Manifolds All of Whose Geodesics Are Closed

ISBN-13: 9783642618789 / Angielski / Miękka / 2011 / 264 str.

A. L. Besse

Opis

X 1 O S R Cher lecteur, J'entre bien tard dans la sphere etroite des ecrivains au double alphabet, moi qui, il y a plus de quarante ans deja, avais accueilli sur mes terres un general epris de mathematiques. JI m'avait parle de ses projets grandioses en promettant d'ailleurs de m'envoyer ses ouvrages de geometrie. Je suis entiche de geometrie et c'est d'elle dontje voudrais vous parler, oh certes pas de toute la geometrie, mais de celle que fait l'artisan qui taille, burine, amene, gauchit, peaufine les formes. Mon interet pour le probleme dont je veux vous entretenir ici, je le dois a un ami ebeniste. En effet comme je rendais un jour visite il cet ami, je le trouvai dans son atelier affaire a un tour. Il se retourna bientot, puis, rayonnant, me tendit une sorte de toupie et me dit: -Monsieur Besse, vous qui calculez les formes avec vos grimoires, que pensez-vous de ceci?)) Je le regardai interloque. Il poursuivit: -Regardez Si vous prenez ce collier de laine et si vous le maintenez fermement avec un doigt place n'importe ou sur la toupie, eh bien la toupie passera toujours juste en son interieur, sans laisser le moindre espace.)) Je rentrai chez moi, fort etonne, car sa toupie etait loin d'etre une boule. Je me mis alors au travail ..."

Spis treści

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 Blaschke Manifolds 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.