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).)

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