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Zbl 0897.53033
Gordon, Carolyn S.; Mao, Yiping; Schueth, Dorothee
Symplectic rigidity of geodesic flows on two-step nilmanifolds. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 30, No. 4, 417-427 (1997). ISSN 0012-9593

The authors consider, in the context of 2-step nilmanifolds, the question of when geodesically conjugate, compact Riemannian manifolds are isometric. Riemannian manifolds $M,M^*$ are $C^k$ geodesically conjugate for some integer $k\ge 0$ if there exists a $C^k$ homeomorphism $F:SM\to SM^*$ such that $F\circ g^t= g^t\circ F$ for all $t$ in $\bbfR$, where $SM$ and $SM^*$ denote the unit tangent bundles of $M$ and $M^*$ and $g^t$ denotes the geodesic flow in both $SM$ and $SM^*$. In this article, the authors assume further that $F$ is the restriction of a conjugacy $F':TM- \{0\}\to TM^*- \{0\}$, but this can always be accomplished, for example by setting $F'(v)= \| v\| \cdot F(v/ \| v\|)$. Let $\Omega$ and $\Omega^*$ denote the symplectic 2-forms defined on the full tangent bundles $TM$ and $TM^* $. A $C^1$ map $F':TM\to TM^*$ is a symplectomorphism if $F'{}^* (\Omega^*) =\Omega$.\par A Lie group $N$ is 2-step nilpotent if $[N,N]$ is contained in the center of $N$. Let $\Gamma$ be a discrete subgroup of $N$ such that the right coset space $\Gamma \setminus N$ is compact. The space $\Gamma \setminus N$ equipped with a Riemannian metric descending from a left invariant Riemannian metric on $N$ is called a compact 2-step Riemannian nilmanifold.\par Theorem. Let $\Gamma \setminus N$ and $\Gamma^* \setminus N^*$ be compact 2-step Riemannian nilmanifolds, and let $F':T(\Gamma \setminus N)- \{0\}\to T(\Gamma^* \setminus N^*)- \{0\}$ be a $C^1$ geodesic conjugacy that is also a symplectomorphism. Then $\Gamma \setminus N$ and $\Gamma^* \setminus N^*$ are isometric.\par It is not known if this conclusion remains valid without the hypothesis that the geodesic conjugacy be a symplectomorphism. We outline the proof. Using a known result about geodesically conjugate compact 2-step Riemannian nilmanifolds, one first reduces to the case that $N^*=N$ and $\Gamma^* =\psi (\Gamma)$, where $\psi: N\to N$ is an automorphism such that $\psi (\gamma)$ is conjugate to $\gamma$ for all $\gamma$ in $\Gamma$ but the conjugating element might depend on $\gamma$. The problem is precisely to show that the conjugating element must be independent of $\gamma$. If ${\cal N}$ denotes the Lie algebra of $N$, then ${\cal N}= {\cal V} \oplus {\cal Z}$, where ${\cal V} ={\cal Z}^\perp$, and $\pi_{\cal V} (\log \Gamma)$ is a vector lattice in ${\cal V}$; here $\log: N\to {\cal N}$ is the inverse of the diffeomorphism $\exp: {\cal N} \to N$ and $\pi_{\cal V}$ denotes the orthogonal projection on ${\cal V}$. Since ${\cal N}$ is a 2-step nilpotent Lie algebra, one may write $d\psi= \exp(\varphi) =\text {Id}+ \varphi$, where $\varphi: {\cal N} \to{\cal N}$ is a derivation such that $\varphi({\cal Z}) \equiv 0$ and $\varphi(V) =[B(V),V] \subseteq {\cal Z}$ for all $V$ in $\pi_{\cal V} (\log \Gamma)$, where $B:\pi_{\cal V} (\log\Gamma) \to {\cal V}$ is some (not uniquely determined) function. The fact that the $C^1$ geodesic conjugacy $F':T(\Gamma \setminus N)- \{0\}\to T(\psi (\Gamma) \setminus N) -\{0\}$ is also a symplectomorphism implies that a suitable choice for $B$ extends to a $C^1$ function $B: {\cal V} -\{0\} \to{\cal V}$ whose differentials $dB_V$ are symmetric linear transformations at all nonzero elements $V$ in ${\cal V}$. A Lie algebra cohomology argument then shows that $B$ must be constant, which shows that $\varphi$ is an inner derivation of ${\cal N}$ and $\psi$ is an inner automorphism of $N$.
[P.Eberlein (Chapel Hill)]

MSC 2000:: *53C22 Geodesics
37D40 Dynamical systems of geometric origin and hyperbolicity
53D25 Geodesic flows
58J50 Spectral problems; spectral geometry; scattering theory
53C30 Homogeneous manifolds

Keywords: symplectic rigidity; 2-step nilmanifolds; geodesic conjugacy

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