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

Fanaï, Hamid-Reza

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

Keywords:

geodesic flow; nilmanifold; two-step nilmanifold; nilmanifold of Heisenberg type; nilmanifold in strong resonance