The geometry of spherical curves and the algebra of quaternions

Contents Introduction 2 Chapter I. The geometry of curves on S 2 3 § 1. The elementary geometry of smooth curves and wavefronts 3 § 2. Contact manifolds, their Legendrian submanifolds and their fronts 9 § 3. Dual curves and derivative curves of fronts 10 § 4. The caustic and the derivatives of fronts 12 Chapter II. Quaternions and the triality theorem 13 § 5. Quaternions and the standard contact structures on the sphere S 3 13 § 6. Quaternions and contact elements of the sphere 5? 15 § 7. The action of quaternions on the contact elements of the sphere 5| 18 § 8. The action of right shifts on left-invariant fields 20 § 9. The duality of j-fronts and fc-fronts of «-Legendrian curves 20 Chapter III. Quaternions and curvatures 22 § 10. The spherical radii of curvature of fronts 22 § 11. Quaternions and caustics 23 § 12. The geodesic curvature of the derivative curve 24 § 13. The derivative of a small curve and the derivative of curvature of the curve 28 Chapter IV. The characteristic chain and spherical indices of a hyper-surface 30 § 14. The characteristic 2-chain 31 § 15. The indices of hypersurfaces on a sphere 33 § 16. Indices as linking coefficients 35 § 17. The indices of hypersurfaces on a sphere as intersection indices 36 § 18. Proofs of the index theorems 38 § 19. The indices of fronts of Legendrian submanifolds on an even-dimensional sphere 40 Chapter V. Exact Lagrangian curves on a sphere and their Maslov indices 44 § 20. Exact Lagrangian curves and their Legendrian lifts 45 V. I. Arnol'd § 21. The integral of a horizontal form as the area of the characteristic chain 48 §22. A horizontal contact form as a Levi-Civita connection and a generalized Gauss-Bonnet formula 49 § 23. Proof of the formula for the Maslov index 52 § 24. The area-length duality 54 §25. The parities of fronts and caustics 56 Chapter VI. The Bennequin invariant and the spherical invariant J + 57 § 26. The spherical invariant J + 58 § 27. The topological meaning of the invariant SJ + 59 Chapter VII. Pseudo-functions 60 §28. The quasi-functions of Chekanov 61 § 29. From quasi-functions on the cylinder to pseudo-functions on the sphere, and conversely 62 § 30. Conjectures concerning pseudo-functions 63 §31. Space curves and Sturm's theorem 66 Bibliography 68