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Differential Geometry, Lie Groups, and Symmetric Spaces
A ninth printing of Helgason's 1978 textbook and reference published by Academic Press, itself a revision of and sequel to his 1962 Differential Geometry and Symmetric Spaces , based in turn on lectures he gave a the University of Chicago in 1958 and later at Columbia and MIT. He begins by explaining differential geometry, emphasizing Riemannian geometry, then applies that to the basic theory of Lie groups and Lie algebras. A two-volume sequence, Groups and Geometric Analysis and Geometric Analysis on Symmetric Spaces have been published in the Society's series Mathematical Surveys and Monographs. Annotation c. Book News, Inc., Portland, OR (booknews.com)
640 pages, Hardcover
First published December 28, 1978
47 people want to read
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October 22, 2023En el capítulo de tensores se menciona la equivalencia de un campo vectorial como derivación
y como vector tangente.
y como vector tangente.
August 18, 2014
Yes, I have read it. From cover to cover.
This book is one of _the_ classics in differential geometry. It is the first and to date only book presenting the complete structure theory and classification of Riemannian symmetric spaces, together with the complete fundamentals in differential geometry and Lie groups needed to develop it. Compiling all this is a tremendous achievement and both, author and book, have received some well-deserved praise for it.
However... there are quite a few downsides to it. The most obvious one is that the book is really hard to read. The proofs are sometimes extremely detailed on technical issues, but often gloss over conceptual details very quickly (the all-too-typical assumption among maths authors that once the reader has read it, he understands and remembers it perfectly henceforth). Also, the proofs contain a few mistakes, which one hardly notices due to the lack of detail, but it becomes clear when one checks the added comments and remarks in the new AMS printing. Another problem is the poor structuring of the content; it is very hard to use the book as a reference, because the index is practically useless and many important results are not in the place you would expect them to be. So unless you know the book really well (from reading it cover to cover, for example...), you will have a hard time finding what you're looking for. (For example, many general facts on symmetric spaces have been postponed to the section on symmetric spaces of non-compact type, even though logically you would expect them in the section on the general structure theory of symmetric spaces).
Nevertheless, it is a very valuable book, albeit one which requires an unpleasant amount of work from its reader; but then, I hear some people say, isn't that what mathematics is all about...?
This book is one of _the_ classics in differential geometry. It is the first and to date only book presenting the complete structure theory and classification of Riemannian symmetric spaces, together with the complete fundamentals in differential geometry and Lie groups needed to develop it. Compiling all this is a tremendous achievement and both, author and book, have received some well-deserved praise for it.
However... there are quite a few downsides to it. The most obvious one is that the book is really hard to read. The proofs are sometimes extremely detailed on technical issues, but often gloss over conceptual details very quickly (the all-too-typical assumption among maths authors that once the reader has read it, he understands and remembers it perfectly henceforth). Also, the proofs contain a few mistakes, which one hardly notices due to the lack of detail, but it becomes clear when one checks the added comments and remarks in the new AMS printing. Another problem is the poor structuring of the content; it is very hard to use the book as a reference, because the index is practically useless and many important results are not in the place you would expect them to be. So unless you know the book really well (from reading it cover to cover, for example...), you will have a hard time finding what you're looking for. (For example, many general facts on symmetric spaces have been postponed to the section on symmetric spaces of non-compact type, even though logically you would expect them in the section on the general structure theory of symmetric spaces).
Nevertheless, it is a very valuable book, albeit one which requires an unpleasant amount of work from its reader; but then, I hear some people say, isn't that what mathematics is all about...?
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February 8, 2024Corollary 4.4. Let G be a connected nilpotent Lie group with Lie algebra Lie(G). Then
the exponential map is a regular mapping of Lie(G) ONTO G.
En el capítulo de tensores se menciona la equivalencia de un campo vectorial como derivación
y como vector tangente.
the exponential map is a regular mapping of Lie(G) ONTO G.
En el capítulo de tensores se menciona la equivalencia de un campo vectorial como derivación
y como vector tangente.
Displaying 1 - 3 of 3 reviews