This book is a translation of an authoritative introductory text
based on a lecture series delivered by the renowned differential
geometer, Professor S S Chern in Beijing University in 1980. The
original Chinese text, authored by Professor Chern and Professor
Wei-Huan Chen, was a unique contribution to the mathematics
literature, combining simplicity and economy of approach with depth
of contents. The present translation is aimed at a wide audience,
including (but not limited to) advanced undergraduate and graduate
students in mathematics, as well as physicists interested in the
diverse applications of differential geometry to physics. In
addition to a thorough treatment of the fundamentals of manifold
theory, exterior algebra, the exterior calculus, connections on
fiber bundles, Riemannian geometry, Lie groups and moving frames,
and complex manifolds (with a succinct introduction to the theory
of Chern classes), and an appendix on the relationship between
differential geometry and theoretical physics, this book includes a
new chapter on Finsler geometry and a new appendix on the history
and recent developments of differential geometry, the latter
prepared specially for this edition by Professor Chern to bring the
text into perspectives.
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