A characterization of Zoll
Riemannian metrics on the 2-sphere
The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element , any geodesic is simple closed with length .
Key words and phrases:Zoll metrics, closed geodesics, Lusternik-Schnirelmann theory
2010 Mathematics Subject Classification:53C22, 58E10
A remarkable class of closed Riemannian manifolds is given by those all of whose geodesics are closed. A detailed account of the state of the art of the research on this subject up to the late 1970s is contained in the celebrated monograph of Besse [Besse:1978pr], while for more recent results we refer the reader to, e.g., [Olsen:2010ne, Radeschi:2017dz, Abbondandolo:2017xz] and references therein. The round -spheres are the simplest examples of manifolds in this class. The first non-trivial example of a 2-sphere of revolution all of whose geodesics are closed was given by Zoll [Zoll:1903by]. The closed geodesics in this example are without self-intersections and have the same length. This is not accidental: a theorem of Gromoll and Grove [Gromoll:1981kl] implies that every 2-sphere all of whose geodesics are closed is Zoll, meaning that all the geodesics are simple closed and have the same length. Our main result, which was claimed by Lusternik in [Ljusternik:1966tk, page 82], shows that the property of being Zoll for a Riemannian 2-sphere can be read off from its simple length spectrum , that is, the set of lengths of its simple closed geodesics.
In a Riemannian 2-sphere such that , every geodesic is simple closed and has length .
Under a weaker assumption on the simple length spectrum, Lusternik also established the following easier statement. We will provide its precise proof in Section LABEL:s:proofs for the reader’s convenience.
Theorem 1.2 ([Ljusternik:1966tk], page 81).
Let be a Riemannian 2-sphere such that has at most two elements. Then, for some , every lies on a simple closed geodesic of of length .
If one further assumes that the sectional curvature of the Riemannian metric takes values inside , Theorems 1.1 and 1.2 are a consequence of the following result of Ballmann, Thorbergsson and Ziller [Ballmann:1983fv, Theorem A]. Consider a Riemannian -sphere, with , whose sectional curvature satisfies . If all the (not necessarily simple) closed geodesics with length in have at most two different length values, for one such length every point of the -sphere lies on a closed geodesic of length . If all the closed geodesics with length in have the same length, then all the geodesics are closed with the same length.
For any sufficiently close to , the ellipsoid of revolution
equipped with the Riemannian metric induced by the ambient Euclidean metric of satisfies the assumptions of Theorem 1.2, but is not a Zoll 2-sphere. Indeed, the meridians of are simple closed geodesics of the same length, and the only other simple closed geodesic is the equator, whose length is . Nevertheless, we do not know whether Theorem 1.2 is optimal.
On a Riemannian 2-sphere such that has at most two elements, is there a length and a point such that every geodesic going through is simple closed with length ?
The proofs of Theorems 1.1 and 1.2 build on the classical minmax recipe due to Lusternik and Schnirelmann [Lusternik:1934km, Ballmann:1978rw] for detecting three simple closed geodesics on every Riemannian 2-sphere. A crucial ingredient for this recipe is a deformation that shrinks emdedded loops without creating self-intersections, which can be obtained by applying Grayson’s curve shortening flow [Grayson:1989ec]. We expect such a flow to be available also in the setting of reversible Finsler metrics. If this were the case, Theorems 1.1 and 1.2 would extend to reversible Finsler metrics on the 2-sphere.
We close the introduction by raising one more question related to Theorem 1.1. Consider the unit tangent bundle equipped with a contact form . The associated Reeb vector field on is defined by and . We say that is reversible when , where is the involution .
Assume that all the periodic orbits of the Reeb vector field of a reversible have the same period. Is a Zoll contact manifold, namely such that all its Reeb orbits are periodic?
We are grateful to Alberto Abbondandolo, who asked us the original question leading to Theorem 1.1, and suggested to convert our homological proof in a cohomological one, which resulted in a dramatic simplification of the exposition. We also thank Wolfgang Ziller for pointing out to us the above mentioned result in [Ballmann:1983fv]. Marco Mazzucchelli is partially supported by the ANR-13-JS01-0008-01 “Contact spectral invariants”. Stefan Suhr is supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”, funded by the Deutsche Forschungsgemeinschaft. Part of this work was carried out during a visit of Marco Mazzucchelli at the Ruhr-Universität Bochum in November 2017, funded by the SFB/TRR 191; both authors wish to thank the university for providing a wonderful working environment.
2. Lusternik-Schnirelmann theory
In their celebrated work [Lusternik:1934km], Lusternik and Schnirelmann showed how to detect three simple closed geodesics on every Riemannian 2-sphere by applying variational methods. The original proof of this fact in [Lusternik:1934km] is known to have a gap. More specifically, the argument requires a deformation of the space of unparametrized embedded circles in the 2-sphere that shrinks all those circles that are not closed geodesics. The deformation provided by Lusternik and Schnirelmann is incomplete. Actually, constructing such a deformation by hand turned out to be highly non-trivial, and several authors proposed their solution in the second half of the 20th century. A particularly elegant one was provided by Grayson [Grayson:1989ec] with its curve shortening flow. In this section, we are going to review the arguments leading to Lusternik-Schnirelmann’s theorem in combination with Grayson’s work. For the topological arguments, we will mainly follow [Ballmann:1978rw].
2.1. Grayson’s curve shortening flow
Let be an oriented Riemannian 2-sphere. We denote by the space of embedded circles , and by the space of constant maps. The group acts on by reparametrizations, i.e.
We consider the space of unparametrized loops endowed with the quotient Whitney topology, and its subsets and . We also consider the length function
which is continuous.
For each parametrized embedded circle , we denote by its positive normal vector field, so that the ordered pair is an oriented orthonormal basis of the tangent space for each . We also denote by the signed curvature of , which is defined by
Up to a sign, both and are independent of the parametrization of ; more precisely, for all , we have
In particular, the product is completely independent of the parametrization of , that is,
Now, let us consider the parabolic partial differential equation
with initial condition . By (2.1), this equation is parametrization invariant. Namely, if is a solution of (2.2), for each the family of curves is a solution of the same equation with initial condition . Therefore, we can view (2.2) as a recipe that prescribes the evolution of unparametrized embedded circles .
The local existence, uniqueness, and continuous dependence on the initial condition in the topology of the solutions of (2.2) is well known by the standard theory of parabolic partial differential equations (see, e.g., [ManteMarti, Theorem 1.1] for a modern account). In his fundamental paper [Grayson:1989ec], Grayson studied the long-term existence and several properties of the solutions of (2.2). Summing up, there is an open neighborhood of and a continuous map encoding the solutions of (2.2), in the sense that . Such map is referred to in the literature as the curve shortening flow, and satisfies the following properties. For each , we denote by the largest extended real number such that .
For all we have
with equality if and only if is a closed geodesic (notice that in the integrand above we have introduced a parametrization of , but the value of the integral is independent of this choice); see [Grayson:1989ec, page 75].
For each , the limit
exists; if then and converges to some constant curve in as ; otherwise, and, for each open neighborhood of the set of simple closed geodesics of length and for all large enough, belongs to ; see [Grayson:1989ec, Theorem 0.1].
Let be an open neighborhood of the subspace of simple closed geodesics of length ; there exists such that, for every compact subset , there exists a continuous function such that for all ; see [Grayson:1989ec, Lemma 8.1].
2.2. The fundamental group of
Let us construct a 2-fold covering map
The idea of this construction goes as follows. Above the subspace of embedded circles , the total space is precisely the space of embedded compact disks in , and the projection sends a compact disk to its boundary curve; above any constant , one element of must be thought as the collapsed disk at the point , whereas the other element must be thought as the compact disk that fills and whose boundary has been collapsed to . Let us now provide the formal construction of this covering space.
For each we define a set of two elements as follows: if , we define to be the set of path-connected components of its complement; otherwise, if , we simply set . We set
We endow with a topology, by defining a fundamental system of open neighborhoods of any point as follows.
Assume first that , so that is a connected component of , and choose an arbitrary point . Let be a sufficiently small open neighborhood of so that, for each , does not lie on the curve . For every such , we set to be the connected component containing .
Assume now that , so that . We choose an arbitrary point , and as before a sufficiently small open neighborhood of such that, for every , does not lie on . For each , we set . For each , if we set to be the connected component containing , whereas if we set to be the connected component not containing .
In both cases, we declare to be an open neighborhood of . With this topology, is a 2-fold covering map with projection .
Let be a constant curve. We employ the covering to define a group homomorphism
as follows. Consider a continuous loop with . We lift to a continuous path such that and the following diagram commutes