Differential Geometry : From Elastic Curves to Willmore Surfaces / by Ulrich Pinkall, Oliver Gross.

Format:

Book

Author/Creator:

Pinkall, Ulrich.

Contributor:

Gross, Oliver.

Series:

Compact Textbooks in Mathematics, 2296-455X

Language:

English

Subjects (All):

Geometry, Differential.

Differential Geometry.

Local Subjects:

Differential Geometry.

Physical Description:

1 online resource (204 pages)

Edition:

1st ed. 2024.

Place of Publication:

Cham : Springer International Publishing : Imprint: Birkhäuser, 2024.

Summary:

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. Moreover, emphasis is given on topics that are useful for applications in science and computer graphics. Most often these applications are concerned with finding the shape of a curve or a surface that minimizes physically meaningful energy. Manifolds are not introduced as such, but the presented approach provides preparation and motivation for a follow-up course on manifolds, and topics like the Gauss-Bonnet theorem for compact surfaces are covered.

Contents:

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.

ISBN:

3-031-39838-6

OCLC:

1455769356