Manifolds all of whose Geodesics are Closed
| Main Author: | |
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| Format: | eBook |
| Language: | English |
| Published: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1978, 1978
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| Edition: | 1st ed. 1978 |
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, A Series of Modern Surveys in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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
- 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