Differential geometry in the large : seminar lectures, New York University, 1946 and Stanford University, 1956


Differential geometry in the large : seminar lectures, New York University, 1946 and Stanford University, 1956

Heinz Hopf ; with a preface by S.S. Chern

(Lecture notes in mathematics, 1000)

Springer-Verlag, c1989

2nd ed.

  • : U.S.
  • : Germany

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Includes bibliographical references


  • Selected topics in geometry : New York University, 1946 / notes by Peter Lax
  • Differential geometry in the large : Stanford University, 1956 / notes by J.W. Gray



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, .


Selected Topics in Geometry.- The Euler Characteristic and Related Topics.- Selected Topics in Elementary Differential Geometry.- The Isoperimetric Inequality and Related Inequalities.- The Elementary Concept of Area and Volume.- Differential Geometry in the Large.- Differential Geometry of Surfaces in the Small.- Some General Remarks on Closed Surfaces in Differential Geometry.- The Total Curvature (Curvatura Inteqra) of a Closed Surface with Riemannian Metric and Poincare's Theorem on the Singularities of Fields of Line Elements.- Hadamard's Characterization of the Ovaloids.- Closed Surfaces with Constant Gauss Curvature (Hilbert's Method) - Generalizations and Problems - General Remarks on Weinqarten Surfaces.- General Closed Surfaces of Genus O with Constant Mean Curvature - Generalizations.- Simple Closed Surfaces (of Arbitrary Genus) with Constant Mean Curvature - Generalizations.- The Congruence Theorem for Ovaloids.- Singularities of Surfaces with Constant Negative Gauss Curvature.

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