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AuthorBesse, Arthur L. author
TitleManifolds all of whose Geodesics are Closed [electronic resource] / by Arthur L. Besse
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1978
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Descript IX, 264 p. online resource


0. Introduction -- A. Motivation and History -- B. Organization and Contents -- C. What is New in this Book? -- D. What are the Main Problems Today? -- 1. Basic Facts about the Geodesic Flow -- A. Summary -- B. Generalities on Vector Bundles -- C. The Cotangent Bundle -- D. The Double Tangent Bundle -- E. Riemannian Metrics -- F. Calculus of Variations -- G. The Geodesic Flow -- H. Connectors -- I. Covariant Derivatives -- J. Jacobi Fields -- K. Riemannian Geometry of the Tangent Bundle -- L. Formulas for the First and Second Variations of the Length of Curves -- M. Canonical Measures of Riemannian Manifolds -- 2. The Manifold of Geodesics -- A. Summary -- B. The Manifold of Geodesics -- C. The Manifold of Geodesics as a Symplectic Manifold -- D. The Manifold of Geodesics as a Riemannian Manifold -- 3. Compact Symmetric Spaces of Rank one From a Geometric Point of View -- A. Introduction -- B. The Projective Spaces as Base Spaces of the Hopf Fibrations -- C. The Projective Spaces as Symmetric Spaces -- D. The Hereditary Properties of Projective Spaces -- E. The Geodesics of Projective Spaces -- F. The Topology of Projective Spaces -- G. The Cayley Projective Plane -- 4. Some Examples of C- and P-Manifolds: Zoll and Tannery Surfaces -- A. Introduction -- B. Characterization of P-Metrics of Revolution on S2 -- C. Tannery Surfaces and Zoll Surfaces Isometrically Embedded in (IR3, can) -- D. Geodesics on Zoll Surfaces of Revolution -- E. Higher Dimensional Analogues of Zoll metrics on S2 -- F. On Conformal Deformations of P-Manifolds: A. Weinsteinโ{128}{153}s Result -- G. The Radon Transform on (S2, can) -- H. V. Guilleminโ{128}{153}s Proof of Funkโ{128}{153}s Claim -- 5. Blaschke Manifolds and Blaschkeโ{128}{153}s Conjecture -- A. Summary -- B. Metric Properties of a Riemannian Manifold -- C. The Allamigeon-Warner Theorem -- D. Pointed Blaschke Manifolds and Blaschke Manifolds -- E. Some Properties of Blaschke Manifolds -- F. Blaschkeโ{128}{153}s Conjecture -- G. The Kรคhler Case -- H. An Infinitesimal Blaschke Conjecture -- 6. Harmonic Manifolds -- A. Introduction -- B. Various Definitions, Equivalences -- C. Infinitesimally Harmonic Manifolds, Curvature Conditions -- D. Implications of Curvature Conditions -- E. Harmonic Manifolds of Dimension 4 -- F. Globally Harmonic Manifolds: Allamigeonโ{128}{153}s Theorem -- G. Strongly Harmonic Manifolds -- 7. On the Topology of SC- and P-Manifolds -- A. Introduction4 -- B. Definitions -- C. Examples and Counter-Examples -- D. Bott-Samelson Theorem (C-Manifolds) -- E. P-Manifolds -- F. Homogeneous SC-Manifolds -- G. Questions -- H. Historical Note -- 8. The Spectrum of P-Manifolds -- A. Summary -- B. Introduction -- C. Wave Front Sets and Sobolev Spaces -- D. Harmonic Analysis on Riemannian Manifolds -- E. Propagation of Singularities -- F. Proof of the Theorem 8. 9 (J. Duistermaat and V. Guillemin) -- G. A. Weinsteinโ{128}{153}s result -- H. On the First Eigenvalue ?1=?12 -- Appendix A. Foliations by Geodesic Circles -- I. A. W. Wadsleyโ{128}{153}s Theorem -- II. Foliations With All Leaves Compact -- Appendix B. Sturm-Liouville Equations all of whose Solutions are Periodic after F. Neuman -- I. Summary -- II. Periodic Geodesics and the Sturm-Liouville Equation -- III. Sturm-Liouville Equations all of whose Solutions are Periodic -- IV. Back to Geometry with Some Examples and Remarks -- Appendix C. Examples of Pointed Blaschke Manifolds -- I. Introduction -- II. A. Weinsteinโ{128}{153}s Construction -- III. Some Applications -- Appendix D. Blaschkeโ{128}{153}s Conjecture for Spheres -- I. Results -- II. Some Lemmas -- III. Proof of Theorem D.4 -- Appendix E. An Inequality Arising in Geometry -- Notation Index

Mathematics Differential geometry Mathematics Differential Geometry


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