A construction of negatively curved manifolds
Let V be a complete Riemannian manifold with finite volume and sectional curvature K satisfying \(-b^ 2<K<0\). It is known that if V is noncompact, then each end of V has a neighborhood of the form \(N\times (0,\infty)\), where N is a compact manifold that finitely covers a manifold diffeomorphic to a quotient of a simply connected nilpotent Lie group by a discrete subgroup. The present paper shows that ends of a different topological type may arise in complete manifolds of finite volume if the curvature is bounded and negative but not strictly negative. Theorem. Let V be a compact manifold of dimension \(n\geq 4\) with \(K\equiv -1\), and let W be a compact, totally geodesic submanifold of V of codimension 2. Then \(V\setminus W\) admits a complete Riemannian metric with finite volume and sectional curvature satisfying \(-b<K<0\) for some positive constant b. The construction of a pair V, W satisfying the hypotheses above is described on page 3 of [\textit{M. Gromov} and \textit{W. Thurston}, Invent. Math. 89, 1-12 (1987; Zbl 0646.53037)]. The end of \(V\setminus W\) in the theorem above has a neighborhood of the form \(U=W\times S^ 1\times (0,\sigma)\) with coordinates (w,\(\theta\),r). If \(\sigma\) is sufficiently small, then the metric with \(K\equiv -1\) is given by \(g_ V=\cosh^ 2(r)g_ W+\sinh^ 2(r)d\theta^ 2+dr^ 2,\) where \(g_ W\) denotes the metric on W. Let f(r) be a positive function with \(f(r)=1\) for \(r>\sigma /2\). Introduce a new metric on \(V\setminus W\) by defining \(g^*_ V=\cosh^ 2(r)g_ W+\sinh^ 2(r)d\theta^ 2+f(r)^ 2dr^ 2\) on U. If f(r) satisfies appropriate further conditions, then the new metric on \(V\setminus W\) has finite volume and sectional curvature satisfying \(- b<K<0\) for some positive constant b.
scientific article; zbMATH DE number 4094307