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**The space of nonpositively curved metrics of a negatively curved manifold.**
*(English)*
Zbl 1318.58007

Let \(M^n\) be a closed smooth manifold of dimension \(n\). We denote by \(\mathcal{M}\) the space of all smooth Riemannian metrics on \(M\) and we consider \(\mathcal{M}\) with the smooth topology. Also, we denote by \(\mathcal{M}^{\mathrm{sec}<0}\) the subspace formed by all negatively curved Riemannian metrics on \(M\). In ibid. 86, No. 2, 273–301 (2010; Zbl 1221.58009)] the authors proved that \(\mathcal{M}^{\mathrm{sec}<0}\) always has infinitely many path-components, provided \(n\geq 10\) and it is non-empty.

Let \(\mathcal{M}^{\mathrm{sec}\leq 0}\) be the subspace of \(\mathcal{M}\) formed by all non-positively curved Riemannian metrics on \(M\). In this paper the authors generalize to \(\mathcal{M}^{\mathrm{sec}\leq 0}\) the results concerning to the space \(\mathcal{M}^{\mathrm{sec}< 0}\), provided \(\pi_1M\) is (word) hyperbolic.

Let \(M^n\) be a closed smooth manifold with hyperbolic fundamental group \(\pi_1 M\). Assume \(\mathcal{M}^{\mathrm{sec}\leq 0}\) is non-empty. Then:

Let \(\mathcal{M}^{\mathrm{sec}\leq 0}\) be the subspace of \(\mathcal{M}\) formed by all non-positively curved Riemannian metrics on \(M\). In this paper the authors generalize to \(\mathcal{M}^{\mathrm{sec}\leq 0}\) the results concerning to the space \(\mathcal{M}^{\mathrm{sec}< 0}\), provided \(\pi_1M\) is (word) hyperbolic.

Let \(M^n\) be a closed smooth manifold with hyperbolic fundamental group \(\pi_1 M\). Assume \(\mathcal{M}^{\mathrm{sec}\leq 0}\) is non-empty. Then:

- (i)
- the space \(\mathcal{M}^{\mathrm{sec}\leq 0}\) has infinitely many components, provided \(n\geq 10\).
- (ii)
- The group \(\pi_1(\mathcal{M}^{\mathrm{sec}\leq 0})\) is not trivial when \(n\geq 12\). In fact it contains the infinite sum \((\mathbb{Z}_2)^\infty\) as a subgroup.
- (iii)
- The groups \(\pi_{2p-4}(\mathcal{M}^{\mathrm{sec}\leq 0})\) are non-trivial for every prime number \(p>2\), and such that \(p<\frac{n+5}{6}\).

Reviewer: Nicolai K. Smolentsev (Kemerovo)

### MSC:

58D17 | Manifolds of metrics (especially Riemannian) |

58B05 | Homotopy and topological questions for infinite-dimensional manifolds |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

53C20 | Global Riemannian geometry, including pinching |