Differential Geometry in the Large: Seminar Lectures New York University 1946 and Stanford University 1956 (Lecture Notes in Mathematics, 1000)

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 .--Provided by publisher These notes consist of two parts: 1) Selected Topics in Geometry, New York University 1946, Notes by Peter Lax. 2) Lectures on Differential Geometry in the Large, Stanford University 1956, Notes by J.W. Gray. They are reproduced here with no essential change. Heinz Hopf was a mathematician who recognized important mathematical ideas and new mathematical phenomena through special cases. In the simplest background the central idea or the difficulty of a problem usually becomes crystal clear. Doing geometry in this fashion is a joy. Hopf's great insight allows this approach to lead to serious mathematics, for most of the topics in these notes have become the starting-points of important further developments. I will try to mention a few. It is clear from these notes that Hopf laid the emphasis on polyhedral differential geometry. Most of the results in smooth differential geometry have polyhedral counterparts, whose understanding is both important and challenging. Among recent works I wish to mention those of Robert Connelly on rigidity, which is very much in the spirit of these notes (cf. R. Connelly, Conjectures and open questions in rigidity, Proceedings of International Congress of Mathematicians, Helsinki 1978, vol. 1, 407-414) - A theory of area and volume of rectilinear'polyhedra based on decompositions originated with Bolyai and Gauss 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 C o n t e n t s , S e l e c t e d t o p i c s i n g e o m e t r y , N e w Y o r k U n i v e r s i t y , 1 9 4 6 / n o t e s b y P e t e r L a x D i f f e r e n t i a l g e o m e t r y i n t h e l a r g e , S t a n f o r d U n i v e r s i t y , 1 9 5 6 / n o t e s b y J . W . G r a y . Heinz Hopf ; With A Preface By S.s. Chern. Also Available In An Electronic Version.