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. One. Differential Invariants of Point and Dual Transformations.- Two. Local Properties of Analytic Transformations at their United Points.- Three. Invariants of Contact and of Osculation. The Concept of Cross-ratio in Differential Geometry.- Four. Principal and Projective Curves of a Surface, and Some Applications.- Five. Some Differential Properties in the Large of Algebraic Curves, their Intersections, and Self-correspondences.- Six. Extensions to Algebraic Varieties.- Seven. Veronese Varieties and Modules of Algebraic Forms.- Eight. Linear Partial Differential Equations.- Nine. Projective Differential Geometry of Systems of Linear Partial Differential Equations.- Ten. Correspondences between Topological Varieties.- Author Index.- Analytic Index.
