Structure of manifolds of nonpositive curvature. I. (English) Zbl 0589.53047

Ballmann, Werner; Brin, Misha; Eberlein, Patrick

Ann. Math. (2) 122, 171-203 (1985).

Let (M,g) be a compact Riemannian manifold, without boundary, of nonpositive sectional curvature. Let u be a vector in the unit tangent bundle SM of M. We define rank(u) to be the dimension of the space of all parallel Jacobi fields along the geodesic \(\gamma_ U: R\to M\) which has initial velocity u. The minimum of rank(u) over all \(u\in SM\) is called the rank of M. Clearly, \(1\leq rank(M)\leq \dim (M)\). From this we conclude that rank(M) can be considered as a measurement of the flatness of M since a parallel field X along \(\gamma_ u\) is a Jacobi field if and only if the sectional curvature of the planes \(\gamma_ u(t)\wedge X(t)\) are zero for all \(t\in {\mathbb{R}}\). The Riemannian manifold (M,g) is flat if \(rank(M)=\dim (M)\). Therefore a compact surface of nonpositive curvature has rank one unless it is a torus or a Klein bottle.
The geodesic flow \(g^ t\) of M acts on the unit tangent bundle SM by translating a unit vector u along the geodesic \(\gamma_ u\) to the vector \(\gamma_ u(t)\). The Liouville measure \(d\mu =dm \times d\lambda\), which is the product of the Riemannian volume dm on M and the Lebesgue measure \(d\lambda\) on the unit sphere is preserved under \(g^ t\). A measure preserving flow \(g^ t\) in a measure space (X,\(\mu)\) (in our case \(X=SM)\) is called ergodic if any measurable subset A, such that \(g^ t(A)=A mod 0\) for any t, has either measure 0 or full measure. A function \(f: X\to {\mathbb{R}}\) is called a first integral if f is constant along orbits.
The main result of this paper can be stated as follows: If M has rank k\(>1\), then (a) \(g^ t\) is not ergodic; moreover there exist k-1 functionally independent differentiable first integrals defined on an open dense, invariant subset of SM. (b) Every geodesic in the universal covering space of M is contained in a k-flat, i.e., a totally geodesic and isometrically embedded copy of the k-dimensional Euclidean space \(e^ k\). (c) The vectors tangent to totally geodesic and isometrically immersed flat k-tori are dense in SM. (d) The fundamental group \(\pi_ 1(M)\) contains infinitely many conjugacy classes of maximal free Abelian subgroups of rank k if M is not flat.

Reviewer: G.Tsagas

MSC:

53C20	Global Riemannian geometry, including pinching
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25	Geodesic flows in symplectic geometry and contact geometry

Keywords:

nonpositive sectional curvature; Jacobi fields; first integrals; geodesic; k-flat; totally geodesic; fundamental group