Manifolds all of whose Geodesics are Closed

Abstract

This is a very important book whose subject matter ranges much more widely than the title suggests. Chapter 0 gives a brief history of the problem, which we now summarise as it gives a good indication of the purpose of the book-essentially an attempted classification of a special class of Riemannian manifolds. More specifically, consider the two assertions:SC: there exists a positive real number I such that for every vector £ in the unit tangent bundle UM of (M, g), the geodesic y with initial velocity vector y(0) = £ is (i) a periodic map with least period /,SC m : there exists a point m in M such that the above property is supposed to hold for every £ in the unit sphere U m at m (i.e. the assumption holds on m only).The whole book is devoted to what may appear to be the narrow question of classifying SC Riemannian manifolds to within isometry and SC" manifolds to within diffeomorphism. It is not difficult to prove that compact symmetric spaces of rank one, namely (S d , can), (RF 1 , can), (CP d , can), (HP d , can), (CaP 2 , can), with their canonical Riemannian structure are examples of SC manifolds.The first attempt to consider non-canonical metrics on S 2 satisfying condition SC appears to go back to Darboux, but he did not establish the existence of such a metric globally. After preliminary attempts by Tannery at the end of the 19th century, the question was settled by Zoll in 1903 who constructed a real analytic Riemannian manifold of revolution (S 2 , g) with the SC property.Natural questions arise such as "Are there many non-isometric SC Riemannian manifolds (S 2 , #)?". "Are there such SC Riemannian structures on the other candidate manifolds, RP d , S d (d ^ 3), CP 6 (d ^ 2), HP 4 (d > 2), CaP 2 ?". Little is known about this problem in general, apart from the case of S d -and even here results are quite recent. The only general result known is due to A. Weinstein (1974). This states that in order that a d-dimensional Riemannian manifold (M,g) satisfy the SC property with / = 2n, it is necessary that its total volume shall be an integral multiple of the volume of (

Erratum to: A Variational Approach to Givental’s Nonlinear Maslov Index

“…For instance the geodesic flow of any P-metric, e.g. the round metric on B = S n , gives rise to such a positive loop (see [Bes78]). Of course, these loops are not contractible by the result of Eliashberg-Kim-Polterovich.…”

“…It follows from Gromov's theorem [Gro78,Gro07] (see also [Pat99]) that if the homology of the loop space grows at most linearly in action then it also grows at most linearly in degree. Using the theory of minimal models by Sullivan [Sul75] and arguing as in the proof of the Bott-Samelson theorem in [Bes78,Chapter 7.D] it follows that the based loop space of a closed manifold with finite fundamental group such that the rational cohomology ring has at least two generators grows at least quadratically.…”

A Remark on H-Contact Unit Tangent Sphere Bundles

“…We may easily check that a 2-stein manifold satisfies the condition (5.2). It is also well-known that every harmonic space is super-Einstein [1].…”

Complete real hypersurfaces in compact rank one symmetric spaces

“…, λ be an eigenvector of A has played an essential role. The global behavior of geodesics on M (c) is understood as those on special Blaschke manifolds (for details see [2]), as stated in §2. The combination of these facts provides us with a general and global treatment of complete real hypersurfaces with two distinct principal curvatures in CROSS.…”