## Structure of manifolds of nonpositive curvature. II.(English)Zbl 0598.53046

[Part I, cf. ibid., 171-203 (1985; Zbl 0589.53047.]
Let N be an m-dimensional differentiable manifold without boundary and let $$B^{\ell}$$ be the unit ball in $${\mathbb{R}}^{\ell}$$. Let V be an open subset of N. A continuous 1-dimensional foliation with $$C^ 1$$-leaves L on V is a partition L of V into 1-dimensional $$C^ 1$$ submanifolds L(x), $$x\in V$$ called leaves, with the following property; for every $$x\in V$$ there is a neighborhood V(x) of x in V and a homeomorphism $$f_ x: B^{m-\ell}\times B\to V(x)$$ such that for any $$y\in B^{m-\ell}$$ the map $$f_ x(y, ):$$ $$B^{\ell}\to V(x)$$ is a $$C^ 1$$-diffeomorphism onto the path component of $$L^ x(f_ x(y,0))\cap V(X)$$ containing $$f_ x(y,0)$$ and such that $$f_ x(y, )$$ depends continuously on y in the $$C^ 1$$-topology.
Let the $$L_ i$$ be 1-dimensional foliations of V, $$i=1,2,...,k$$. The foliations $$L_ 1,L_ 2,...,L_{\kappa}$$ are called transverse if for every $$x\in V$$ the tangent spaces $$T_ x(L_ i(x))$$, $$i=1,2,...,k$$ span a subspace of $$T_ xN$$ of dimension $$\ell_ 1+\ell_ 2+...+\ell_{\kappa}$$. This theory of foliations is used for the investigation of the structure of (non-flat) manifolds of non-positive curvature. The main result of this paper can be stated as follows. Let M be a quotient of M of finite volume, and assume that the sectional curvature of M is bounded from below by $$-\alpha^ 2$$. Then there is a $$g^ t$$-invariant open and dense subset R of SM and k-1 independent $$C^ 1$$ first integrals $$\Phi_ i: {\mathbb{R}}\to {\mathbb{R}}$$, $$1\leq i\leq k- 1$$, such that each $$v\in R$$ has a neighborhood R(v) in R with the following property: If $$v^*\in SF_{R(v)}(v')$$, then $$\Phi_ i(V^*)=\Phi_ i(v')$$ for all i if and only if $$v^*$$ is parallel to v’ in F(v’). (For notations see part I.) (Recall that $$SF_{R(v)}$$ is the foliation induced by SF in R(v).)
Reviewer: G.Tsagas

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C12 Foliations (differential geometric aspects) 57R30 Foliations in differential topology; geometric theory

Zbl 0589.53047
Full Text: