Orbits of the geodesic flow and chains on the boundary of a Grauert tube (Q1598163)

Let \((X,g)\) be a compact, connected \(C^\omega\) Riemannian manifold. For \(\varepsilon>0\) sufficiently small there is a canonical integrable complex structure defined on the tube \(T^{*\varepsilon} (X)=\{\xi\in T^*X: |\xi|< \varepsilon\}\) with the property that the level sets of the length function are strictly pseudoconvex domains on a Stein manifold. It is known that two such domains \(\Omega_1,\Omega_2\) in a Stein manifold are biholomorphic iff their boundaries are CR diffeomorphic. Let \(M^\varepsilon\) denote the boundary of the tube \(T^{*\varepsilon}(X)\). The submanifolds \(M^\varepsilon\) are not CR equivalent for different values of \(\varepsilon\). The goal of this article is to understand the behavior of certain CR invariants of \(M^\varepsilon\) as \(\varepsilon\to 0\). There are two natural families of curves on the manifolds \(M^\varepsilon\). The first family consists of the chains defined by Chern and Moser, which now have several equivalent descriptions. If \(M\) is any CR manifold, then for each \(v\in TM\) transverse to the maximal complex subspace there is a unique chain with tangent vector \(v\). For the manifolds \(M^\varepsilon\) there is also a natural second family of curves, the orbits of the geodesic flow, which are also invariant under the identity component of the holomorphic automorphism group of \(T^{*\varepsilon}(X)\). The main results of this article are the following: (1) lf the orbits of the geodesic flow are all chains for sufficiently small \(\varepsilon>0\), the Riemannian manifold \((X,g)\) is Einstein. (2) If \((X,g)\) is harmonic (stronger than Einstein), then the orbits of the geodesic flow are chains. From statement (2) one obtains new examples of pseudo-Einstein CR manifolds, namely the harmonic, nonsymmetric Damek-Ricci spaces constructed from representations of the real negative definite Clifford algebra \(C\ell(p)\), \(p\neq 1,3\) or 7.