Lectures on Differential Geometry
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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assume basis becomes called Chapter Chern choice Choose closed compact complex complex manifold complex structure condition connection consider constant containing coordinate neighborhood coordinate system corresponding covering curvature defined Definition denote derivative determined diffeomorphisms differential form direct discussion dual elements equality equation equivalent Example exists expressed exterior derivative exterior differential fact fiber finite formula frame field fundamental geometry given hand side Hence holds holomorphic identity implies induced integral Lemma length Lie group line segment linear linearly independent m-dimensional matrix means metric natural normal obtain Obviously operator orientation positive Proof properties prove Remark respect result Riemannian manifold satisfies smooth function smooth manifold smooth tangent vector structure submanifold sufficient Suppose surface tangent vector field tensor Theorem transformation unique variation vector bundle vector space zero