Intro -- Preface -- Acknowledgement -- Contents -- Part I Curves -- 1 Curves in Rn -- 1.1 What is a Curve in Rn? -- 1.2 Length and Arclength -- 1.3 Unit Tangent and Bending Energy -- 2 Variations of Curves -- 2.1 One-Parameter Families of Curves -- 2.2 Variation of Length and Bending Energy -- 2.3 Critical Points of Length and Bending Energy -- 2.4 Constrained Variation -- 2.5 Torsion-Free Elastic Curves and the Pendulum Equation -- 3 Curves in R2 -- 3.1 Plane Curves -- 3.2 Area of a Plane Curve -- 3.3 Planar Elastic Curves -- 3.4 Tangent Winding Number -- 3.5 Regular Homotopy -- 3.6 Whitney-Graustein Theorem -- 4 Parallel Normal Fields -- 4.1 Parallel Transport -- 4.2 Curvature Function of a Curve in Rn -- 4.3 Geometry in Terms of the Curvature Function -- 5 Curves in R3 -- 5.1 Total Torsion of Curves in R3 -- 5.2 Elastic Curves in R3 -- 5.3 Vortex Filament Flow -- 5.4 Total Squared Torsion -- 5.5 Elastic Framed Curves -- 5.6 Frenet Normals -- Part II Surfaces -- 6 Surfaces and Riemannian Geometry -- 6.1 Surfaces in Rn -- 6.2 Tangent Spaces and Derivatives -- 6.3 Riemannian Domains -- 6.4 Linear Algebra on Riemannian Domains -- 6.5 Isometric surfaces -- 7 Integration and Stokes' Theorem -- 7.1 Integration on Surfaces -- 7.2 Integration Over Curves -- 7.3 Stokes' Theorem -- 8 Curvature -- 8.1 Unit Normal of a Surface in R3 -- 8.2 Curvature of a Surface -- 8.3 Area of Maps Into the Plane or the Sphere -- 9 Levi-Civita Connection -- 9.1 Derivatives of Vector Fields -- 9.2 Equations of Gauss and Codazzi -- 9.3 Theorema Egregium -- 10 Total Gaussian Curvature -- 10.1 Curves on Surfaces -- 10.2 Theorem of Gauss and Bonnet -- 10.3 Parallel Transport on Surfaces -- 11 Closed Surfaces -- 11.1 History of Closed Surfaces -- 11.2 Defining Closed Surfaces -- 11.3 Boy's Theorem -- 11.4 The Genus of a Closed Surface -- 12 Variations of Surfaces.

12.1 Vector Calculus on Surfaces -- 12.2 One-Parameter Families of Surfaces -- 12.3 Variation of Curvature -- 12.4 Variation of Area -- 12.5 Variation of Volume -- 13 Willmore Surfaces -- 13.1 The Willmore Functional -- 13.2 Variation of the Willmore Functional -- 13.3 Willmore Functional Under Inversions -- A Some Technicalities -- A.1 Smooth Maps -- A.2 Function Toolbox -- B Timeline -- References -- Index.