Nonpositively curved manifolds of higher rank. (English) Zbl 0585.53031

Ballmann, Werner

Ann. Math. (2) 122, 597-609 (1985).

The author investigates the structure of the universal covering space \(\tilde M\) of a complete Riemannian manifold M with nonpositive sectional curvature bounded from below by some constant \(-a^ 2\). The rank of a unit tangent vector v of M is defined to be the dimension of the space of parallel Jacobi vector fields along the geodesic \(\gamma_ v\) with the initial velocity v. Further, the rank of the Riemannian manifold M is defined as \[ rank(M)=\min \{rank(v)| \quad v\in SM\}. \] The main theorem says that if M has finite volume, then \(\tilde M\) is a space of rank one, or a symmetric space, or a product of such spaces. Corollary: Suppoe that M has finite volume. If M is irreducible and \(rank(M)=k\geq 2\), then M is a locally symmetric space of noncompact type of rank k.

Reviewer: O.Kowalski

MSC:

53C20

Global Riemannian geometry, including pinching

Keywords:

nonpositive sectional curvature; Jacobi vector fields; space of rank one; finite volume; locally symmetric space