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Studies in global geometry and analysis. (English) Zbl 0175.48102

MAA Studies in Mathematics. Vol. 4. Buffalo, N. Y.: Mathematical Association of America; Englewood Cliffs, N. J.: Distributed by Prentice-Hall, Inc. (1967).
The volumes of the MAA studies are intended to present a field of current interest in mathematical research to a general audience of college teachers (i.e., nonspecialists and not very research oriented). Some articles in the present volume reach this goal.
The contents are:
Marton Morse, What is Analysis in the Large? p. 5–15. (An introduction to Morse theory without proofs. The bibliography is particularly inadequate for the intended audience.)
S. S. Chern, Curves and Surfaces in Euclidean Space, p. 16–56. (Avery good survey of global differential geometry in 2 or 3 dimensions with a particularly good discussion of the theorems connected with Fenchel’s theorem about the total curvature of a space curve and an adequate treatment of rigidity theorems for convex surfaces. The proof of Hadamard’s theorem from the Gauss-Bonnet formula is correct only if the surface is assumed to be imbedded; in the full Hadamard theorem, \(K>0\) implies that the immersed surface is imbedded, see H. Hopf’s NYU Lecture Notes on differential geometry.)
H. Flanders, Differential Forms, p. 57–95. (A very careful introduction to differential forms and their applications. This article is perhaps the most adequate in the book for the purpose intended. Most of the material is mentioned in a hand-waving manner in Chern’s paper.)
S. Kobayashi, On conjugate and cut loci, p. 96–122. (A survey of the modern theory of the cut locus of interest both to the general reader and the specialist and with an adequate bibliography.)
L. Cesari, Surface Area, p. 123–146. (A report on the Cesari theory of Lebesgue area without proofs. Unfortunately, examples are also missing including the Schwarz-Peano example of the polyhedral approximation of the cylinder.)
L. A. Santalò, Integral Geometry, p. 147–194. (A reasonably modern introduction to Integral Geometry in the Blaschke style with many applications and an extensive bibliography.)
Reviewer: H. Guggenheimer

MSC:

53-06 Proceedings, conferences, collections, etc. pertaining to differential geometry
58-06 Proceedings, conferences, collections, etc. pertaining to global analysis
00B15 Collections of articles of miscellaneous specific interest
00A35 Methodology of mathematics
53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces
58A10 Differential forms in global analysis
53A45 Differential geometric aspects in vector and tensor analysis
53C20 Global Riemannian geometry, including pinching
53C22 Geodesics in global differential geometry
53C65 Integral geometry
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C05 Connections (general theory)