##
**Hard and soft packing radius theorems.**
*(English)*
Zbl 0846.53042

For a compact metric space \(X\), let \(\text{pack}_q X\) denote the \(q\)th packing radius of \(X\), that is
\[
2 \text{pack}_qX = \max_{(x_1, \dots, x_q) \in X^q} \min_{i < j} \text{dist} (x_i, x_j),
\]
where the maximum is taken over all configurations of \(q\) points in \(X\), and the minimum is the minimum of all pairwise distances in such a configuration. Now assume that \(X\) is an \(n\)-dimensional Alexandrov space with curvature \(\geq 1\). Then \(\text{pack}_qX \leq \text{pack}_q S^n_1\) for all integers \(q \geq 2\), where \(S^n_1\) is the unit sphere. In particular, this inequality hold for any closed Riemannian \(n\)-manifold with sectional curvature \(\geq 1\) (or Ricci curvature \(\geq n - 1)\). The authors prove the following geometric (resp. topological) join theorem:

If \(\text{pack}_qX = \text{pack}_q S^n_1\) (resp. \(\text{pack}_q X > {\pi \over 4})\) for some \(2 \leq q \leq n + 2\), then \(X\) is isometric (resp. homeomorphic) to the \((q - 1)\)-fold spherical suspension \(\sum_1^{q - 1} E \equiv S^{q - 2}_1* E\) [resp. a \((q - 1)\)-fold topological suspension \(\sum^{q - 1} E \cong S^{q - 2}* E]\) of an \((n - q + 1)\)-dimensional Alexandrov space \(E\) with curve \(E \geq 1\).

The key tool in the proof of the geometric version of the theorem is a rigidity distance comparison theorem [see K. Grove and S. Markvorsen, J. Am. Math. Soc. 8, No. 1, 1-28 (1995; Zbl 0829.53033) appendix], whereas the critical point theory for distance functions in Alexandrov spaces [see Yu. Burago, M. Gromov and G. Perelman, Russ. Math. Surv. 47, No. 2, 1-58 (1992); translation from Usp. Mat. Nauk 47, No. 2(284), 3-51 (1992; Zbl 0802.52018)] is used to prove the topological version.

Also, the following packing radius sphere theorem is proved: If \(n \geq 3\), then any closed Riemannian \(n\)-manifold \(M\) with sectional curvature \(\geq 1\) and \(\text{pack}_{n - 1} M > {\pi \over 4}\) is diffeomorphic to \(S^n\). The proof is based on ideas from Alexandrov geometry, namely the global Riemannian problem associated with the theorem is changed to a local problem in Alexandrov geometry. This result generalizes differentiable sphere theorems of Y. Otsu, K. Shiohama and T. Yamaguchi [Invent. Math. 98, No. 2, 219-228 (1989; Zbl 0688.53016)], C. Plaut [Spaces of Wald curvature bounded from below, J. Geom. Anal., to appear), K. Shiohama and T. Yamaguchi [in Geometry of manifolds, Coll. Pap. 35th Symp. Differ. Geom., Matsumoto/Japan 1988, Perspect. Math. 8, 345-350 (1989; Zbl 0697.53041)], F. H. Wilhelm jun. [Invent. Math. 107, No. 3, 653-668 (1992; Zbl 0739.53036); Indiana Univ. Math. J. 41, No. 1, 1119-1142 (1992; Zbl 0771.53028)], J.-Y. Wu [in Proc. Symp. Pure Math. 54, Part 3, 685-692 (1993; Zbl 0804.53063)], since the hypotheses of these results imply that \(M\) is close to \(S^n_1\) in the Gromov-Hausdorff topology and consequently that \(\text{pack}_{n - 1} M > {\pi \over 4}\). Two corollaries of the authors’ version of the differentiable sphere theorem are pointed out.

If \(\text{pack}_qX = \text{pack}_q S^n_1\) (resp. \(\text{pack}_q X > {\pi \over 4})\) for some \(2 \leq q \leq n + 2\), then \(X\) is isometric (resp. homeomorphic) to the \((q - 1)\)-fold spherical suspension \(\sum_1^{q - 1} E \equiv S^{q - 2}_1* E\) [resp. a \((q - 1)\)-fold topological suspension \(\sum^{q - 1} E \cong S^{q - 2}* E]\) of an \((n - q + 1)\)-dimensional Alexandrov space \(E\) with curve \(E \geq 1\).

The key tool in the proof of the geometric version of the theorem is a rigidity distance comparison theorem [see K. Grove and S. Markvorsen, J. Am. Math. Soc. 8, No. 1, 1-28 (1995; Zbl 0829.53033) appendix], whereas the critical point theory for distance functions in Alexandrov spaces [see Yu. Burago, M. Gromov and G. Perelman, Russ. Math. Surv. 47, No. 2, 1-58 (1992); translation from Usp. Mat. Nauk 47, No. 2(284), 3-51 (1992; Zbl 0802.52018)] is used to prove the topological version.

Also, the following packing radius sphere theorem is proved: If \(n \geq 3\), then any closed Riemannian \(n\)-manifold \(M\) with sectional curvature \(\geq 1\) and \(\text{pack}_{n - 1} M > {\pi \over 4}\) is diffeomorphic to \(S^n\). The proof is based on ideas from Alexandrov geometry, namely the global Riemannian problem associated with the theorem is changed to a local problem in Alexandrov geometry. This result generalizes differentiable sphere theorems of Y. Otsu, K. Shiohama and T. Yamaguchi [Invent. Math. 98, No. 2, 219-228 (1989; Zbl 0688.53016)], C. Plaut [Spaces of Wald curvature bounded from below, J. Geom. Anal., to appear), K. Shiohama and T. Yamaguchi [in Geometry of manifolds, Coll. Pap. 35th Symp. Differ. Geom., Matsumoto/Japan 1988, Perspect. Math. 8, 345-350 (1989; Zbl 0697.53041)], F. H. Wilhelm jun. [Invent. Math. 107, No. 3, 653-668 (1992; Zbl 0739.53036); Indiana Univ. Math. J. 41, No. 1, 1119-1142 (1992; Zbl 0771.53028)], J.-Y. Wu [in Proc. Symp. Pure Math. 54, Part 3, 685-692 (1993; Zbl 0804.53063)], since the hypotheses of these results imply that \(M\) is close to \(S^n_1\) in the Gromov-Hausdorff topology and consequently that \(\text{pack}_{n - 1} M > {\pi \over 4}\). Two corollaries of the authors’ version of the differentiable sphere theorem are pointed out.

Reviewer: M.Craioveanu (Timişoara)

### MSC:

53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |

53C20 | Global Riemannian geometry, including pinching |