Introduction to Differential W-Geometry

Ideas recently put forward by Y. Matsuo and the author are summarized on the example of the simplest ($W_3$) generalization of two-dimensional gravity. These notes are based on lectures given at the workshop `` Strings, Conformal Models and Topological Field Theories'', Cargese 12-21 May 1993; and at the meeting ``String 93'', Berkeley, 24-29 May 1993.