Torus actions, maximality, and non-negative curvature
The question of which closed smooth manifolds admit a Riemannian metric with positive or non-negative sectional curvature is a classical long-standing problem in Riemannian geometry. In the 1990's, Grove advocated for a specialization of this problem in the presence of a large symmetry group. A prototypical result in this direction is [\textit{K. Grove} and the second author, J. Pure Appl. Algebra 91, No. 1--3, 137--142 (1994; Zbl 0793.53040)], which bounds the rank of a torus acting on a closed simply connected manifold with positive sectional curvature and establishes an equivariant diffeomorphism classification when the bound is saturated. Many other topological classification results of varying strength have been established over the years and a generalisation of the result of Grove-Searle to the case of non-negative curvature was proposed in [\textit{F. Galaz-Garcia} and the second author, Proc. Am. Math. Soc. 139, No. 7, 2559--2564 (2011; Zbl 1220.53044)] as the Maximal Symmetry Rank Conjecture. The paper under review considers closed simply connected non-negatively curved Riemannian manifolds admitting an isometric \emph{isotropy-maximal} torus action. The main result of the paper (Theorem A) gives bounds for the rank of the torus in terms of the dimension and a classification up to equivariant diffeomorphism (any such manifold is equivariantly diffeomorphic to the quotient of a product of spheres of dimension larger or equal than 3 by a free linear action of a torus). Consequences of this result for the Maximal Symmetry Rank Conjecture in low dimensions are also discussed. The type of torus actions considered in the paper imply that there exists a sub-torus \(T'\) of the torus \(T\) acting on the non-negatively curved manifold \(M^n\), such that the quotient \(X^{2m}=M/T'\) is an even-dimensional orbifold with non-negative curvature (in the Alexandrov sense) and an isometric action of an \(m\)-torus \(T^m=T/T'\). Under a smoothness assumption, a classification of \(X\) up to equivariant diffeomorphism was given in [\textit{M. Wiemeler}, J. Lond. Math. Soc., II. Ser. 91, No. 3, 667--692 (2015; Zbl 1321.57043)], while a classification of such so-called \emph{torus orbifolds } (without any assumption on curvature) up to equivariant rational homotopic equivalence was given in [\textit{F. Galaz-García} et al., Int. Math. Res. Not. 2018, No. 18, 5786--5822 (2018; Zbl 1418.57021)]. In either of these results, the description of the orbit space \(X/T^m\) (as a manifold with corner and a stratification induced by the isotropy data) is an important tool. In the paper under review, the authors establish that under additional technical assumptions the orbit space \(M/T = X/T^m\) uniquely determines the equivariant diffeomorphism type of \(M\) (Theorem 3.15) and that these additional assumptions are always satisfied when \(M\) has non-negative sectional curvature (Theorem 6.1).
scientific article; zbMATH DE number 7436420
scientific article