Lectures on Classical Differential Geometry

PREFACE BIBLIOGRAPHY CHAPTER 1. CURVES 1-1 Analytic representation 1-2 "Arc length, tangent " 1-3 Osculating plane 1-4 Curvature 1-5 Torsion 1-6 Formulas of Frenet 1-7 Contact 1-8 Natural equations 1-9 Helices 1-10 General solution of the natural equations 1-11 Evolutes and involutes 1-12 Imaginary curves 1-13 Ovals 1-14 Monge CHAPTER 2. ELEMENTARY THEORY OF SURFACES 2-1 Analytical representation 2-2 First fundamental form 2-3 "Normal, tangent plane" 2-4 Developable surfaces 2-5 Second fundamental form 2-6 Euler's theorem 2-7 Dupin's indicatrix 2-8 Some surfaces 2-9 A geometrical interpretation of asymptotic and curvature lines 2-10 Conjugate directions 2-11 Triply orthogonal systems of surfaces CHAPTER 3. THE FUNDAMENTAL EQUATIONS 3-1 Gauss 3-2 The equations of Gauss-Weingarten 3-3 The theorem of Gauss and the equations of Codazzi 3-4 Curvilinear coordinates in space 3-5 Some applications of the Gauss and the Codazzi equations 3-6 The fundamental theorem of surface theory CHAPTER 4. GEOMETRY ON A SURFACE. 4-1 Geodesic (tangential) curvature 4-2 Geodesics 4-3 Geodesic coordinates 4-4 Geodesics as extremals of a variational problem 4-5 Surfaces of constant curvature 4-6 Rotation surfaces of constant curvature 4-7 Non-Euclidean geometry 4-8 The Gauss-Bonnet theorem CHAPTER 5. SOME SPECIAL SUBJECTS 5-1 Envelopes 5-2 Conformal mapping 5-3 Isometric and geodesic mapping 5-4 Minimal surfaces 5-5 Ruled surfaces 5-6 lmaginaries in surface theory SOME PROBLEMS AND PROPOSITIONS APPENDIX: The method of Pfaffians in the theory of curves and surfaces ANSWERS TO PROBLEMS INDEX