Manifolds all of whose geodesics are closed

Arthur L. Besse

[目次]

  • 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 Kahler 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.

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[目次]

  • 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 Kahler 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.

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この本の情報

書名 Manifolds all of whose geodesics are closed
著作者等 Besse, Arthur L.
Besse A. L.
Besse A L
シリーズ名 Ergebnisse der Mathematik und ihrer Grenzgebiete
出版元 Springer-Verlag
刊行年月 1978
版表示 Softcover reprint of the original 1st ed. 1978
ページ数 ix, 262 p.
大きさ 25 cm
ISBN 0387081585
3540081585
9783642618789
NCID BA01409680
※クリックでCiNii Booksを表示
言語 英語
出版国 ドイツ
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