15307498
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
121836338
Wave front evolution and equivariant Morse lemma
120329630
Weak asymptotics for the numbers of solutions of diophantine problems
122507359
The Theory of Singularities and its Applications
Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.
122633217
Poisson structures on the plane and other powers of volume forms
The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.
122136489
Spaces of functions with moderate singularities
118067094
Singularities of smooth mappings with additional structures : collected papers