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Sobolev spaces on Riemannian manifolds. (English) Zbl 0866.58068
Lecture Notes in Mathematics. 1635. Berlin: Springer. x, 116 p. (1996).
The nice little monograph has been written with the dual aim of providing a global study of Sobolev spaces in the context of Riemannian manifolds and of demonstrating that the seemingly obvious idea that which is valid for Euclidean spaces must be valid also for Riemannian manifolds is completely false. The objectives have been achieved rather well. As an added bonus the author gives details of some of his unpublished results in collaboration with M. Vaugon in the last (fifth) Chapter which is devoted to the study of the influence of symmetries on Sobolev embeddings.
The first four Chapters consider the following: 1. The recent developments of Anderson and Anderson-Cheeger on harmonic coordinates and some packing results. 2. Sobolev spaces on Riemannian manifolds and some density problems. 3. Sobolev embeddings and 4. The best constants problems. In Chapter 4 the author improves some of his earlier results including some with M. Vaugon. At the end there is a comprehensive bibliography with 124 entries.
The English, though pleasant to read and adequately precise for the presentation of rather technical results, does falter at times, but even this gives the presentation a charm which is often lacking in texts of this kind.

MSC:
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58J05 Elliptic equations on manifolds, general theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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