Differential Geometry in the Large

1. Front Matter

   Pages I-VII

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2. Selected Topics in Geometry

   1. Front Matter

      Pages 1-1

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   2. New York University 1946 Notes by Peter Lax

      1. The Euler Characteristic and Related Topics

         • Heinz Hopf

         Pages 3-29

      2. Selected Topics in Elementary Differential Geometry

         • Heinz Hopf

         Pages 30-46

      3. The Isoperimetric Inequality and Related Inequalities

         • Heinz Hopf

         Pages 47-57

      4. The Elementary Concept of Area and Volume

         • Heinz Hopf

         Pages 58-75

3. Differential Geometry in the Large

   1. Front Matter

      Pages 77-80

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   2. Stanford University 1956 Notes by J.W. Gray

      1. Introduction

         • Heinz Hopf

         Pages 81-81

      2. Differential Geometry of Surfaces in the Small

         • Heinz Hopf

         Pages 82-99

      3. Some General Remarks on Closed Surfaces in Differential Geometry

         • Heinz Hopf

         Pages 100-106

      4. The Total Curvature (Curvatura Inteqra) of a Closed Surface with Riemannian Metric and Poincaré’s Theorem on the Singularities of Fields of Line Elements

         • Heinz Hopf

         Pages 107-118

      5. Hadamard’s Characterization of the Ovaloids

         • Heinz Hopf

         Pages 119-122

      6. Closed Surfaces with Constant Gauss Curvature (Hilbert’s Method) — Generalizations and Problems — General Remarks on Weinqarten Surfaces

         • Heinz Hopf

         Pages 123-135

      7. General Closed Surfaces of Genus O with Constant Mean Curvature — Generalizations

         • Heinz Hopf

         Pages 136-146

      8. Simple Closed Surfaces (of Arbitrary Genus) with Constant Mean Curvature — Generalizations

         • Heinz Hopf

         Pages 147-162

      9. The Congruence Theorem for Ovaloids

         • Heinz Hopf

         Pages 163-173

      10. Singularities of Surfaces with Constant Negative Gauss Curvature

          • Heinz Hopf

          Pages 174-184

4. Back Matter

   Pages 185-189

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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, .

• Gaussian curvature
• Mean curvature
• Riemannian manifold
• curvature
• differential geometry
• differential geometry of surfaces

• Eidgenössische Technische Hochschule Zürich, Germany

  Heinz Hopf