On manifolds with positive curvature

The symmetry rank of a Riemannian manifold is the rank of the isometry group. We determine precisely which closed simply connected 5-manifolds admit positively curved metrics with (almost maximal) symmetry rank two. We also determine the precise Euler characteristic and the fundamental groups of all closed positively curved n-manifolds with almost maximal symmetry rank [(n−1)/2] (n≠ 6, 7).

Positively Curved Manifolds with Almost Maximal Symmetry Rank

A π1 -invariant torus-action on a manifold M is a T k -action on the universal covering which extends to the action of a semi-direct product π1 ( M )×ρ T k . In particular, the T k -action is the lift of a T k -action on M if ρ is the identity map. The main result asserts that if a compact manifold M n of positive sectional curvature admits a π1 -invariant isometric T k -action, then the fundamental group has a cyclic subgroup of index ≤ w ( n ). This refines the main result in [Ro1].

Positive curvature, local and global symmetry, and fundamental groups

Let M be a closed simply connected manifold and 0 < 
$ \delta \le 1 $
 . Klingenberg and Sakai conjectured that there exists a constant 
$ {i_0} = {i_0}(M,\delta) > 0 $
 such that the injectivity radius of any Riemannian metric g on M with 
$ \delta \le {K_g} \le 1 $
 can be estimated from below by i

 0. We study this question by collapsing and Alexandrov space techniques. In particular we establish a bounded version of the Klingenberg-Sakai conjecture: Given any metric d

 0 on M, there exists a constant 
$ {i_0} = {i_0}(M,{d_0},\delta) > 0 $
 , such that the injectivity radius of any 
$ \delta $
 -pinched d

 0-bounded Riemannian metric g on M (i.e., 
$ {\rm dist}_g \le d_0 $
 and 
$ \delta \le K_g \le 1) $
 can be estimated from below by i

 0. We also establish a continuous version of the Klingenberg-Sakai conjecture, saying that a continuous family of metrics on M with positively uniformly pinched curvature cannot converge to a metric space of strictly lower dimension.

/pdf/collapsing-vs-positive-pinching-34dcqghfy5.pdf

Collapsing vs. positive pinching

In this paper, we study the fundamental groups of closed manifolds of positive sectional curvature which admit compatible local isometric torus T -actions. We explore relations between basic properties of an isometric T -action and the structure of the fundamental group of M . Using these relations, we prove several results on the fundamental groups.

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/2005/0009/0004/AJM-2005-0009-0004-a006.pdf

On Fundamental Groups of Positively Curved Manifolds with Local Torus Actions

Let M be a closed 5-manifold of pinched curvature 0 < δ ≤ secM ≤ 1. We prove that M is homeomorphic to a spherical space form if M satisfies one of the following conditions: (i) δ = 1/4 and the fundamental group is a non-cyclic group of order ≥ C, a constant. (ii) The center of the fundamental group has index ≥ w(δ), a constant depending on δ. (iii) The ratio of the volume and the maximal injectivity radius is < ǫ(δ). (iv) The volume is less than ǫ(δ) and the fundamental group π1(M) has a center of index at least w, a universal constant, and π1(M) is either isomorphic to a spherical 5-space group or has an odd order.

/pdf/collapsed-manifolds-with-pinched-positive-sectional-jtazu7kc32.pdf

Collapsed manifolds with pinched positive sectional curvature

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Collapsed manifolds with pinched positive sectional curvature