Circle foliations by geodesics revisited: Periods of flows whose orbits are all closed

We consider a [Formula: see text] free action [Formula: see text] on an [Formula: see text]-dimensional [Formula: see text] compact connected manifold [Formula: see text]. The manifold [Formula: see text] is foliated by circles if all orbits induced by [Formula: see text] are closed. We handle the periods of circle foliations, especially when the circles are closed geodesics with respect to some Riemannian metric. Then, it is known that the periods of [Formula: see text] are exactly the same if and only if there exists a Riemannian metric on [Formula: see text] with respect to which the circles are embedded as totally geodesic submanifolds of [Formula: see text] [A. W. Wadsley, Geodesic foliations by circles, J. Differ. Geom. 10 (1975) 541–549.]. It automatically follows a classical result [A. Besse, Manifolds All of Whose Geodesics are Closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 93 (Springer-Verlag, New York, 1978)] that all periods are exactly the same if all the geodesic flows on the cosphere bundle [Formula: see text] are periodic (the so-called “[Formula: see text]-manifold”). Our purpose here is to adapt these results to Reeb or Hamiltonian flows on contact manifolds. Then [Formula: see text]-circles are considered as totally geodesic submanifolds. Consequently, all periods are exactly the same if the contact manifold is connected and all orbits on the contact manifold are closed. We also present concrete examples of periodic flows, all of whose orbits are closed, such as Harmonic oscillators, Lotka–Volterra systems, and others. Lotka–Volterra systems, Reeb flows, and some geodesic flows have non-trivial periods, whereas the periods of Harmonic oscillators and similar systems can be easily obtained through direct calculations. As an application to quantum mechanics, we examine the spectrum of semiclassical Schrödinger operators [Formula: see text] If the spectrum of [Formula: see text] near [Formula: see text] consists of eigenvalues, we denote such a set as [Formula: see text] for small [Formula: see text]. For fixed constants [Formula: see text] and [Formula: see text], the cluster of the spectrum near [Formula: see text] (the so-called essential difference spectrum) is defined by [Formula: see text] where the notation [Formula: see text] denotes the set of cluster (accumulation) points. Under suitable assumptions, we find that [Formula: see text] In the former case, at least one of the Hamiltonian flows is not periodic on the energy surface [Formula: see text]. In the latter case, all Hamiltonian flows on [Formula: see text] are periodic with a common period [Formula: see text]. This fact represents one of the semiclassical analogies of the Helton-Guillemin theorem [V. Guillemin, Lectures on spectral theory of elliptic operators, Duke Math. J. 44 (1977) 487–517; W. Helton, An operator algebra approach to partial differential equations; Propagation of singularities and spectral theory, Indiana Univ. Math. J. 26 (1977) 997–1018.].