Author(s): A.V. Pogorelov
Publisher: Noordhoff
Title page
INTRODUCTION
PART ONE Theory of Curves
CHAPTER 1. The concept of curve
1. Elementary curve
2. Simple curve
3. General curve
4. Regular curve. Analytic definition of a curve
5. On the implicit representatian of a curve
6. Singular points on regular plane curves
7. Singular points on analytic curves, defined by equations in the implicit form
8. Asymptotes to curves
EXERCISES FOR CHAPTER 1
PROBLEMS AND THEOREMS FOR CHAPTER 1
CHAPTER II. Concepts for curves which are related to the concept of contact
1. Vector functions of a scalar argument
2. Tangent to a curve
3. The osculating plane to a curve
4. Contact of curves
5. Envelope of a family of curves, depending on a parameter
EXERCISES FOR CHAPTER II
PROBLEMS AND THEOREMS FOR CHAPTER II
CHAPTER III. Fundamental concepts for curves which are related to the concepts of curvature and torsion
1. Concept of arc length of a curve
2. Arc length of a smooth curve. Natural parametrization of a curve
3. Curvature of a curve
4. Torsion of a curve
5. The Frenet formulas. Natural equations of a curve
6. Plane curves
EXERCISES FOR CHAPTER III
PROBLEMS AND THEOREMS FOR CHAPTER III
PART Two Theory of surfaces
CHAPTER IV. Concept of surface
1. Elementary surface. Simple surface. General surface
2. Regular surface. Analytic definition of a surface
3. Special parametrizations of a surface
4. Singular points on regular surfaces
EXERCISES AND PROBLEMS FOR CHAPTER IV
CHAPTER V. Fundamental concepts for surlaces which are related to the concept of contact
1. Tangent plane to a surface
2. Lemma on the distance from a point to a surface. Contact of a curve with a surface
3. Osculating paraboloid. Classification of points on a surface
4. Envelope of a family of surfaces, depending on one or two parameters
5. Envelope of a family of surfaces, depending on one parameter
EXERCISES FOR CHAPTER V
PROBLEMS AND THEOREMS FOR CHAPTER V
CHAPTER VI. First quadratic form of a surface and concepts related to it
1. Length of a curve on a surface
2. Angle between curves on a surface
3. Surface area
4. Conformal mapping
5. Isometric surfaces. Bending of surfaces
EXERCISES FOR CHAPTER VI
PROBLEMS AND THEOREMS FOR CHAPTER VI
CHAPTER VII. Second quadratic form of a surface and questions about surface theory related to it
1. Curvature of a curve lying on a surface
2. Asymptotic directions. Asymptotic curves. Conjugate directions. Conjugate nets on a surface
3. Principal directions on a surface. Lines of curvature
4. Relation between the principal curvatures of a surface and the normal curvature in an arbitrary direction. Mean and Gaussian curvatures of a surface
5. Ruled surfaces
6. Surfaces of revolution
EXERCISES FOR CHAPTER VII
PROBLEMS AND THEOREMS FOR CHAPTER VII
CHAPTER VIII. Fundamental equations of the theory of surfaces
1. The Gauss formula for total curvature of a surface
2. Derived formulas
3. The Peterson-Codazzi formulas
4. Existence and uniqueness of a surface with givcn first and second quadratic forms
PROBLEMS AND THEOREMS FOR CHAPTER VIII
CHAPTER IX. Intrinsic geometry of surfaces
1. Geodesic curvature of a curve on a surface
2. Geodesic curves on a surface
3. Semigeodesic parametrization of a surface
4. Shortest curves on a surface
5. The Gauss-Bonnet theorem
6. Surfaces with constant Gaussian curvature
PROBLEMS AND THEOREMS FOR CHAPTER IX
BIBLIOGRAPHY
INDEX
