# Table of Contents

*Part I. Synthetic Methods and Results*

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1. Spherical geometry

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2. Euclid

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3. The theory of parallels

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4. Non-Euclidean geometry I

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5. Non-Euclidean geometry II

*Part II. Development Differential Geometry:*

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6. Curves

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7. Curves in space

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8. Surfaces

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9. Curvature for surfaces

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10. Metric equivalence of surfaces

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11. Geodesics

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12. The Gaussâ€“Bonnet theorem

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13. Constant curvature surfaces

*Part III. Recapitulation*

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14. Abstract surfaces

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15. Modelling non-Euclidean geometry

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16. Coda: generalizations.

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# Reviews:

*From the Mathematical Reviews*:

Differential geometry is a subject of basic importance for all mathematicians, regardless of their special interests, and it also furnishes essential ideas and tools to physicists and engineers. Its close ties with important portions of algebra, topology, non-Euclidean geometry, analysis generally (in particular with the theory of partial differential equations), mechanics and the general theory of relativity makes this importance more valuable. In recent years, it has become a luxury to offer a course in differential geometry in an undergraduate curriculum. And when such a course exists, its students often arrive with a modern introduction to analysis and linear algebra, but without having seen geometry since high school. The modern subject turns on problems that have emerged from the new foundations that are far removed from the ancient roots of geometry, and when we teach the new and cut off the past, students are left to find their own way to a meaning of “geometry” in differential geometry. This book is an attempt to carry the reader from the familiar Euclidean world to the recent state of development of differential geometry. The main theoretical thread that runs through this vast historical period in this book is the search for a proof of Euclid’s Parallel Postulate, and the eventual emergence of a new and non-Euclidean geometry. Another theme that also emerges is the importance of properties of a surface that are intrinsic, that is, independent of the manner in which the surface is embedded in space. This gave rise to the fruitful idea, due to Gauss and later developed in full generality by Riemann, of dealing with intrinsic differential geometry, that is, to geometrical questions that concern only geometry in the surface as evidenced by the nature of the length measurements on it. The book consists of three parts (it is written in sonata-allegro form). Part A (prelude) consists of five chapters and opens with a small dose of spherical geometry, in which some of the important ideas of non-Euclidean geometry are touched on. Chapters 2 and 3 treat Book I of Euclid’s Elements and the criticism of Euclid’s theory of parallels. Chapters 4 and 5 deal with an exposition of synthetic non-Euclidean geometry as introduced by Saccheri, Bolyai, and Lobachevskii. Part B consists of nine chapters and begins with Chapter 6, which presents the theory of plane curves in such a way as to give the basic general motivations once for all for the underlying concepts of differential geometry so that the concepts can be introduced without much motivation in the more complicated cases of space curves and surfaces. Chapter 7 deals with space curves. The concepts of arc length, curvature, and torsion are introduced and it is shown that these quantities form a complete set of invariants in the sense that any two curves for which these quantities are the same differ at most by a rigid motion. Chapter 8 presents the basic theory of surfaces in three-dimensional space and is followed by Chapter 8b on map projections, a particular application of definitions and apparatus associated to the sphere. Chapters 9 and 10 develop the analogue of curvature for curves on a surface, an intrinsic feature of the surface, the Gaussian curvature. In Chapter 11, the author introduces geodesics. Integrals of the Gaussian curvature are computed in Chapter 12 leading to the Gauss-Bonnet theorem and its global consequences. And, finally, in Chapter 13 an analytic recipe for a non-Euclidean plane, that is, a complete, simply connected surface of constant negative Gaussian curvature, is obtained. The final Part C of the book comprises the last three chapters. Chapter 14 has as its purpose the discussion of Hilbert’s famous theorem on the nonexistence in three-dimensional space of a complete, simply connected surface with constant negative Gaussian curvature. This leads to something more general, abstract surfaces. In the very interesting Chapter 15, the author constructs the models of non-Euclidean geometry and discusses their properties. Chapter 16 presents Riemann’s visionary lecture of 1854 along with the developments it motivated in differential geometry up to the turn of the century. This includes the modern idea of an n-dimensional manifold, Riemannian and Lorentz metrics, vector and tensor fields, the Riemann curvature tensor, covariant differentiation, and Levi-Civita parallelism. A number of problems are formulated at the end of the chapters. Interesting diversions are offered, such as Huygens’ pendulum clock and mathematical cartography; however, the focus of the book is on the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. In summary, the author has succeeded in making differential geometry an approachable subject for advanced undergraduates.

—*Reviewed by Andrzej Bucki*

*From Zentralblatt:*

This is a book for advanced undergraduates carrying “the reader from the familiar Euclid to the state of development of differential geometry at the beginning of the twentieth century”. The historical path is followed through the parallel postulate problem, non-Euclidean geometry (Saccheri, Gauss, Lobachevskij, Bolyai), Gauss’ intrinsic geometry of surfaces, constant-curvature surfaces according to Minding, modeling the non- Euclidean plane (Beltrami, Poincare, Hilbert), abstract surfaces, Riemannian manifolds, curvature and its constancy, and the Levi-Civita connection.

The subject is treated deductively based on definitions, theorems (their authors are mostly indicated), proofs (often near the original ones). All this is alternated with historical stories and linking comments. Sometimes the material “is a bit out of the historical sequence” and then it is specially noted. In a very lively manner the spherical and hyperbolic geometries, the classical theory of curves and surfaces and a great part of Riemannian geometry are presented, as well as some applications (the tautochrone and accurate clock of Huygens, map projections and mathematical cartography, Lorentz manifolds as space-time models). Characteristic of the author’s care to put the material in a wide historical perspective is the heading “Euclid revisited” (“I: The Hopf-Rinow Theorem” about geodesical completeness in Ch. 11 “Geodesics”; “II: Uniqueness of lines” (the case of nonpositive Gaussian curvature) in Ch. 12 “The Gauss-Bonnet Theorem”; “III: Congruences” in Ch. 13 “Constant-curvature surfaces”).

The historical stories contain several well-known details, e.g. Gauss’ student Waechter is mentioned as the first who established that the horosphere carries Euclidean planimetry, Rodrigues is named as the inventor of the Gauss map, Jacobi as the first who used the term “geodesic line” etc. The role of Dorpat (now Tartu, Estonia) University as an outpost and of Bartels as a pioneer in the development of differential geometry is brought up, in particular an achievement of Senff’s and Minding’s student K. Peterson (Anticipation of Mainardi- Codazzi equations). But the {it variable axiom systema} and formulae equivalent to the Frenet-Serret apparatus, given by Bartels and published in Senff’s prize work in 1831, are left aside, like the preliminary version of the fundamental theorem of surfaces by Peterson (in 1853).

A lot of additional information is contained in the exercises which are given after each of the 17 chapters (in total 167). There are some Appendices: “The Elements: Book I” (a survey), “On Euclidean rigid motions”, “Notes on selected exercises”. Riemann’s {it Habilitationsvortrag} is added in the (slightly modified) translation of M. Spivak.

*–Reviewed by Ue.Lumiste (Tartu)*