Differential Geometry : From Elastic Curves to Willmore Surfaces

Differential Geometry From Elastic Curves to Willmore Surfaces

This open access book covers the main topics for a course on the differential geometry of curves and surfaces. Unlike the common approach in existing textbooks, there is a strong focus on variational problems, ranging from elastic curves to surfaces that minimize area, or the Willmore functional. Mo...

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Detalles Bibliográficos
Autor principal: Pinkall, Ulrich (-)
Otros Autores: Gross, Oliver
Formato: Libro electrónico
Idioma:Inglés
Publicado: Cham : Springer International Publishing 2024.
Edición:1st ed. 2024.
Colección:Compact Textbooks in Mathematics,
Materias:
Geometry, Differential.
Differential Geometry.
Geometria diferencial
Llibres electrònics
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009805099806719
Tabla de Contenidos:
  • 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.

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