On the Topology of the Space of Negatively Curved Metrics

We show that the space of negatively curved metrics of a closed negatively curved Riemannian n-manifold, n ≥ 10, is highly non-connected. Section 0. Introduction. Let M be a closed smooth manifold. We denote by MET (M) the space of all smooth Riemannian metrics on M and we consider MET (M) with the smooth topology. Note that the space MET (M) is contractible. A subspace of metrics whose sectional curvatures lie in some interval (closed, open, semi-open) will be denoted by placing a superscript on MET (M). For example, MET (M) denotes the subspace of MET (M) of all Riemannian metrics on M that have all sectional curvatures less that ǫ. Thus saying that all sectional curvatures of a Riemannian metric g lie in the interval [a, b] is equivalent to saying that g ∈ MET a≤ sec≤ (M). Note that if I ⊂ J then MET (M) ⊂ MET (M). Note also that MET (M) is the space of hyperbolic metrics Hyp (M) on M . An open problem posed by K. Burns and A. Katok ([2], Question 7.1) about closed negatively curved manifolds M is the following: is the space MET (M) of negatively curved metrics on M path connected? In dimension two, Hamilton’s Ricci flow [8] shows that Hyp (M2) is a deformation retract of MET (M2). But Hyp (M2) fibers over the Teichmüller space T (M2) ∼= R6μ−6 (μ is the genus of M2), with contractible fiber D = R+ ×DIFF (M2) [5]. Therefore Hyp (M2) and MET (M2) are contractible. In this paper we prove that, for n ≥ 10, MET (Mn) is never path-connected; in fact, it has infinitely many path-components. Moreover we show that all the groups π2p−4(MET (Mn)) are non-trivial for every prime number p > 2, and such that p < n+5 6 . (In fact, these groups contain the infinite sum (Zp) ∞ of Zp = Z/pZ’s, and hence they are not finitely generated. Also, the restriction on n = dimM can be improved to p ≤ n−2 4 . See Remarks 1 below.) We also show that π1(MET (Mn)) is not finitely generated when n ≥ 14. These results about πk are true for each path component of MET (Mn); i.e. relative to any base point. Before we state our Main Theorem, we ∗The first author was partially supported by a NSF grant. The second author was supported in part by research grants from NSF and CNPq, Brazil.