a triangulation $\mathcal{T}$ whose smallest angle is at least $\pi /6 $ and whose edges have length between $\eta $ and $2\eta .$ We then show that given any $k = 1,2,3\ldots$, there is a subdivision $\mathcal{T} _{k}$ of $\mathcal{T}$ whose edges have length in $\left[ \frac{\eta}{10 k},\frac{2\eta}{10 k} \right] $ and whose minimum angle is also $\pi /6$. These results are used in the proof of the following theorem in [10]: For any $k\in R,v>0,$ and $D>0,$ the class of closed Riemannian $4$-manifolds with sectional curvature $\geq k,$ volume $\geq v,$ and diameter $\leq D$ contains at most finitely many diffeomorphism types. Additionally, these results imply that for any $\varepsilon >0$, if $\eta >0$ is sufficiently small, any $\eta $-Gromov subset of a compact Riemannian $2$-manifold admits a geodesic triangulation $\mathcal{T}$ for which all side lengths are in $\left[ \eta \left( 1-\varepsilon \right) ,2\eta \left( 1+\varepsilon \right) \right] $ and all angles are $\geq \frac{\pi }{6}-\varepsilon .$
We provide an alternative proof that Crosscaps are diffeomorphically stable.
Given $k\in \mathbb{R},$ $v,$ $D>0,$ and $n\in \mathbb{N},$ let $\left\{ M_{\alpha }\right\} _{\alpha =1}^{\infty }$ be a Gromov-Hausdorff convergent sequence of Riemannian $n$--manifolds with sectional curvature $\geq k,$ volume $>v,$ and diameter $\leq D.$ Perelman's Stability Theorem implies that all but finitely many of the $M_{\alpha }$s are homeomorphic. The Diffeomorphism Stability Question asks whether all but finitely many of the $ M_{\alpha }$s are diffeomorphic. We answer this question affirmatively in the special case when all of the singularities of the limit space occur along smoothly and isometrically embedded Riemannian manifolds of codimension $\leq 3$. We then describe several applications. For instance, if the limit space is an orbit space whose singular strata are of codimension at $\leq 3,$ then all but finitely many of the $M_{\alpha }$s are diffeomorphic.
We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds $M$ with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by Grove and Petersen and by Sill, Wilhelm, and the author by showing any such $M$ that has volume sufficiently close to this upper bound is diffeomorphic to the standard sphere $S^{n}$ or a standard lens space $S^n/\mathbb{Z}_m$ where $m\in\{2,3,\ldots\}$ is no larger than an a priori constant.
If $\pi :M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from manifolds with positive Ricci curvature to manifolds with nonpositive Ricci curvature.
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 sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.
We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp's theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.