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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