Bonn Wilhelm Klingenberg June,1977 vii From the Preface to the German Edition This book has its origins in a one-semester course in differential geometry which 1 have given many times at Gottingen, Mainz, and Bonn. Spivak: Differential Geometry I, Publish.

At the same time I would like to commend the editors of Springer-Verlag for their patience and good advice. In addition to the usual topics, it has a nice discussion of vector bundles, tubular neighborhoods and Morse theory. 1 wish to thank David for his work and his enthusiasm during the whole period of our collaboration. I trust that the merits of the German edition have survived whereas at the same time the efforts of David helped to elucidate the general conception of the Course where we tried to put Geometry before Formalism without giving up mathematical rigour. There are some e1aborations and several new figures have been added. Upon David Hoffman fell the difficult task of transforming the tightly constructed German text into one which would mesh well with the more relaxed format of the Graduate Texts in Mathematics series. Read reviews and buy Differential Geometry - (Dover Books on Mathematics) by Heinrich W Guggenheimer (Paperback) at Target. 342) that in a surface in which every parametrized geodesic is dened for all time (a complete surface), every two points are in fact joined by a geodesic of least length. MIT Press, Cambridge, Mass., 1958, and for the topology of surfaces: Massey, Algebraic Topology, Springer-Verlag, New York, 1977. Comprehensive Introduction to Differential Geometry, third edition, volume 1, Publish or Perish, Inc., 1999, p. Lectures on ordinary differential equations. Suitable references for ordin ary differential equations are Hurewicz, W. This English edition could serve as a text for a first year graduate course on differential geometry, as did for a long time the Chicago Notes of Chern mentioned in the Preface to the German Edition. This book may not be reproduced in any form without the permission of the publishers.

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