The present volume contains, together with numerous addition and extensions, the course of lectures which I gave at Pavia (26 September till 5 October 1955) by invitation of the "Centro Internazionale Mate- matico Estivo". The treatment has the character of a monograph, and presents various novel features, both in form and in substance; these are indicated in the notes which will be found at the beginning and end of each chapter, Of the nine parts into which the work is divided, the first four are essentially differential in character, the next three deal with algebraic geometry, while the last two are concerned with certain aspects of the theory of differential equations and of correspondences between topo- logical varieties. A glance at the index will suffice to give a more exact idea of the range and variety of the contents, whose chief characteristic is that of establishing suggestive and sometimes unforeseen relations between apparently diverse subjects (e. g. differential geometry in the small and also in the large, algebraic geometry, function theory, topo- logy, etc. ); prominence is given throughout to the geometrical view- point, and tedious calculations are as far as possible avoided. The exposition has been planned so that it can be followed without much difficulty even by readers who have no special knowledge of the subjects treated.

Table of Contents:
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.