A. L. Besse

Manifolds all of whose Geodesics are Closed (eBook, PDF)

• Format: PDF

Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 264
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9783642618765
• Artikelnr.: 53133160

Inhaltsangabe

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. Weinstein's Result.- G. The Radon Transform on (S2, can).- H. V. Guillemin's Proof of Funk's Claim.- 5. Blaschke Manifolds and Blaschke's Conjecture.- A. Summary.- B. Metric Properties of a Riemannian Manifold.- C. The Allamigeon-Warner Theorem.- D. Pointed BlaschkeManifolds and Blaschke Manifolds.- E. Some Properties of Blaschke Manifolds.- F. Blaschke's Conjecture.- G. The Kähler 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: Allamigeon'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. Weinstein's result.- H. On the First Eigenvalue ?1=?12.- Appendix A. Foliations by Geodesic Circles.- I. A. W. Wadsley'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. Weinstein's Construction.- III. Some Applications.- Appendix D. Blaschke's Conjecture for Spheres.- I. Results.- II. Some Lemmas.- III. Proof of Theorem D.4.- Appendix E. An Inequality Arising in Geometry.- Notation Index.