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Transformation Groups in Differential Geometry
Given a mathematical structure, one of the basic associated mathematical objects is its automorphism group. The object of this book is to give a biased account of automorphism groups of differential geometric struc tures. All geometric structures are not created equal; some are creations of ~ods while others are products of lesser human minds. Amongst the former, Riemannian and complex structures stand out for their beauty and wealth. A major portion of this book is therefore devoted to these two structures. Chapter I describes a general theory of automorphisms of geometric structures with emphasis on the question of when the automorphism group can be given a Lie group structure. Basic theorems in this regard are presented in §§ 3, 4 and 5. The concept of G-structure or that of pseudo-group structure enables us to treat most of the interesting geo metric structures in a unified manner. In § 8, we sketch the relationship between the two concepts. Chapter I is so arranged that the reader who is primarily interested in Riemannian, complex, conformal and projective structures can skip §§ 5, 6, 7 and 8. This chapter is partly based on lec tures I gave in Tokyo and Berkeley in 1965.
182 pages, Hardcover
First published September 1, 1972
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July 10, 2018
When a book is published in Springer's "Classics in Mathematics" series, it is fair to assume that you have an exquisitely written book with relevant content (there are exceptions to confirm the rule, though). Kobayashi's book is no exception, this book is a true gem. The sections are clearly structured, as are the statements of theorems and proofs. Though the text is not exactly aimed supporting geometric intuition, the content can be readily understood, given that one has the necessary foundations to read this text. Those would be, for the most part, knowledge corresponding to a solid first course in Riemannian geometry, and in some chapters some knowledge of complex geometry and characteristic classes (the famous textbooks by Kobayashi & Nomizu will be a good reference).
As for the content, the book presents the theory of transformation groups from a principle bundle/G-structure point of view, which is very useful knowledge in contemporary differential geometry. The main examples are introduced early on to supply some intuition for the later parts of the text. A lot of results concern the dimensionality of automorphism groups, or the question whether a given geometric structure has a Lie group as automorphism group. A large part is devoted to transformations of complex manifolds. Perhaps a small point of disappointment is the final chapter on Cartan geometry, which is quite sketchy in several parts (though fleshing out the details can often require an excessive amount of work, as you see by looking at more modern surveys of Cartan geometry).
As for the content, the book presents the theory of transformation groups from a principle bundle/G-structure point of view, which is very useful knowledge in contemporary differential geometry. The main examples are introduced early on to supply some intuition for the later parts of the text. A lot of results concern the dimensionality of automorphism groups, or the question whether a given geometric structure has a Lie group as automorphism group. A large part is devoted to transformations of complex manifolds. Perhaps a small point of disappointment is the final chapter on Cartan geometry, which is quite sketchy in several parts (though fleshing out the details can often require an excessive amount of work, as you see by looking at more modern surveys of Cartan geometry).
Displaying 1 of 1 review