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Geometry of 2-step nilpotent groups with a left invariant metric. (English) Zbl 0820.53047

In this paper the author studies the differential geometry of simply connected, 2-step, nilpotent Lie groups \(N\) with a left invariant Riemannian metric. In particular he considers properties of closed geodesics in a compact nilmanifold \(\Gamma \setminus N\), where \(\Gamma\) is a discrete cocompact subgroup of \(N\). Some of the results he obtains are the following:
1) There is an obstruction (resonance) to the density in \(T_ 1(\Gamma\setminus N)\) of the set of vectors \(P\) that are periodic with respect to the geodesic flow. In particular \(P\) is not always dense in \(T_ 1(\Gamma \setminus N)\), but \(P\) is dense in \(T_ 1(\Gamma \setminus N)\) for any \(\Gamma\) if \(N\) is of Heisenberg type.
2) Every free homotopy class of closed curves in \(\Gamma \setminus N\) contains a closed geodesic of largest period. Define the maximal length spectrum of \(\Gamma \setminus N\) to be the collection with multiplicities of these largest periods. If \(\Gamma \setminus N\), \(\Gamma^* \setminus N^*\) are compact 2-step nilmanifolds with the same marked maximal length spectrum, then it is shown that \(\Gamma \setminus N\), \(\Gamma^* \setminus N^*\) are equivalent up to isometry and \(\Gamma\)-almost inner automorphism.
Reviewer: H.Özekes (Muǧla)

MSC:

53C22 Geodesics in global differential geometry
22E25 Nilpotent and solvable Lie groups
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