Applicable Differential Geometry (London Mathematical Society Lecture Note Series Book 59) by M. Crampin (PDF)

Description

This is an introduction to geometrical topics that are useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing readers for the study of modern treatments of mechanics, gauge fields theories, relativity and gravitation. The order of presentation corresponds to that used for the relevant material in theoretical physics: the geometry of affine spaces, which is appropriate to special relativity theory, as well as to Newtonian mechanics, is developed in the first half of the book, and the geometry of manifolds, which is needed for general relativity and gauge field theory, in the second half. Analysis is included not for its own sake, but only where it illuminates geometrical ideas. The style is informal and clear yet rigorous; each chapter ends with a summary of important concepts and results. In addition there are over 650 exercises, making this a book which is valuable as a text for advanced undergraduate and postgraduate students.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Disclaimer: This review is provided by someone who already has a working knowledge of differential geometry.The book introduces the reader to several abstract constructions on a general smooth manifold firstly on affine spaces. This is the viewpoint the book sticks to for the first 9 chapters. Starting in Chapter 10, these ideas are generalized to arbitrary smooth manifolds. I think this approach facilitates internalizing and learning the material immensely. I believe I would have had a much smoother and easier time understanding the core concepts in differential geometry if I had come across this book early in my graduate school days.The book has a good deal of exercises spread around in each chapter. Some of these exercises bolster the understanding of the material, while some of them are designed to get the readers to add some material which are alluded to or used later in the book. The exercises are usually not difficult and are concise — they should be considered as part of the textbook and tackled as soon as possible.The final few chapters of the book delve more into the theory of classical mechanics formulated in a coordinate-independent manner and finish with principal bundle structures. The exposition is again quite succinct and is at least as entertaining to read and study as the classic books of Kobayashi and Nomizu.

⭐This is a really fantastic book for anyone wanting to understand this subject from a practical and not abstract viewpoint

⭐I bought this book direct from CUP in 2002 for £33. At that time, I was impressed because it agreed with some of my prejudices, but over time, I have found more and more things not to like.[This is a review of

⭐by Crampin and Pirani, the 1987 reprint with corrections, ISBN 0521231906. My copy says it was “transferred to digital reprinting 1999”, printed in the USA.]This book has about 400 pages, equivalent to 224 A4 pages. The structure of the book is as follows.* Pages 1-163 (163 pages): Non-metric single-chart spaces.* Pages 164-235 (72 pages): Metric single-chart spaces.* Pages 236-382 (147 pages): Manifolds (with lots of charts).I mention this because the modern presentation of differential geometry generally starts with manifolds defined in terms of atlases of coordinate charts. This book doesn’t define manifolds until 62% of the way through the book.* Chapter 1 (21 pages) defines a flat affine space in terms of an abstract set with a difference operation by a linear space, which is probably the best way to define an affine space, but most DG authors don’t define affine spaces at all. There’s no real need for them.* Chapter 2 defines tangent and cotangent spaces, induced maps, smooth maps and parallelism on a flat affine space. The Christoffel symbol for parallel transport along a coordinate grid is calculated. Covariant derivatives are expressed in terms of the Christoffel symbol. (This is all done in the old-fashioned tensor calculus way.)* Chapter 3 is about flows, Lie transport and Lie derivatives, commutators of Lie derivatives (i.e. Lie bracket). This, of course, does not assume a connection. So in my opinion, this topic should be presented before parallelism.* Chapter 4 is on volume elements, exterior algebra and alternating tensors. This is presented in terms of coordinates, but still only in single-chart flat space.* Chapter 5 builds on exterior algebra to present exterior calculus, including the exterior derivative, which is expressed in terms of p-forms. Lie derivatives are then applied to p-forms.* Chapter 6 is about Frobenius’s Theorem, which gives necessary and sufficient conditions for integrability of a distribution.* Chapter 7 introduces the metric (and pseudo-metric) tensor on affine spaces, first as a constant metric tensor, and then generalising to a metric (or pseudo-metric) tensor field. From this, distance along curves and the Christoffel symbol for the Levi-Civita connection are derived.* Chapter 8 is about isometries, Killing fields, and groups of rotations and Lorentz transformations.* Chapter 9 is about the geometry of classical (hyper)surfaces embedded in Euclidean space, defining exterior forms, the Levi-Civita connection and the Riemann curvature tensor on such surfaces. This is all very much in the classical manner, i.e. up to about 1925.* Chapter 10 finally introduces manifolds! Also general tensors on manifolds, Frobenius’s Theorem on manifolds, and Riemannian metrics on manifolds.* Chapter 11 (30 pages) presents general affine connections (in the absence of a Riemannian metric) as parallel transport along curves, together with covariant derivatives and the Christoffel symbol for the connection. Then this is applied to torsion and Riemannian curvature, the Bianchi identities, connection forms and curvature forms. This is all done with heavy use of coordinates. The equation for geodesics in terms of the Christoffel symbol is given, and normal coordinates. Then the torsion-free Levi-Civita connection is introduced. Then the Ricci tensor, the curvature scalar and a short section on conformal geometry.* Chapter 12 is about Lie groups, the Lie algebra of a Lie group, left-invariant forms, and the exponential map.* Chapter 13 introduces tangent and cotangent bundles, plus the exponential map and Jacobi fields, which are expressed in terms of the Riemann curvature tensor (which does not require a metric). This is then applied to Euler-Lagrange and Hamiltonian mechanics.* Chapter 14 presents general fibre bundles. The purpose of this is clearly to allow the theory of connections to be formulated in full generality in the final chapter. The presentation of fibre bundles is fairly standard, starting from fibrations, adding a structure group, specialising to vector bundles, defining associated tensor bundles, frame bundles and principal bundles.* Chapter 15 finally presents general Koszul-style connections on vector bundles, connection forms, curvature forms, connections on principal fibre bundles, and horizontal lift maps.Some comments.* The authors say a few times that they are against the use of coordinates, and then they proceed to use coordinates anyway throughout the book. It’s unavoidable really because differential geometry is defined in terms of atlases of coordinate charts.* The print is too tiny. They seem to have photographically scaled down a 10 point TeX font to get it onto smaller pages. TeX fonts really don’t look pretty when they are scaled like that.* My overwhelming sense of this book is that it is old-fashioned, like something between the time of Darboux and the time of Levi-Civita.* My interest in this book almost completely disappeared when I obtained a copy of the first edition of Frankel’s

⭐, which is now in a

⭐. Frankel just completely blows away Crampin and Pirani in every way, and it’s half the price, and it’s much bigger and better, and it covers lots of physics and does everything in a very modern way, and it is very nicely typeset and printed! So I don’t know why anyone would buy Crampin and Pirani now.* They really, really need to halve the price. Something under $50 would be better.

⭐I think the book by Crampin and Pirani may serve as an example of a thoughtfully written and useful textbook.It treats those parts of differential geometry which are important in application (as the title indicates), especially in physics and related subjects.It is written in a clear and comprehensible style and may also be used by beginners not being exposed to lectures on the subject.Perhaps it should be mentioned that the book does not contain integration theory of forms.However, I do not regard this as a substantial drawback, there are many standard sources (Flanders, Spivak, …) to fill this gap after becoming familiar with the rich stuff which IS there.(No book on the subject gives you all you might want and the one under consideration gives you quite a lot.)

⭐Differential Geometry is a difficult subject to get to grips with. The old OU MSc course was based on this book, and as the course has been abandoned by the OU I’m trying to study it without tutor support.The book is fairly difficult to follow. Text is pretty dense, diagrams are sparse. The subject is fairly comprehensively covered though – both from a coordinate free modern viewpoint and a coordinate independent tensor approach. If this book is mastered then the Diff Geom that is used in branches of physics such as general relativity should be a breeze.

⭐Since this is a textbook for post-graduate mathematics and other students applying mathematics in their studies and researches, it is not possible to give more than a tentative view after using it for a few days and having worked through only the first two chapters.It is intended to be capable of use without reference to other books etc., and was written for a module in an MSc programme in mathematics. The notation takes some getting used to especially if you have not previously studies any tensor calculus.All I can say is that I will continue to work through it as differential geometry is a subject I enjoyed at under-graduate level.

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