Differential Geometry in the Large

1. Front Matter

   Pages N2-VII

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2. Selected Topics in Geometry

   1. Front Matter

      Pages 1-1

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   2. The Euler Characteristic and Related Topics

      • Heinz Hopf

      Pages 3-29

   3. Selected Topics in Elementary Differential Geometry

      • Heinz Hopf

      Pages 30-46

   4. The Isoperimetric Inequality and Related Inequalities

      • Heinz Hopf

      Pages 47-57

   5. 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. Introduction

      • Heinz Hopf

      Pages 81-81

   3. Differential Geometry of Surfaces in the Small [summary sketch]

      • Heinz Hopf

      Pages 82-99

   4. Some General Remarks on Closed Surfaces in Differential Geometry

      • Heinz Hopf

      Pages 100-106

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

      • Heinz Hopf

      Pages 107-118

   6. Hadamard’s Characterization of the Ovaloids

      • Heinz Hopf

      Pages 119-122

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

      • Heinz Hopf

      Pages 123-135

   8. General Closed Surfaces of Genus O with Constant Mean Curvature - Generalizations

      • Heinz Hopf

      Pages 136-146

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

      • Heinz Hopf

      Pages 147-162

   10. The Congruence Theorem for Ovaloids

       • Heinz Hopf

       Pages 163-173

   11. Singularities of Surfaces with Constant Negative Gauss Curvature

       • Heinz Hopf

       Pages 174-184

4. Back Matter

   Pages 185-187

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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 mathema­ tical 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 ma­ thematics, for most of the topics in these notes have become the star­ ting-points of important further developments. I will try to mention a few. It is clear from these notes that Hopf laid the emphasis on poly­ hedral differential geometry. Most of the results in smooth differen­ tial 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 ri­ gidity, Proceedings of International Congress of Mathematicians, Hel­ sinki 1978, vol. 1, 407-414 ) • A theory of area and volume of rectilinear'polyhedra based on de­ compositions originated with Bolyai and Gauss.

• Geometrie
• Geometry
• Globale Differentialgeometrie
• curvature
• differential geometry
• Gaussian curvature
• Mean curvature
• Riemannian manifold

• Eidgenössische Technische Hochschule Zürich, Switzerland

  Heinz Hopf

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