Differential Geometric Structures - Dover Books on Mathematics | Advanced Math Textbook for Geometry & Topology Studies | Perfect for University Courses & Self-Learning

Walter Poor's text, Differential Geometric Structures, is truly unique among the hundreds of currently available volumes on topological manifolds, smooth manifolds, and Riemannian geometry. Poor's book offers a treatment of Fiber Bundles and their applications to Riemannian geometry that goes well beyond a cursory introduction, and it does so while assuming minimal background on the part of the reader. I am unaware of any book that directly rivals it.Eventually, any serious student of Riemannian geometry will want to study fiber bundles. I will appeal to authority to support this claim; in the opening to Chapter 11 of his highly respected graduate text, Differentiable Manifolds 2nd ed., Lawrence Conlon writes:"A good command of the theory of principal bundles is essential for mastery of modern differential geometry (cf. [22], [23]). In recent years, principal bundles have also become central to key advances in mathematical physics (Yang-Mills theory) which have in turn generated exciting new mathematics (e.g. S. K. Donaldson's work on differentiable 4-manifolds [7], [11]). This chapter will be a brief introduction to principal bundles."Conlon's citations [22] and [23] refer to the two-volume set Foundations of Differential Geometry I & II by Kobayashi and Nomizu, considered for many years the definitive introduction to Ehresmann's theory of connections on principal fiber bundles in book form. Many readers have struggled with these two volumes since their appearance in the 1960s; the learning curve is steep because Kobayashi and Nomizu assume that their reader is already familiar with the material covered in the books by Chevalley (The Theory of Lie Groups), Steenrod (The Topology of Fiber Bundles), and Pontrjagin (Topological Groups), among others. It is clear that a more accessible study of fiber bundle theory would be widely appreciated. But despite the importance of bundle theory to modern geometry, it is surprising how few of the standard texts offer more than a brief introduction.Poor's book is the exception. It offers a solid introduction to bundle theory that is accessible to any reader who has had a first course in the theory of smooth manifolds. In the Preface, Poor states:" The prerequisites are the basic facts about manifolds as presented in the books by Auslander and Mackenzie, Boothby, Brickell and Clark, Hu, Lang, Matsushima, Milnor, Singer and Thorpe, or Warner;....." The preceding list of references includes many popular texts that were available in 1981 when Poor's book was published. Readers today may well have taken a first course in manifolds out of any one of a number of more recent texts; I still recommend Boothby's book to all of my students because I think it remains one of the clearest and most thorough introductions to Riemannian geometry available to this day. Others may have different choices. But the point is that an introductory course in smooth manifolds is adequate preparation for studying Poor's book. I would respectfully suggest, however, that a reader who has also studied (the pre-bundle version of) some elementary Riemannian geometry (Riemannian metric tensor, Levi-Civita connection, Riemann, Ricci and scalar curvature tensors, geodesics, etc.) will stand to benefit more from Chapter 3 in Poor, in which the bundle approach to furnishing a manifold with a Riemannian metric is studied. I should remark that Poor states in his Preface that almost all references to elementary manifold theory refer to Frank Warner's book, Foundations of Differentiable Manifolds and Lie Groups; he has also followed Warner in most of the basic notation in the book. It can therefore be helpful to have a copy of Warner on hand when reading Poor's book; at the time of this review, Warner's book, first published in 1971, is still in print.A list of the chapter titles in Poor's book may help inform the prospective reader who is considering buying it. As the titles reflect, after an introduction grounded in fiber bundle theory, Poor proceeds to study many of the topics found in standard differential geometry texts: Killing fields, bi-invariant metrics, complex manifolds, holonomy, etc. Poor also writes about some topics found in relatively few of the standard texts, such as Chern's formula for the Laplacian, Cartan connections, and spin structures. I have included page numbers to indicate the length of each chapter; these page numbers come from the 1981 McGraw-Hill hardback edition. The new Dover edition appears to be paginated differently.Chapter 1: An Introduction to Fiber Bundles (page 1)Chapter 2: Connection Theory for Vector Bundles (page 40)Chapter 3: Riemannian Vector Bundles (page 112)Chapter 4: Harmonic Theory (page 150)Chapter 5: Geometric Vector Fields on Riemannian Manifolds (page 166)Chapter 6: Lie Groups (page 185)Chapter 7: Symmetric Spaces (page 221)Chapter 8: Symplectic and Hermitian Vector Bundles (page 243)Chapter 9: Other Differential Geometric Structures (page 274)It is extremely gratifying to see Poor's 1981 book once again available through Dover. This book seemed to fly under the radar when it first appeared; for some reason that I never understood, it was not cited one tenth as often as it deserved. For readers who have the prerequisites in hand, Poor's book is an extremely accessible and well written introduction to fiber bundles in Riemannian geometry and additional topics as discussed above.