Some Properties of Differentiable Varieties and Transformations

One. Differential Invariants of Point and Dual Transformations.- § 1. Local metrical study of point transformations.- § 2. Some topologico-differential invariants.- § 3. Projective construction of the above invariants.- § 4. Local metrical study of the dual transformations.- § 5. Calculation of the first order differential invariants just considered.- § 6. Some particular transformations. Relations between densities.- § 7. The curvature of hypersurfaces and of Pfaffian forms.- Historical Notes and Bibliography.- Two. Local Properties of Analytic Transformations at their United Points.- § 8. Coefficients of dilatation and residues of transformations in the analytic field.- § 9. Transfer to the Riemann variety.- § 10. Formal changes of coordinates.- § 11. Formal reduction to the canonical form for the arithmetically general transformations.- § 12. The case of arithmetically special transformations.- § 13. Criteria of convergence for the reduction procedure in the general case.- § 14. Iteration and permutability of analytic transformations.- § 15. On the united points of cyclic transformations.- § 16. Arithmetically general transformations not representable linearly.- Historical Notes and Bibliography.- Three. Invariants of Contact and of Osculation. The Concept of Cross-ratio in Differential Geometry.- § 17. Projective invariants of two curves having the same osculating spaces at a point.- § 18. A notable metric case.- § 19. An important extension.- § 20. Projective invariants of contact of differential elements of any dimension.- § 21. Two applications.- § 22. On certain varieties generated by quadrics.- § 23. The notion of cross-ratio on certain surfaces.- § 24. Applications to various branches of differential geometry.- § 25. Some extensions.- Historical Notes and Bibliography.- Four. Principal and Projective Curves of a Surface, and Some Applications.- § 26. Some results of projective-differential geometry.- § 27. The definition and main properties of the principal and projective curves.- § 28. Further properties of the above curves.- § 29. The use of the Laplace invariants and of the infinitesimal invariants.- § 30. Some classes of surfaces on which the concept of cross-ratio is particularly simple.- § 31. Point correspondences which conserve the projective curves.- § 32. Point correspondences which preserve the principal lines.- § 33. On the plane cone curves of a surface.- Historical Notes and Bibliography.- Five. Some Differential Properties in the Large of Algebraic Curves, their Intersections, and Self-correspondences.- § 34. The residues of correspondences on curves, and a topological invariant of intersection of two curves on a surface which contains two privileged pencils of curves.- § 35. A complement of the correspondence principle on algebraic curves.- § 36. A geometric characterization of Abelian integrals and their residues.- § 37. The first applications.- § 38. The equation of Jacobi, and some consequences.- § 39. The relation of Reiss, and some extensions.- § 40. Further algebro-differential properties.- Historical Notes and Bibliography.- Six. Extensions to Algebraic Varieties.- § 41. Generalizations of the equation of Jacobi.- § 42. Generalizations of the relation of Reiss.- § 43. The residue of an analytic transformation at a simple united point.- § 44. Some important particular cases.- § 45. Relations between residues at the same point.- § 46. The total residues of correspondences of valency zero on algebraic varieties.- § 47. The residues at isolated united points with arbitrary multiplicities.- § 48. Extensions to algebraic correspondences of arbitrary valency.- § 49. Applications to algebraic correspondences of a projective space into itself.- Historical Notes and Bibliography.- Seven. Veronese Varieties and Modules of Algebraic Forms.- § 50. n-regular points of differentiable varieties.- § 51. Some special properties of n-regular points of differentiable varieties.- § 52. On the freedom of hypersurfaces having assigned multiplicities at a set of points.- § 53. On the effective dimension of certain linear systems of hypersurfaces.- § 54. Two relations of Lasker concerning modules of hypersurfaces.- § 55. Some important criteria for a hypersurface to belong to a given module.- § 56. Some properties of the osculating spaces at the points of a Veronese variety Vd(n).- § 57. The ambients of certain subvarieties of Vd(n).- § 58. The isolated multiple intersections of d primals on Vd(n).- § 59. The regular multiple intersections on Vd(n).- § 60. A special property of the space associated with an isolated intersection on Vd(n) in the simple case.- § 61. On a theorem of Torelli and some complements.- Historical Notes and Bibliography.- Eight. Linear Partial Differential Equations.- § 62. Preliminary observations.- § 63. The reduction of differential equations to a canonical form.- § 64. Remarks on the solution of the differential equations.- § 65. The construction of the conditions of integrability.- § 66. The conditions of compatibility for a system of linear partial differential equations in one unknown.- § 67. The analytic case where the characteristic hypersurfaces intersect regularly.- § 68. An extension to the non-analytic case.- § 69. Some remarks on sets of linear partial differential equations in several unknowns.- § 70. The solution of a system of homogeneous equations.- § 71. The resolving system associated with a general set of m differential equations in m unknowns.- Historical Notes and Bibliography.- Nine. Projective Differential Geometry of Systems of Linear Partial Differential Equations.- § 72. r-osculating spaces to a variety.- § 73. Surfaces representing Laplace equations.- § 74. The hyperbolic case.- § 75. The parabolic case.- § 76. Surfaces representing differential equations of arbitrary order.- § 77. Varieties of arbitrary dimension representing Laplace equations.- § 78. Generalized developables.- § 79. Varieties of arbitrary order representing differential equations of arbitrary order.- § 80. The postulation of varieties by conditions on their r-osculating spaces.- Historical Notes and Bibliography.- Ten. Correspondences between Topological Varieties.- § 81. Products of topological varieties.- § 82. Correspondences and relations.- § 83. Inverse correspondences.- § 84. Homologous correspondences.- § 85. Topological invariants of correspondences between topological varieties.- § 86. Arithmetic and algebraic invariants.- § 87. Geometric invariants.- § 88. i-correspondences on topological varieties.- § 89. Semiregular correspondences and their products.- § 90. Characteristic integers of a semi-regular correspondence.- § 91. Involutory elementary s-correspondences.- § 92. Algebraic and skew-algebraic involutory transformations.- § 93. An extension of Zeuthen’s formula to the topological domain.- § 94. One-valued elementary correspondences.- § 95. Correspondences represented by differentiable varieties.- Historical Notes and Bibliography.- Author Index.- Analytic Index.