This is a review text file submitted electronically to MR. Reviewer: W.M.Goldman Reviewer number: 014521 Address:

Math Dept. University of Maryland,College Park, Maryland 20742 Author: R.W.Sharpe Short title: Differential Geometry: Cartan’s generalization of Klein’s Erlangen program Control number: 1 453 120 Primary classification: 53C Secondary classification(s): Review text:

This book discusses geometric structures infinitesimally and locally modeled on classical geometries. A “classical geometry” is — according to Klein’s Erlangen program — the geometry of a homogeneous space G/H of a Lie group G. For Klein, the fundamental entities of the geometry are just classes of objects on G/H invariant under G. The author calls such a geometry a Klein geometry. A natural question is how (and whether) a given smooth manifold M can be modeled (infinitesimally, locally, or globally) on a Klein geometry. For example Euclidean geometry is the geometry of ℝ^{n} invariant under the group G of rigid motions of ℝ^{n}, and a Euclidean structure is just a flat Riemannian metric. Of closed surfaces, only the torus and Klein bottle can be locally modeled (or uniformized) by ℝ^{2} but every surface has an “infinitesimal Euclidean” — that is, Riemannian — geometry.

A Cartan geometry is a structure infinitesimally modeled on a Klein geometry. For example, Riemannian metrics are infinitesimally Euclidean geometries. A Riemannian structure is locally Euclidean if and only if the metric is flat — that is, if the Riemann curvature tensor vanishes. In a similar way, a general Cartan geometry possesses an infinitesimal invariant whose vanishing detects whether the infinitesimal model given by the Cartan geometry is an exact local model of the geometry. This invariant results by viewing the Cartan geometry as a connection on a certain principal H-bundle P over M — when M is G/H itself, the bundle is just the Lie group G, regarded as an H-bundle over G/H by the quotient map G ⟶ G/H. The Cartan geometry is then given by an H-equivariant homomorphism from the tangent bundle of P into the tangent bundle of G, interpreted as a connection. The local invariants of the geometric structure derive from the curvature of this connection.

This far-reaching theory generalizes (in the context of the theory of Lie groups and Lie algebras) the classical theory of “moving frames.” Points of the fiber of P ⟶ M over x ∈ M represent the infinitesimal “coordinate frames” at x. Globally defined objects on P are thus objects expressed in “all possible” coordinate frames and those which are H-equivariant are those which are invariant under all “allowable” changes of coordinates.

The first chapter, “In the Ashes of the Ether: Differential Topology,” discusses differentiable manifolds, fiber bundles, vector bundles, tensor fields and exterior differential calculus. The second chapter, “Looking for the Forest in the Leaves: Foliations,” discusses integrability conditions for vector bundles (the Frobenius theorem), foliations and holonomy. The third chapter, “The Fundamental Theorem of Calculus,” provides the main technical tools: Lie algebras, Maurer-Cartan forms on Lie groups and the Structure Equation. As its title suggests, the chapter deals with the integration of derivatives of functions. However, the functions take values in noncommutative Lie groups, and their derivatives (the “Darboux derivatives”) are exterior 1-forms ω with values in a Lie algebra. The structure equation dω + \frac{1}{2}[ω, ω] = 0 satisfied by a 𝔤-valued 1-form ω which is the “Darboux derivative” of a G-valued mapping is the genesis of the curvature tensor. The uniqueness (up to right-translations in G) of two functions whose Darboux derivatives are equal, is the origin of holonomy.

This material is developed carefully and the treatment is quite readable. (I found this section of the book very suitable for an introductory course on Lie groups and Lie algebras.)

After these introductory chapters, the book moves into the general theory of geometric structures. The fourth chapter, “Shapes Fantastic: Klein Geometries,” discusses Klein’s unification of geometries by group theory. The chapter discusses various motivating examples of Klein geometries: Euclidean, hyperbolic, elliptic geometry, similarity geometry (later called Weyl geometry), the geometry of Minkowski space, affine and real projective geometry, inversive geometry (later called Möbius geometry), Lie sphere geometry, and complex projective geometry. Klein united these disparate, yet interrelated, geometries by the algebraic properties of their symmetry groups.

The general theory developed in the first three chapters provides a modern language for the theory of homogeneous spaces. Careful attention is paid to the relationship between the geometry of the homogeneous space G/H and algebraic properties of the Lie algebras 𝔤 and 𝔥 corresponding to G and H respectively. With a view towards generalizing the Erlangen program, the author emphasizes the interpretation of G/H as the base of a principal H-bundle (whose total space is G). Local sections of this bundle — called gauges — are essential in computations. However, the choice of gauge is noncanonical and eventually the result of computations must be H-invariant. The results of the third chapter are used to discuss the infinitesimal geometry associated with a Klein geometry. The consequences of effectiveness — that H contain no subgroup normal in G — are discussed in detail, as well as the infinitesimal analogue of a gauge, which by the Fundamental Theorem of the third chapter, is equivalent to a gauge.

The fifth chapter, “Shapes High Fantastical: Cartan geometries,” is the book’s centerpiece. The theory of the preliminary four chapters synthesize into Cartan’s generalization of the Erlangen program. Specifically, the Cartan geometry (or Cartan connection, is given by a 1-form ω on the total space of the principal H-bundle P, with values in 𝔤. Cartan geometries are “lumpy” or “inhomogeneous” versions of Klein geometries, where curvature — the measure of lumpiness — may be defined as the Darboux derivative of an infinitesimal gauge.

With respect to a local gauge, curvature is a function Ω on M with values in the vector space Hom(Λ^{2}(𝔤/𝔥), 𝔤), which enjoys the natural structure of an H-module. Here is where representation theory enters the picture. Special properties of the geometry reflect the structure of the representation of H. For example, generalizing the notion of torsion of an affine connection, one says that a Cartan geometry is torsionfree if the curvature assumes values in the submodule Hom(Λ^{2}(𝔤/𝔥), 𝔥). The two Bianchi identities generalize to the identity dΩ = [ω, Ω] further constraining the values of the curvature. The geometry is infinitesimally effective if 𝔥 contains no nontrivial ideal of 𝔤. A geometry is first-order if the isotropy representation 𝔥 ⟶ 𝔤𝔩(𝔤/𝔥) is injective. A generalization of the Ricci contraction is an H-module homomorphism Hom(Λ^{2}(𝔤/𝔥), 𝔥) ⟶ (𝔤/𝔥)^{∗} ⊗ (𝔤/𝔥)^{∗}. The general condition that a general Cartan geometry be normal is that its curvature assume values in the kernel of this homomorphism.

When the pair (𝔤, 𝔥) is reductive — that is, when there is an Ad(H)- invariant complement to 𝔥 inside 𝔤 — covariant differentiation may be defined. Parallel transport and holonomy can be defined and the image of the curvature tensor is the Lie algebra of the holonomy representation.

When the infinitesimal obstruction (curvature) vanishes, the Cartan geometry is locally Klein. Then the tangential maps defined by ω integrate to rigid local coordinate charts. The corresponding process of development unrolls M — or rather its universal covering space — into the model space. Under the global hypothesis of completeness, M itself may be realized as a Clifford-Klein form, that is, a quotient of G/H by a discrete subgroup of G acting properly on G/H.

A particularly interesting discussion concerns mutations of Cartan geometries: two infinitesimal model geometries (with the same isotropy group H) defined by 𝔤_{1} and 𝔤_{2} respectively are mutants of each other if there is an H-equivariant isomorphism 𝔤_{1} ⟶ 𝔤_{2} as H-modules, (but not necessarily as Lie algebras). The extra flexibility of Cartan geometries provided by curvature suggests that geometries with mutant models should really be regarded as the same geometry. The basic example of mutations are the classical triumvirate of Klein geometries: the elliptic plane, the Euclidean plane, and the hyperbolic plane. In particular constant curvature geometries are mutants of flat geometries, and allowing mutation often simplifies curvature.

The general theory of Cartan geometries developed in these first five chapters specializes to three specific examples — Riemannian geometry, conformal geometry, and projective geometry — covered in the final three chapters of the book.

First Riemannian geometry is developed as a Cartan geometry. That a Cartan geometry modeled on Euclidean geometry is identical to the geometry arising from a field of positive definite inner products on the tangent spaces is far from obvious. This is proved, and the special properties of Riemannian curvature (Weyl curvature, Ricci curvature and scalar curvature) are derived from representation theory. Riemannian submanifolds (second fundamental form, principal curvatures) are treated from the point of view of the “locally ambient geometry Riemannian geometry” on a germ of a neighborhood of a submanifold in a Riemannian manifold. The chapter concludes with a discussion of isoparametric submanifolds and umbilic hypersurfaces.

The chapter on conformal geometry is the longest chapter and contains the most advanced results. Conformal geometry (like projective geometry) is not first-order, but closely relates to Riemannian geometry. Several notions of “conformal structure” are discussed: conformal classes of Riemannian metrics, Cartan geometries modeled on S^{n} with its conformal group (Möbius geometry,) Cartan geometries modeled on ℝ^{n} with its conformal group (Weyl geometry). The foundations of this theory are discussed in detail, especially the relations between these notions of conformal structure. The conformal theory of immersed submanifolds is discussed, with particular attention to curves and surfaces. The Willmore form is shown to be the intrinsic curvature associated to a Möbius geometry. This nontrivial application of the general methods developed earlier in the book well illustrates the power of this approach.

The chapter on projective geometry discusses geodesics, and the projective geometry associated to a Riemannian geometry. Cartan’s geometricization of second-order ordinary differential equations which are cubic polynomials in the first derivative, in terms of projective connections is developed as an exercise. Beltrami’s characterization of Riemannian manifolds whose geodesics are straight line as those of constant curvature is discussed.

The book concludes with five appendices. The first relates the theory to the more familiar theory of connections (“Ehresmann connections”). The second appendix is a mechanical application to rolling without slipping or twisting, related to the Levi-Civita connection. The last three appendices cover one-dimensional Klein geometries, the representation theory needed for the differential geometry of surfaces, and a characterization of principal bundles.

The author has provided a real service to the mathematical community by making this subject accessible with such an inviting text written at an elementary level. Valuable to a wide audience of mathematicians and physicists, this book should stimulate further research into hitherto unexplored applications of the ideas of Klein and Cartan.