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A comprehensive introduction to differential geometry. Vol. 2. (English) Zbl 0202.52201
Waltham, Mass.: Brandeis University. ix, 425 p. (not consecutively paged) (loose errata) (1970).
Summary: Differential Geometry has been variously described, at informal gatherings as “Differentiable manifolds and a little bit more”, or as “the attempt to find out what Élie Cartan really said!”. In opening the second volume of this work one is immediately struck by the fact that whereas the first volume was almost entirely on differentiable manifolds, in this volume we come to the “little bit more”. This volume could also be described in terms of the second of the above definitions, in that it tells its readers what Gauss, Riemann and Élie Cartan really said. Indeed, the title of one chapter is “What did Riemann say”, and evidently it is not so easy to find out what he did say. The history of the subject is quite dominant in this volume, with a pre-Gauss period, the subject as it developed under Gauss, then in historical order to Riemann, Ricci and Levi-Civita (with a break to introduce Koszul’s version of the 1950s), Cartan and Ehresmann.
Chapter 1 starts in a very innocent-looking way with the theory of curves in Euclidean 3-space. It is on page 46 that the author goes back to his chapter on Lie groups in the first volume to examine the rôle played by the theory of Lie groups in this theory of curves in space. The chapter ends with the classification of curves under particular groups of motions. Chapter 2 gives an account of the contributions of Euler and Meusnier.
Chapter 3A is a guide, section by section, to the fundamental paper by Gauss, written in Latin, of which an English translation has appeared under the title “General investigations of curved surfaces” which the author describes as the single most important work in the history of differential geometry. Chapter 3B is the substance of the Gauss memoir in modern language, starting with the Gauss and the Weingarten maps, which he denotes respectively by $$n$$ and $$dn$$. Since one sees the Weingarten map denoted by other letters like $$L$$ and $$S$$, one is reminded of still another definition of differential geometry as “what remains invariant under changes of notation”. Since $$n$$ is at any rate the first letter of the word “normal”, may the $$n$$ and $$dn$$ prevail!

Chapter 4 is again divided into two parts with (4A) devoted to a translation of Riemann’s actual address, and (4B) devoted to an explanation of it. The origin of what is called Riemannian Geometry is explained, with the reason which led Riemann to consider the well known Riemann normal coordinates. The idea of what has since become known as a Finsler metric is already implicit in the address, and the author gives an addendum on Finsler metrics. In the third and fourth sections of the chapter there is an indication of how near Riemann came to using what was soon to become the tensor calculus, – in his investigation of the curvature tensor.
In Chapter 5, the essentials of the tensor calculus are given, and the first definition of a connection, as an assignment of $$n^3$$ numbers to each coordinate system, with a certain law of transformation governing the passage from the set of $$n^3$$ numbers attached to one coordinate system to those of another system.
In Chapter 6 the contents are practically the same as for Chapter 5, with the Koszul symbolism replacing the tensor symbolism. A (Koszul) connection on a $$C^\infty$$ manifold $$M$$ is a function $$V$$ which associates a $$C^\infty$$ vector field $$\nabla_XY$$, to any two $$C^\infty$$ vector fields $$X$$ and $$Y$$ and which satisfies four conditions. This chapter ends with an invariant formulation of the first variation formula.
There are two addenda, one on connections having the same geodesics, and the other on Riemann’s invariant definition of the curvature tensor. The Koszul notation first appeared in the literature in 1954, about 80 years after the developments treated in Chapter 5.
Cartan’s contributions in the field of differential geometry appeared during the period 1920–1940. In his extensive use of the theory of moving frames, which is the subject matter of Chapter 7, he was influenced by Darboux. The first step emphasized in this chapter is the importance of the passage from the natural frame determined by a coordinate system to a general frame. This is the chapter in which we first meet the “structure equations” of a space, and one starts off on page 4 with the equations of structure of Euclidean space. Matrix notation proves convenient in this context. Moving frames with dual 1-forms are then defined on a manifold $$M$$, with the structure equations of a Riemannian manifold. A little later (page 27) the author comes back to the main assertions made by Riemann in this Habilitation lecture (4B–24). He proves a theorem on the exponential map and a corollary on the sectional curvatures of two Riemannian manifolds determining a local isometry (the curvature determines the metric). Whereas the first part of this chapter is restricted to Riemannian spaces, the remainder of this chapter is more general, and there are torsion forms as well as connection forms. There is an addendum on Riemannian manifolds with constant curvature, and another on Cartan’s treatment of normal coordinates.
To continue with history, it was the decade 1940–50 that saw the notion of fibre bundle come to fruition with its application to the modern formulation of the theory of connections. After some notes in the Comptes Rendus, Ehresmann put matters in a definite form as far as fibre bundles are concerned in the proceedings of a colloquium in Brussels in 1950. These matters are treated in Chapter 8. Since the idea behind the modern treatment of connection theory is to obtain a bundle whose sections are just the moving frames on $$M$$, the author imitates the construction of a tangent bundle. He defines a principal bundle over $$M$$ with group $$G$$ (with $$G$$ a Lie group), as a triple consisting of the (i) total space $$P$$ of the principal bundle, which is a $$C^\infty$$ manifold, (ii) the projection $$\pi$$ from $$P$$ to $$M$$, (iii) the action of $$G$$. The notion of fundamental vector field, section, and a Cartan connection (as an assignment of an $$n\times n$$ matrix of 1-forms to every section satisfying a certain essential condition) are introduced in preparation for the definition of a connection (in the Ehresmann sense) as a Lie algebra-valued 1-form $$\omega$$ on $$P$$ satisfying two conditions. It is proved that the $$\omega$$ determines a distribution $$H$$ called a horizontal distribution, which in turn leads to the definition of basic vector fields. Structural equations with torsion and curvature forms follow. The chapter ends with another look at Koszul’s connection $$\nabla$$ and the classical connection determined by functions on a coordinate neighbourhood, all in relation to the Ehresmann connection. The relationship between the various definitions of connection are summarized in a diagram. Four addenda are added to this chapter, on the tangent bundle of the frame bundle $$F(M)$$, on complete, connections, on connections in vector bundles, and on flat connections.
From the beginning of the first volume to the end of the second is quite an excursion, of over 1000 pages. The reader may be of the generation of the reviewer who was, in the early 1920s, trying to cope with the “debauch of indices” and at the same time trying to make out what Élie Cartan was saying in those long papers which were appearing at that time in the Annales de l’École Normale. Or he may be a young man with all the sophisticated modern treatises at his disposal. In either case, he should have a different outlook on the subject by the time he has finished. The author might have called these notes “Differential Geometry in 1970 as I see it”. By making them available in 1970 he has earned the gratitude of quite a variety of readers.
Reviewer: Evan Tom Davies

##### MSC:
 53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry