The geometry of spherical curves and the algebra of quaternions

Abstract

Contents Introduction Chapter I. The geometry of curves on §1. The elementary geometry of smooth curves and wavefronts §2. Contact manifolds, their Legendrian submanifolds and their fronts §3. Dual curves and derivative curves of fronts §4. The caustic and the derivatives of fronts Chapter II. Quaternions and the triality theorem §5. Quaternions and the standard contact structures on the sphere §6. Quaternions and contact elements of the sphere §7. The action of quaternions on the contact elements of the sphere §8. The action of right shifts on left-invariant fields §9. The duality of j-fronts and k-fronts of i-Legendrian curves Chapter III. Quaternions and curvatures §10. The spherical radii of curvature of fronts §11. Quaternions and caustics §12. The geodesic curvature of the derivative curve §13. The derivative of a small curve and the derivative of curvature of the curve Chapter IV. The characteristic chain and spherical indices of a hypersurface §14. The characteristic 2-chain §15. The indices of hypersurfaces on a sphere §16. Indices as linking coefficients §17. The indices of hypersurfaces on a sphere as intersection indices §18. Proofs of the index theorems §19. The indices of fronts of Legendrian submanifolds on an evendimensional sphere Chapter V. Exact Lagrangian curves on a sphere and their Maslov indices §20. Exact Lagrangian curves and their Legendrian lifts §21. The integral of a horizontal form as the area of the characteristic chain §22. A horizontal contact form as a Levi-Civita connection and a generalized Gauss-Bonnet formula §23. Proof of the formula for the Maslov index §24. The area-length duality §25. The parities of fronts and caustics Chapter VI. The Bennequin invariant and the spherical invariant §26. The spherical invariant §27. The topological meaning of the invariant Chapter VII. Pseudo-functions §28. The quasi-functions of Chekanov §29. From quasi-functions on the cylinder to pseudo-functions on the sphere, and conversely §30. Conjectures concerning pseudo-functions §31. Space curves and Sturm's theorem Bibliography