Handbook of Finsler Geometry

It was over three years ago, at the annual meeting of the American Ma- ematical Society in San Diego, California, that Dr. Paul Roos of Kluwer asked me to poll Finsler geometers around the world as to their interest in writing a HANDBOOK OF FINSLER GEOMETRY. The result of that query was a resounding a?rmation, and here at long last, is the ?nal result. You have in your hands,the most complete and authoritative exposition of state-of-the-art Finsler geometry that is possible, today. Each of the eleven partsiscompletely independent of the rest, and each has been written with mathematics and science students in mind. These articles are accessible! P. L. Antonelli Edmonton, Alberta, Canada June, 2003 xv ACKNOWLEDGEMENTS The editors would like to express their sincere thanks to Vivian Spak, who typeset this book, and to Scott Berard, who kept our computers running. xvii HANDBOOK OF FINSLER GEOMETRY Volume 2 PART 1 Complex Finsler Geometry TadashiAikou Contents 1K¨ ¨ ahler Fibrations 9 1. 1 Fibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1. 2 LocalTreatments. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1. 3 Bott Connections. . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1. 3. 1 StructureTensors. . . . . . . . . . . . . . . . . . . . . . . 14 1. 3. 2 Bott Connections. . . . . . . . . . . . . . . . . . . . . . . 16 1. 4 K¨ahlerFibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2Complex Finsler Bundles 23 2. 1 Vector Bundles Over Complex Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2. 2 ComplexFinslerMetrics. . . . . . . . . . . . . . . . . . . . . . . 27 2. 2. 1 Complete Circular Domains and Minkowski Functionals . 27 r r?1 2. 2. 2 Complex Finsler Metrics on C and Kahler ¨ ¨ Metrics on P 30 2. 2. 3 Complex Finsler Metric on Vector Bundles . . . . . . . .