Elementary Topics in Differential Geometry

1. Front Matter

   Pages i-xiii

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2. Graphs and Level Sets

   • J. A. Thorpe

   Pages 1-5

3. Vector Fields

   • J. A. Thorpe

   Pages 6-12

4. The Tangent Space

   • J. A. Thorpe

   Pages 13-15

5. Surfaces

   • J. A. Thorpe

   Pages 16-22

6. Vector Fields on Surfaces; Orientation

   • J. A. Thorpe

   Pages 23-30

7. The Gauss Map

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   Pages 31-37

8. Geodesics

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   Pages 38-44

9. Parallel Transport

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   Pages 45-52

10. The Weingarten Map

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    Pages 53-61

11. Curvature of Plane Curves

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    Pages 62-67

12. Arc Length and Line Integrals

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    Pages 68-81

13. Curvature of Surfaces

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    Pages 82-94

14. Convex Surfaces

    • J. A. Thorpe

    Pages 95-107

15. Parametrized Surfaces

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    Pages 108-120

16. Local Equivalence of Surfaces and Parametrized Surfaces

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    Pages 121-131

17. Focal Points

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    Pages 132-138

18. Surface Area and Volume

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    Pages 139-155

19. Minimal Surfaces

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    Pages 156-162

20. The Exponential Map

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    Pages 163-176

In the past decade there has been a significant change in the freshman/ sophomore mathematics curriculum as taught at many, if not most, of our colleges. This has been brought about by the introduction of linear algebra into the curriculum at the sophomore level. The advantages of using linear algebra both in the teaching of differential equations and in the teaching of multivariate calculus are by now widely recognized. Several textbooks adopting this point of view are now available and have been widely adopted. Students completing the sophomore year now have a fair preliminary under­ standing of spaces of many dimensions. It should be apparent that courses on the junior level should draw upon and reinforce the concepts and skills learned during the previous year. Unfortunately, in differential geometry at least, this is usually not the case. Textbooks directed to students at this level generally restrict attention to 2-dimensional surfaces in 3-space rather than to surfaces of arbitrary dimension. Although most of the recent books do use linear algebra, it is only the algebra of ~3. The student's preliminary understanding of higher dimensions is not cultivated.

• Differentialgeometrie
• Isometrie
• Minimal surface
• curvature
• differential geometry

• Queens College, City University of New York, Flushing, USA

  J. A. Thorpe

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