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Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book III is aimed at the first-year graduate level but is certainly accessible to advanced undergraduates. It deals with invariance theory and discusses invariants both of Weyl and not of Weyl type; the Chern-Gauss-Bonnet formula is treated from this point of view. Homothety homogeneity, local homogeneity, stability theorems, and Walker geometry are discussed. Ricci solitons are presented in the...

Book V completes the discussion of the first four books by treating in some detail the analytic results in elliptic operator theory used previously. Chapters 16 and 17 provide a treatment of the techniques in Hilbert space, the Fourier transform, and elliptic operator theory necessary to establish the spectral decomposition theorem of a self-adjoint operator of Laplace type and to prove the Hodge Decomposition Theorem that was stated without proof in Book II. In Chapter 18, we treat the de Rham complex and the Dolbeault complex, and discuss spinors. In Chapter 19, we discuss complex geometry...