This book explores geometric structures on manifolds locally modeled on a classical geometry. It surveys the theory, with a special emphasis on affine and projective geometry and is based on examples. The author tries to present examples as a way to suggest the general theory. The book is suitable as a graduate textbook and contains many exercises. Material from beginning graduate courses in topology, differential geometry, and algebra is assumed. The relationship between Lie groups and Lie algebras is heavily used.
The book is divided into three parts. Part 1 describes affine and projective geometry and provides some of the main background on these extensions of Euclidean geometry. As noted by Lie and Klein, most classical geometries can be modeled in projective geometry. This part is entitled “Affine and projective geometry” and has four chapters. The first one introduces affine geometry as the geometry of parallelism. Chapter 2 develops the geometry of projective space, viewed as the compactification of affine space. Ideal points arise as “where parallel lines meet.” Chapter 3 discusses the Cayley-Beltrami-Klein (CBK) model for hyperbolic geometry. In Chapter 4, the author develops notions of convexity. The CBK metric on hyperbolic space is a special case of the Hilbert metric on properly convex domains.
Part 2 entitled “Geometric manifolds” globalizes these geometric notions to manifolds, introducing locally homogeneous geometric structures in the sense of J. H. C. Whitehead and Charles Ehresmann in Chapter 5. The author associates to every transformation group (G,X)
 a category of geometric structures on manifolds locally modeled on the geometry of X
 invariant under the group G
. Chapter 6 discusses other examples of geometric structures. Chapter 7 deals with the general classification of (G,X)
-structures from the point of view of developing maps. The main result is an important observation due to W. P. Thurston that the deformation space of marked (G,X)
-structures on a fixed topology Σ
 is itself “locally modeled” on the quotient of the space Hom(π1(Σ),G))
 by the group Inn(G)
 of inner automorphisms of G
. Chapter 8 introduces the important notion of completeness, for taming the developing map. Chapter 8 also introduces some of the basic examples in the theory, for example Bieberbach’s theorems.
Part 3 entitled “Affine and projective structures” has seven chapters. Chapter 9 begins the classification of affine structures on surfaces. Chapter 10 offers a detailed study of left-invariant affine structures on Lie groups, and Chapter 11 discusses the question of whether, for a closed orientable affine manifold, completeness is equivalent to parallel volume. Finally, Chapter 12 expounds the notions of “hyperbolicity” of Vey and Kobayashi. Chapter 13 summarizes some aspects of the now blossoming subject of ℝP2
-structures on surfaces, in terms of the explicit coordinates and deformations which extend some of the classic geometric constructions on the deformation space of hyperbolic structures on closed surfaces. Chapter 14 describes the classical subject of ℂP1
-manifolds, which traditionally identify with projective structures on Riemann surfaces. Chapter 15 surveys known results, and the many open questions, in dimension three.
The last 45 pages of a book enclose appendices about: transformation groups, affine connections, representations of nilpotent groups, 4-dimensional filiform nilpotent Lie algebras, semicontinuous functions, SL(2,ℂ)
 and O(3,1)
, Lagrangian foliations of symplectic manifolds. The book is profusely illustrated.
Reviewer: Andrzej Szczepański (Gdańsk)