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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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