Lectures On Differential Geometry (University Mathematics) Paperback – March 15, 2000
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"it is highly recommended for all mathematicians, from the advanced undergraduate student to the experienced professor." -- Prof. F Hirzebruch Max-Planck Institute, Bonn
- Item Weight : 1.39 pounds
- Paperback : 356 pages
- ISBN-10 : 9810234945
- ISBN-13 : 978-9810234942
- Product Dimensions : 6.22 x 0.83 x 9.21 inches
- Publisher : Wspc (March 15, 2000)
- Language: : English
- Best Sellers Rank: #8,749,564 in Books (See Top 100 in Books)
- Customer Reviews:
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With a willingness to dig in, Chern's exposition proves to be a satisfying excursion. Well worth the effort ! I note that part one (topology) of Simmon's text Introduction to Topology and Modern Analysis (1963, McGraw-Hill) is a helpful prerequisite to Chern (or, Buck's Advanced Calculus). Allowing prerequisites, Chern comes together most beautifully !
(1) Finsler Geometry (chapter eight, sixty pages). That is good ! Why is that good ? Chern writes "Solid State physics involves lattices and the geometry is naturally Finsler." (page 266). How do we begin ? Chern also writes: "The starting point for Finsler geometry is the first notion of the integral calculus, the calculation of arc-lengths."
(2) You will need solid acquaintance with definitions and terminology, such as: homeomorphism, isomorphism, equivalence relation, open sets, automorphism, Jacobian matrix, partitions. Assimilation of this terminology carries one quite far !
(3) I like to let S.S. Chern speak through his words: "Vector bundles and connections form the mathematical basis of gauge fields in physics." (page 69, chapter four) and "the simplest device connecting the local and global properties of a manifold is the integration of exterior differential forms on the manifold." (page 85) and "the relationship between integrals over a domain and integrals over its boundary is at the heart of calculus" (page 92), or "the concept of moving frames originates in mechanics." (page 205), finally: "Topologically, the m-dimensional torus and the 2m-dimensional real torus are homeomorphic. Yet, the former has a complex-manifold structure and thus has richer contents." (page 225). Every one of those statements is amplified aplenty throughout Chern's marvelous discourse.
(4) There is much here, and, of such continuing fascination, that it renders nearly impossible my the desire to isolate my favorite part(s) ! However, I would be remiss if I neglected to highlight Chern's exposition of Submanifolds (pages 18-28). Beautiful ! Also, allow me to accentuate the discussion of "r-dimensional distributions" (page 83) where Chern starts from Frobenius Theorem (local) and proceeds to integral manifolds (global). Thus, another trek proceeding from the 'local' to the 'global.'
(5) You do not get far without proofs. The preface includes physicist K.S.Lam extolling the virtues of these lectures for the benefit of physicists (Lam helped Chern prepare the chapter of Finsler Geometry). While the lectures are beneficial to that audience (physicists), I fear that the proofs may well be unfamiliar territory. A glance at the proof of theorem #2.3 (page 190) provides evidence of Chern's clarity of approach. That same clarity in the proof of theorem #1.1 (page 134): "There exists a Riemannian metric on any m-dimensional smooth manifold." Chern writes: "In the context of fiber bundles, the existence of a Riemannian metric implies the existence of a positive-definite smooth section of the bundle of symmetric-covariant tensors of order-two on M." Beautiful !
(6) Concluding my review: Advanced undergraduates will find these lectures challenging. Card-carrying Physicists might even find these lectures challenging. With prerequisites firmly in hand (Simmons or Buck) along with a certain degree of mathematical sophistication (for the understanding of proofs), the lectures are inspiring !
That is, the lectures are introductory, though, hardly elementary or necessarily easy !
Yet, ever so satisfying and enriching !
The point is: as an introductory text, the various ideas and structures are not well motivated. They may be economical in the way of the presentation. However, it never seems natural from the point of view of a beginner. It is more natural to start with Riemannian geometry and then proceed to the more general concept of vector bundles and connections. It is in Riemannian geometry, that it is natural to first introduce the concept of a geodesic, and this leads, though a lot of books dont do it this way, to the concept of Levi -Civita connection and therefore holonomy and curvature. The general concept of vector bundles and connections before introducing the Riemannian geometry, makes a complex subject even more abstract and though maybe economical from the point of view of the writers, are formidable for a reader.
Even the presentation of specific facts, the book should emphassize, for the benefit of the reader, the structrual (pictorial) aspects more than it does, to illuminate the essence of the formulas, for example, the way it introduces the theta forms on frame bundle omits entirely in mentioning that the essence of thse forms is simply the concept of a coframe. It merely constructs these forms using local coordinates, which seems to be quite tricky to get to its bottom.
rather condensed. However, it would not be beyond comprehension
if the crucial pictures are established. It is my personal opinion that the first crucial place where it should be understood without any compromise is the section on the frame bundle. Later chapters build on this. Previous chapters are
synthesized here. To any readers who are interested, you are invited to discuss this book. My email address is firstname.lastname@example.org (Notice there are two "l" in "topollogy")