Search

We prove that the group of isometries preserving a metric foliation on a closed Alexandrov space $X$, or a singular Riemannian foliation on a manifold $M$ is a closed subgroup of the isometry group of $X$ in the case of a metric foliation, or of the isometry group of $M$ for the case of a singular Riemannian foliation. We obtain a sharp upper bound for the dimension of these subgroups and show that, when equality holds, the foliations that realize this upper bound are induced by fiber bundles whose fibers are round spheres or projective spaces. Moreover, singular Riemannian foliations that realize the upper bound are induced by smooth fiber bundles whose fibers are round spheres or projective spaces., Comment: 20 pages, we thank gold essen.trinken in Karlsruhe for its excellent working conditions

We extend basic results in $3$-manifold topology to general three-dimensional Alexandrov spaces (or Alexandrov $3$-spaces for short), providing a unified framework for manifold and non-manifold spaces. We generalize the connected sum to non-manifold $3$-spaces and prove a prime decomposition theorem, exhibit an infinite family of closed, prime non-manifold $3$-spaces which are not irreducible, and establish a conjecture of Mitsuishi and Yamaguchi on the structure of closed, simply-connected Alexandrov $3$-spaces with non-negative curvature. Additionally, we define a notion of generalized Dehn surgery for Alexandrov $3$-spaces and show that any closed Alexandrov $3$-space may be obtained by performing generalized Dehn surgery on a link in $S^3$ or the non-trivial $S^2$-bundle over $S^1$. As an application of this result, we show that every closed Alexandrov $3$-space is homeomorphic to the boundary of a $4$-dimensional Alexandrov space., Comment: 24 pages, 6 figures

We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply-connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely-many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions., Comment: Minor changes, 29 pages

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature., Comment: Final version. To appear in the Bulletin of the London Mathematical Society

Under mild topological restrictions, this article establishes that a smooth, closed, simply connected manifold of dimension at most seven which can be decomposed as the union of two disk bundles must be rationally elliptic. In dimension five, such manifolds are classified up to diffeomorphism, while the same is true in dimension six when either the second Betti number vanishes or the third Betti number is non-trivial., Comment: 38 pages, final version. To appear in the Indiana University Journal of Mathematics

We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus Central Limit Theorem for Fr\'echet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case., Comment: Typos corrected

We classify closed, simply-connected cohomogeneity-one Alexandrov spaces in dimensions $5$, $6$ and $7$. We show that every closed, simply-connected smooth $n$-orbifold, $2\leq n\leq 7$ with a cohomogeneity one action is equivariantly homeomorphic to a smooth good orbifold of cohomogeneity one., Comment: 47 pages

We prove that sufficiently collapsed, closed and irreducible three-dimensional Alexandrov spaces are modeled on one of the eight three-dimensional Thurston geometries. This extends a result of Shioya and Yamaguchi, originally formulated for Riemannian manifolds, to the Alexandrov setting., Comment: We have added a proof of the fact that a closed collapsing Alexandrov 3-space cannot admit hyperbolic geometry (see Remark B). This improves our main Theorem (Theorem A) by ruling out the appearance of hyperbolic geometry. 24 pages, 3 Tables

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of $\mathsf{RCD}^{*}(K,N)$-spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An $\mathsf{RCD}^{*}(K,N)$-space is a metric measure space with an upper dimension bound $N$ and weighted Ricci curvature bounded below by $K$ in a generalized sense. In particular, this shows that if $(M,g)$ has Ricci curvature bounded below by $K\in \mathbb{R}$ and dimension $N$, then the quotient space is an $\mathsf{RCD}^{*}(K,N)$-space. Additionally, we tackle the same problem for the $\mathsf{CD}/\mathsf{CD}^*$ and $\mathsf{MCP}$ curvature-dimension conditions. We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogenous manifolds to $\mathsf{RCD}^{*}(K,N)$-spaces with essential minimal dimension $n\leq N$; a structure theorem for $\mathsf{RCD}^{*}(K,N)$-spaces admitting actions by large (compact) groups; and geometric rigidity results for orbifolds such as Cheng's Maximal Diameter and Maximal Volume Rigidity Theorems. Finally, in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of (super-)Ricci flows., Comment: 51 pages. Comments welcome!

We obtain a topological and equivariant classification of closed, connected three-dimensional Alexandrov spaces admitting a local isometric circle action. We show, in particular, that such spaces are homeomorphic to connected sums of some closed 3-manifold with a local circle action and finitely many copies of the suspension of the real projective plane., Comment: 23 pages, Final version to appear at the Kyoto Journal of Mathematics

A gap in the proof of the main result in reference [1] in our original submission propagated into the constructions presented in the first version of our manuscript. In this version we give an alternative proof for the existence of Riemannian metrics with positive Ricci curvature on an infinite subfamily of closed, simply-connected smooth manifolds with a cohomogeneity two torus action and recover some of our original results. Namely, we show that, for each $n\geqslant 1$, there exist infinitely many spin and non-spin diffeomorphism types of closed, smooth, simply-connected $(n+4)$-manifolds with a smooth, effective action of a torus $T^{n+2}$ and a metric of positive Ricci curvature invariant under a $T^{n}$-subgroup of $T^{n+2}$. As an application, we show that every closed, smooth, simply-connected $5$- and $6$-manifold admitting a smooth, effective torus action of cohomogeneity two supports metrics with positive Ricci curvature., Comment: 11 pages, to appear in Proceedings of the American Mathematical Society

We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $\mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $\mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of $\mathbb{R}P^2$. We then classify closed three-dimensional $\mathsf{CD}^*(0,3)$-Alexandrov spaces., Comment: 16 pages

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type., Comment: 29 pages; Final version. To appear in Math. Z

We prove a structure theorem for closed topological manifolds of cohomogeneity one; this result corrects an oversight in the literature. We complete the equivariant classification of closed, simply connected cohomogeneity one topological manifolds in dimensions $5$, $6$, and $7$ and obtain topological characterizations of these spaces. In these dimensions, these manifolds are homeomorphic to smooth manifolds., Comment: 22 pages. Added results on the homeomorphism type in the 7-dimensional case

We show that for any point $p$ in a closed Riemannian manifold $M$, there exists at least one point $q\in M$ such that $p$ is critical for the distance function from $q$. We also show that such a point $q$ cannot always be reached with geodesic loops based at $q$ with midpoint $p$., Comment: 7 pages, 2 figures. Minor changes. Accepted for publication in Calculus of Variations and Partial Differential Equations

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjecture in the presence of a slice-maximal torus action is proved., Comment: A gap in the first version has been fixed and the paper has been substantially expanded. Several new results have been added. The title and abstract have been changed accordingly. 36 pages, 1 figure

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspension of the real projective plane; we use this to classify, up to homeomorphism, closed, positively curved Alexandrov spaces of dimension three. Second, we classify closed three-dimensional Alexandrov spaces of nonnegative curvature. Third, we study the well-known Poincar\'e Conjecture in dimension three, in the context of Alexandrov spaces, in the two forms it is usually formulated for manifolds. We first show that the only three-dimensional Alexandrov space that is also a homotopy sphere is the 3-sphere; then we give examples of closed, geometric, simply connected three-dimensional Alexandrov spaces for five of the eight Thurston geometries, proving along the way the impossibility of getting such examples for the Nil, $\widetilde{\mathrm{SL}_2(\mathbb{R})}$ and Sol geometries. We conclude the paper by proving the analogue of the geometrization conjecture for closed three-dimensional Alexandrov spaces., Comment: 13 pages, a section on the geometrization of closed three-dimensional Alexandrov spaces has been added

We determine the structure of the fundamental group of the regular leaves of a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold. We also study closed singular Riemannian foliations whose leaves are homeomorphic to aspherical or to Bieberbach manifolds. These foliations, which we call A-foliations and B-foliations, respectively, generalize isometric torus actions on Riemannian manifolds. We apply our results to the classification problem of compact, simply connected Riemannian 4- and 5-manifolds with positive or nonnegative sectional curvature., Comment: 23 pages, 1 figure. Final version. Accepted for publication by the Journal of Topology

We point out that a 4-dimensional topological manifold with an Alexandrov metric (of curvature bounded below) and with an effective, isometric action of the circle or the 2-torus is locally smooth. This observation implies that the topological and equivariant classifications of compact, simply connected Riemannian 4-manifolds with positive or nonnegative sectional curvature and an effective isometric action of a circle or a 2-torus also hold if we consider Alexandrov manifolds instead of Riemannian manifolds., Comment: 10 pages, title changed, results are now proved for Alexandrov 4-manifolds, added examples, simplified proofs

We classify closed, simply connected $n$-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions $2\leq n\leq 6$. In dimensions $3k$, $k=1,2$ there is only one such manifold and it is diffeomorphic to the product of $k$ copies of the 3-sphere.

We show that any effective isometric torus action of maximal rank on a compact Riemannian manifold with positive (sectional) curvature and maximal symmetry rank, that is, on a positively curved sphere, lens space, complex or real projective space, is equivariantaly diffeomorphic to a linear action. We show that a compact, simply connected Riemannian 4- or 5-manifold of quasipositive curvature and maximal symmetry rank must be diffeomorphic to the 4-sphere, complex projective plane or the 5-sphere.

We show that a closed, simply-connected, non-negatively curved 5-manifold admitting an effective, isometric $T^2$ action is diffeomorphic to one of $S^5$, $S^3\times S^2$, $S^3\tilde{\times} S^2$ (the non-trivial $S^3$-bundle over $S^2$) or the Wu manifold $SU(3)/SO(3)$., Comment: We fix an omission in a previous posting (arXiv:0906.3870v1 [math.DG])

Let $M^n$, $n \in \{4,5,6\}$, be a compact, simply connected $n$-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $M^n$ by a torus $T^{n-2}$ is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

For an Alexandrov space (with curvature bounded below), we determine the maximal dimension of its isometry group and show that the space is isometric to a Riemannian manifold, provided the dimension of its isometry group is maximal. We also determine a gap in the possible dimensions of the isometry groups and show that if the Alexandrov space is symmetric, then it is isometric to a Riemannian manifold., Comment: Final version of the paper, accepted for publication at the Bulletin of the London Mathematical Society. It improves the proof of the Principal Orbit Theorem in section 2, and completes the missing cases in section 6 on dimension gaps

Let $G$ be a compact Lie group acting effectively by isometries on a compact Riemannian manifold $M$ with nonempty fixed point set $Fix(M,G)$. We say that the action is \emph{fixed point homogeneous} if $G$ acts transitively on a normal sphere to some component of $Fix(M,G)$, equivalently, if $Fix(M,G)$ has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group., Comment: 12 pages

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure., Comment: To appear in Geom. Dedicata. Main difference with v1: "Linear" circle actions on $S^2\times S^2$ and $CP^2\#\pm CP^2$ are now called "extendable actions". Typos corrected

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric $T^{n-1}$ action. In contrast to the 1- and 2-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of $RP^2$, which is not a manifold., Comment: 15 pages; Proposition 2.2 in v1 (now Proposition 4) has been modified. An omission in the classification of cohomogeneity one 4-manifolds was corrected

We classify closed, simply-connected non-negatively curved 5-manifolds admitting an (almost) effective, isometric $T^3$ or $T^2$ action. As a direct consequence, we show that for any manifold, of dimensions up to and including 9 under the same hypotheses, the maximal symmetry rank is equal to $[2n/3]$ and the free rank is less than or equal to one half that value., Comment: This paper has been revised and split into two papers. The first one ("Low-dimensional manifolds with non-negative curvature and maximal symmetry rank", Proc. A.M.S. 139 (2011), no. 7, 2559--2564) treats maximal symmetry rank. The second one (arXiv:1111.3183v1 [math.DG]) treats almost maximal sym. rank in dimension 5 and fixes an omission in the withdrawn submission (arXiv:0906.3870v1 [math.DG])