Differential Geometry in the Large: Seminar Lectures New York University 1946 and Stanford University 1956

Heinz Hopf (auth.)

These notes consist of two parts: Selected in York 1) Geometry, New 1946, Topics University Notes Peter Lax. by Differential in the 2) Lectures on Stanford Geometry Large, 1956, Notes J.W. University by Gray. are here with no essential They reproduced change. Heinz was a mathematician who mathema- Hopf recognized important tical ideas and new mathematical cases. In the phenomena through special the central idea the of a or difficulty problem simplest background is becomes clear. in this fashion a crystal Doing geometry usually lead serious allows this to to – joy. Hopf’s great insight approach for most of the in these notes have become the st- thematics, topics I will to mention a of further try ting-points important developments. few. It is clear from these notes that laid the on Hopf emphasis po- differential Most of the results in smooth differ- hedral geometry. whose is both t1al have understanding geometry polyhedral counterparts, works I wish to mention and recent important challenging. Among those of Robert on which is much in the Connelly rigidity, very spirit R. and in – of these notes (cf. Connelly, Conjectures questions open International of Mathematicians, H- of gidity, Proceedings Congress sinki vol. 1, 407-414) 1978, .

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Table of contents :
Front Matter….Pages 1-1
The Euler Characteristic and Related Topics….Pages 3-29
Selected Topics in Elementary Differential Geometry….Pages 30-46
The Isoperimetric Inequality and Related Inequalities….Pages 47-57
The Elementary Concept of Area and Volume….Pages 58-75
Front Matter….Pages 77-80
Introduction….Pages 81-81
Differential Geometry of Surfaces in the Small….Pages 82-99
Some General Remarks on Closed Surfaces in Differential Geometry….Pages 100-106
The Total Curvature (Curvatura Inteqra) of a Closed Surface with Riemannian Metric and Poincaré’s Theorem on the Singularities of Fields of Line Elements….Pages 107-118
Hadamard’s Characterization of the Ovaloids….Pages 119-122
Closed Surfaces with Constant Gauss Curvature (Hilbert’s Method) — Generalizations and Problems — General Remarks on Weinqarten Surfaces….Pages 123-135
General Closed Surfaces of Genus O with Constant Mean Curvature — Generalizations….Pages 136-146
Simple Closed Surfaces (of Arbitrary Genus) with Constant Mean Curvature — Generalizations….Pages 147-162
The Congruence Theorem for Ovaloids….Pages 163-173
Singularities of Surfaces with Constant Negative Gauss Curvature….Pages 174-184