國立臺灣大學數學系圖書室

This is the sixth edition of the famous book by Jost (as a complement to this review, readers are encouraged to consult reviews of the earlier editions). It presents both Riemannian geometry and some analytical methods to tackle and solve geometric problems, involving geometric flows that have received new attention with Perelman's resolution of the Poincaré conjecture in 2003. The present book is divided into ten chapters, each of them including very interesting perspectives on the domain and exercises. Below is a short description of each chapter.
  Chapter 1 concerns Riemannian manifolds, with an emphasis on geodesics. Namely, after defining differentiable manifolds, tangent space, submanifolds and Riemannian metrics, the author defines geodesics and proves their local existence and uniqueness. The main task of this chapter is the proof of the following global existence theorem: On a compact Riemannian manifold, the homotopy class of paths joining two points contains a minimizing geodesic. This theorem is proved via two different analytic methods: the first uses the local existence of geodesics; the second uses an adapted heat flow. This latest method is a very representative example of one of the objectives of this book: that is, using deep analytic methods to solve geometric problems. This chapter ends with the Hopf-Rinow theorem on the equivalent definitions of complete manifolds and the existence of global geodesics.
  Chapter 2 is about Lie groups and vector bundles. In particular, the integration of vector fields is dealt with here, and Spin structures are defined.
  Chapter 3 concerns the Laplace operator and harmonic differential forms. For the Laplacian acting on functions, one can find, for instance, Weyl's asymptotic formula and the minoration of the spectrum via Cheeger's constant. For the Laplacian on forms, the author concentrates on Hodge's theorem; here again, he gives two different proofs of it. The first proof is by minimizing the Dirichlet energy; the second proof is via a heat flow.
  Chapter 4 introduces the fundamental concepts of connection and curvature. In a very general way, these concepts are defined on vector bundles, which leads to the Yang-Mills functional, for which some examples of existence of minimizers are given. After introducing the Levi-Civita connection, the author proves the Weitzenböck formula and gives an application to the finiteness of dimension of Killing fields in the compact setting. A section is devoted to the Bochner method and some of its applications, like Lichnerowicz's minorations of λ1 and the upper bound of the dimension of the first de Rham cohomology group. Then the famous minoration of λ1 by Li and Yau is proved. The end of this chapter is devoted to submanifolds, including the Gauss equation and the Theorema Egregium on minimal surfaces.
  Chapter 5 focuses on some applications of geodesics and Jacobi fields. After some nice applications of the variational formulae (geodesics on manifolds with nonpositive curvature; Synge's theorem on positively curved manifolds), Jacobi fields and conjugate points are introduced and their basic properties are proved. Many applications are provided: the Gauss lemma, minimizing distance geodesics, the Morse index theorem, the Bonnet-Myers theorem, the structure of manifolds with constant curvature, the Rauch comparison theorem, and a section about the geometry of manifolds of nonpositive sectional curvature.
  Before going into more advanced topics (in the five following chapters), the author provides ``a short survey on curvature and topology'', devoted to issues on the relation between these two topics. In this survey are discussed: the 1/4-pinched theorem, the use of the Ricci flow and its recent application to Thurston's geometrization, manifolds with nonnegative Ricci curvature, flat manifolds, finiteness and convergence theorems.
  Chapter 6 is devoted to symmetric spaces and Kähler manifolds. The first two sections deal with the first properties of complex and Kähler manifolds. The rest of this chapter concerns (locally) symmetric spaces, including homogeneous spaces: Killing fields, curvature tensor, adjoint representation, structure, explicit examples (SL(n,R)/SO(n,R)).
  Chapter 7 deals with Morse theory and Floer homology. Morse theory yields a relation between the critical set of a Morse function f:X→R (that is, a function with nondegenerate critical points only) and the topology of X. One striking result is the expression of the Euler characteristic χ(X)=∑(−1)μ(p)μ(p), where the sum is taken on all critical points of f and μ(p) is the Morse index at a critical point p (that is, the dimension of the negative space of the Hessian at p). In the same spirit, Floer homology deals with pairs of critical points, a context in which a homology can be defined. The following tools and concepts are presented: the Morse function, the Morse index, stable/unstable manifolds, the Morse lemma, Betti numbers, the Euler characteristic, and gradient flow. An application of gradient flow and the Palais-Smale condition to the existence of a nontrivial closed geodesic on a compact manifold concludes this chapter.
  Chapter 8 is devoted to harmonic maps. The different associated notions are defined: (weak) harmonic maps and (un)stable harmonic maps. The author's emphasis is on harmonic maps with values in a compact manifold with nonpositive sectional curvature: here, the stability of harmonic maps, the existence in any homotopy class (Eells and Sampson's result) and the regularity of minimizers in a given homotopy class are proved. Applications of the Bochner technique are also presented.
  Chapter 9 is devoted to harmonic maps on Riemann surfaces (that is, in two real dimensions). The first part of this chapter presents the complex structure (conformal or not) of a Riemann surface and some properties of the Dirichlet energy (conformal invariance, for instance). The second part is devoted to the proof that any smooth function from a Riemann surface to a compact Riemannian manifold with vanishing π2 is homotopic to a harmonic map (and even a minimizing one in a homotopy class): more generally, via blow-up and compactness arguments, the absence of a harmonic map in a homotopy class for a general target N yields the existence of a minimal 2-sphere v:S2→N (which does not hold in the case π2(N)=0). Chapter 9 concludes with regularity results for harmonic maps, in particular the continuity of weak harmonic maps when the target has bounded geometry.
  Chapter 10 presents three variational problems arising from quantum field theory. The first two functionals are the Ginzburg-Landau and Seiberg-Witten functionals; the associated Euler-Lagrange equations are computed and the functionals are rewritten as expressions involving the curvature and/or the topology of the associated manifolds. The author also proves pointwise control of some solutions of the Euler-Lagrange equation in terms of the topology or the curvature. The third functional is a Dirac-Dirichlet functional, and the solutions of the associated Euler-Lagrange equation are Dirac-harmonic maps; many properties of these maps are given, in particular the influence of topology on the existence of such maps.
  The book concludes with two Appendices. Appendix A recalls properties of Sobolev spaces, and existence and regularity for solutions to elliptic and parabolic linear equations. Appendix B lists some topological results about fundamental groups and covering spaces.
  {For the fifth edition see [J. Jost, Riemannian geometry and geometric analysis, Universitext, Springer, Berlin, 2008; MR2431897].}

Reviewer: Robert, Frédéric [form MathSciNet]