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Riemannian geometry. Translated from the Portuguese by Francis Flaherty. (English) Zbl 0752.53001

Mathematics: Theory & Applications. Boston, MA etc.: Birkhäuser. xiii, 300 p. (1992).
This is a translation of the second edition of the Portuguese original of this book. The first edition is available in Portuguese only. A review for that edition can be found in Zbl 0505.53001. Besides the numerous corrections and modifications throughout the text, the second edition differs from the first one in the following modifications: To chapter 4 a rapid introduction to the study of tensors on a Riemannian manifold has been added, emphasizing the coordinate-free description. The main application of this part is the introduction of the fundamental equations of an isometric immersion in Chapter 6. Chapter 8 has been augmented by a discussion of the isometries of the hyperbolic space and their relation to conformal transformations of Euclidean spaces. A proof of the Theorem of Liouville for conformal transformations can be found in that section. Finally, Chapter 13 has been entirely rewritten. The sphere theorem, which is the main subject of this chapter is discussed in more detail. The part of Morse Theory, which is used in the proof of that theorem, is quoted from other work. But for the benefit of readers less familiar with Morse Theory, the proof of the sphere theorem in even dimension is presented in an independent manner, because this can be done without Morse Theory.

MSC:

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 0505.53001
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