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

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
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