Series: Encyclopaedia of Mathematical Sciences. VOL. 28
(Geometry I),
VINITI, Moscow, (1988), 298 pp. 62 figs. (in Russian).
Since the early work of Gauss and Riemann, differential geometry has
grown into a vast network of ideas and approaches, encompassing local
considerations such as differential invariants and jets as well as
global ideas, such as Morse theory and characteristic classes. In this
volume of the Encyclopaedia, the authors give a tour of the
principal areas and methods of modern differential geomerty. The book
is structured so that the reader may choose parts of the text to read
and still take away a completed picture of some area of differential
geometry. Beginning at the introductory level with curves in Euclidian
space, the sections become more challenging, arriving finally at the
advanced topics which form the greatest part of the book:
transformation groups, the geometry of differential equations,
geometric structures, the equivalence problem, the geometry of
elliptic operators. Several of the topics are approaches which are now
enjoying a resurgence, e.g. G-structures and contact geometry. As an
overview of the major current methods of differential geometry, this
book is a map of these different ideas which explains the
interesting points at every stop. The authors' intention is that
the reader should gain a new understanding of geometry from the
process of reading this survey.