Preface The 1990 American Mathematical Society Summer Institute on Differen- tial Geometry took place at University of California, Los Angeles from July 9 to July 27, 1990. This was the largest AMS Summer Institute ever. There were 426 registered participants and 270 lectures. The organizing commit- tee for the Institute consisted of Robert Bryant, Duke University Eugenio Calabi, University of Pennsylvania S. Y. Cheng, University of California, Los Angeles H. Blaine Lawson, State University of New York, Stony Brook H. Wu, University of California, Berkeley and as co-chairmen, Robert E. Greene, University of California, Los Angeles, and S. T. Yau, Harvard Uni- versity. In the years since the previous AMS Summer Institute on Differential Geometry in 1973, the field has undergone a remarkable expansion, both in the number of people working in the area and in the number and scope of the topics under investigation. Even in the context of the rapid growth of mathematics as a whole during this period, the growth of geometry is striking. It is our hope that the three volumes of these proceedings, taken as a whole, will provide a broad overview of geometry and its relationship to mathematics in toto, with one obvious exception the geometry of complex manifolds and the relationship of complex geometry to complex analysis were the subject of one week of the (three-week) 1989 AMS Summer Institute on Several Complex Variables and Complex Geometry. While some topics in complex geometry arose naturally in the 1990 Summer Institute and are covered in these Proceedings, the coverage of this subject in 1989 justified a reduced emphasis in 1990. Thus the reader seeking a complete view of geometry would do well to add the second volume on complex geometry from the 1989 Proceedings to the present three volumes. Each week of the 1990 Summer Institute was given a general emphasis as to subject and the Proceedings volumes are organized in the same way. While overlap is natural, and indeed inevitable, the subjects of the volumes are as follows: XX I

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