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Rigidity of the geodesic flow on certain nilmanifolds of rank two. (Rigidité du flot géodésique de certaines nilvariétés de rang deux.) (French) Zbl 0902.58027

Séminaire de théorie spectrale et géométrie. Année 1996-1997. St. Martin D’Hères: Univ. de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 15, 25-36 (1997).
A closed Riemannian manifold \((M,g)\) is called \(C^k\)-geodesically rigid iff any closed Riemannian manifold \((N,h)\) for which there exists a homeomorphism \(F: S_g(M)\to S_h(N)\) of class \(C^k\) between the tangent unit sphere bundles commuting with the geodesic flow must be isometric to \((M,g)\). A two-step nilpotent group is a nilpotent group \(N\) whose derived group \([N,N]\) is contained in the center of \(N\). A two-step nilmanifold is the quotient by a cocompact lattice of a two-step nilpotent group equipped with a left-invariant metric. There are two well-studied algebraically characterized subclasses of two-step nilmanifolds, the nilmanifolds “strongly in resonance” and the even more special nilmanifolds of “Heisenberg type”. A result of C. Gordon and Y.-P. Mao [‘Geodesic conjugacies of two-step nilmanifolds’, Preprint] asserts that Heisenberg type nilmanifolds are \(C^0\)-geodesically rigid in the class of all nilmanifolds and that nilmanifolds strongly in resonance are \(C^2\)-geodesically rigid in the class of nilmanifolds. It is natural to conjecture that nilmanifolds strongly in resonance are \(C^0\)-geodesically rigid in the class of nilmanifolds.
In the present paper, it is shown that the only known example of a nilmanifold strongly in resonance which is not of Heisenberg type is indeed \(C^0\)-geodesically rigid in the class of nilmanifolds. The methods of proof are essentially those of Gordon and Mao.
For the entire collection see [Zbl 0882.00016].
Reviewer: C.Bär (Freiburg)

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
53C22 Geodesics in global differential geometry
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