This book is based on the manuscripts for a course on the theory of connections which I gave at Nagoya University in the winter of 1955, and is presented here as an introduction to ipodern differential geometry.
I wish to express my sincere gratitude to Professors Y. Kawada and K. Yano who have kindly suggested me to rearrange the lecture notes as a book for publication. It is my pleasant duty to acknowledge many helpful suggestions which Mr. N. Ivvahori has made for the improvement of the final redaction. My thanks go also to Mr. H. Ozeki who has kindly helped me by frequent discussions on the content and rendered a lot of service in proofreading.