Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension

Abstract

Let M n , n ∈ {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, nonnegatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.The classification of (compact) Riemannian manifolds (M n , g) with positive or nonnegative (sectional) curvature is a notoriously difficult problem. One of the few successes in this quest occurs when one considers such manifolds equipped with an effective action by a suitably large group G of isometries. The ambiguity of the term "suitably large" allows various classification results to be achieved (see, for example, [BB, GS1, GS2, GWZ, Ve, Wa, Wi1, Wi2] and the surveys [Gr,Wi3]).There are, in fact, two parts to the classification program. First is the topological classification, the goal of which is to determine, up to diffeomorphism, all possible positively or non-negatively curved manifolds on which G can act. The second part is the equivariant classification, where the goal is to determine all possible actions of G on a given positively or non-negatively curved manifold up to equivariant diffeomorphism.This approach to the classification problem is inspired by the work of Hsiang and Kleiner [HK]. They showed that a simply connected, four-dimensional manifold (M 4 , g) with positive curvature admitting an effective, isometric circle action must be homeomorphic to either S 4 or to CP 2 . If (M 4 , g) is assumed to have only non-negative curvature then, by Kleiner [Kl] and Searle and Yang [SY], M 4 must be homeomorphic to one of S 4 , CP 2 , CP 2 # ± CP 2 or S 2 × S 2 . In both situations, homeomorphism is improved to diffeomorphism by appealing to work of Fintushel [Fi1, Fi2], Pao [Pao] and Perelman's proof of the Poincaré conjecture [Pe1, Pe2].The existence of an effective, isometric circle action is equivalent to the rank of the isometry group Iso(M n , g) being positive. The success in dimension four suggests that it may be beneficial to consider the topological classification when "largeness" of our group G of isometries refers to the symmetry rank of (M n , g), defined as the rank of Iso(M n , g) and denoted symrank(M n , g). Indeed, Grove and Searle [GS1] showed that if (M n , g) is positively curved, then symrank(M n , g) ⌊ n+1 2 ⌋. Moreover, if the symmetry

Torus manifolds and non-negative curvature

“…Furthermore, a simply connected non-negatively curved four-dimensional torus manifold is diffeomorphic to one of the manifolds in the above list (see [11,13,16,22]). Moreover, by [6] or [10], the T 2 -actions on these spaces are always equivalent to a torus action as described above.…”

A Note on Maximal Symmetry Rank, Quasipositive Curvature, and Low Dimensional Manifolds

“…(b) Topological classification: Classify, up to diffeomorphism, all manifolds in M n with symmetry rank k ≤ K. These problems have received particular attention when M n is the class of compact, positively curved n-manifolds or the class of compact, simply connected nmanifolds of nonnegative curvature (cf. [12,13,14,17,23,26,41,42,50,51]). In a curvature-free setting, analogs of problems (a), (b) and (c) for compact, simply connected smooth n-manifolds, 3 ≤ n ≤ 6, have also been extensively studied (cf.…”

Torus actions on rationally elliptic manifolds

“…On the other hand, since a maximal effective torus action is of rank two (i.e. of cohomogeneity two), the classification of such actions up to equivariant diffeomorphism follows from the results in [14] and [18].…”