Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces. (English) Zbl 1192.53040

Kapovitch, Vitali; Petrunin, Anton; Tuschmann, Wilderich

Ann. Math. (2) 171, No. 1, 343-373 (2010).

A closed smooth manifold \(M\) of dimension \(m\) is said to be almost nonnegatively curved if it can Gromov-Hausdorff converge to a single point under a lower curvature bound, i.e., if it admits a sequence of Riemannian metrics with sectional curvatures bounded from below by \(-1/n\) and diameters bounded from above by \(1/n\). The authors show:
Theorem A. Let \(M\) be a closed almost nonnegatively curved manifold. Then a finite cover of \(M\) is a nilpotent space. The authors also provide an affirmative answer to a conjecture of K. Fukaya and T. Yamaguchi [Ann. Math. (2) 136, No. 2, 253–333 (1992; Zbl 0770.53028)].
Theorem B. Let \(M\) be an almost nonnegatively curved \(m\)-manifold. Then \(\pi_1(M)\) contains a nilpotent subgroup of index at most \(C(m)\). The authors also establish a fibration theorem.
Theorem C. Let \(M\) be an almost nonnegatively curved manifold. Then a finite cover \(\tilde M\) of \(M\) is the total space of a fiber bundle \(F\rightarrow\tilde M\rightarrow N\) over a nilmanifold \(N\) with simply connected fiber \(F\). Moreover, the fiber \(F\) is almost nonnegatively curved in a generalized sense.
The authors use Alexandrov geometry and the gradient flow of the square of a distance function. They use “limit fundamental groups of Alexandrov spaces” and discuss some open questions.

Reviewer: Peter B. Gilkey (Eugene)

MSC:

53C20

Global Riemannian geometry, including pinching

Keywords:

Alexandrov gometry; gradient flow; almost nonnegatively curved manifold; nilpotent space

Citations:

Zbl 0770.53028

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