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Existence of flat tori in analytic manifolds of nonpositive curvature. (English) Zbl 0760.53032
The aim of the paper is the proof of the following Theorem: Let $$M^ n$$ be a compact real analytic manifold with sectional curvature $$K\leq 0$$. Let $$k\leq n$$. If there exists a totally geodesic isometric immersion of the Euclidean $$\mathbb{R}^ k$$ into $$M$$ (“$$k$$-flat”), then $$M$$ contains also a totally geodesic isometric immersion of some flat $$k$$-dimensional torus (“closed $$k$$-flat”).
It is well known (Gromoll/Wolf, Lawson/Yau) that there is a closed $$k$$- flat in a compact $$M$$ with $$K\leq 0$$ if and only if the fundamental group $$\pi_ 1(M)$$ contains a subgroup isomorphic to $$Z^ k$$. Hence in the analytic case, the existence of any $$k$$-flat is equivalent to this topological condition. The statement has been proved by the second author for $$k\geq n-2$$. For $$k=1$$, it is just the (well known) existence of a closed geodesic.
As in the case of geodesics, the authors use arguments from the theory of dynamical systems. Here, the $$\mathbb{R}$$-action on the unit tangent bundle $$T^ 1M$$ ( geodesic flow) has to be replaced with the natural $$\mathbb{R}^ k$$-action (via geodesic flow and parallel transport) on the Stiefel bundle $$St(M)$$ of orthonormal $$k$$-frames over $$M$$ which leaves the set of $$k$$-frames tangent to $$k$$-flats invariant. To get a natural choice of such $$k$$-frames, the authors consider so called well structured (“w.s.”) $$k$$-flats which contain a full flag of singular subflats. Namely, if $$F$$ is a maximal flat (necessarily embedded) in the universal cover of $$M$$ (we may assume that there are maximal flats of dimension $$k$$), then a subflat $$F'\subset F$$ is called singular if the union of all flats parallel to $$F'$$ is larger than $$F$$. Using induction over $$\dim M$$, one may assume that w.s. $$k$$-flats exist. Now the theory of subanalytic sets is applied to show that there is a closed $$\mathbb{R}^ k$$-invariant analytic submanifold $$V$$ of $$St(M)$$, such that any $$k$$-frame in $$V$$ is tangent to the singular flag of some w.s. $$k$$-flat. The $$\mathbb{R}^ k$$- action on $$V$$ is normally hyperbolic, and the theory of Hirsch, Pugh and Shub can be applied to show the existence of a closed $$k$$-flat.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 34C25 Periodic solutions to ordinary differential equations 14P15 Real-analytic and semi-analytic sets
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##### References:
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