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Differential Geometry

From Elastic Curves to Willmore Surfaces

  • Textbook
  • Open Access
  • © 2024

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Overview

Authors:
  1. Ulrich Pinkall

    1. Department of Mathematics, Technical University of Berlin, Berlin, Germany

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    You can also search for this author in PubMed   Google Scholar

  2. Oliver Gross

    1. Department of Mathematics, Technical University of Berlin, Berlin, Germany

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  • This book is open access, which means that you have free and unlimited access
  • Focusses on variational problems
  • Includes a large number of graphical illustrations
  • Presents historic backgrounds and notes

Part of the book series: Compact Textbooks in Mathematics (CTM)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xi
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  2. Curves

    1. Front Matter

      Pages 1-1
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    2. Curves in \(\mathbb {R}^n\)

      • Ulrich Pinkall, Oliver Gross
      Pages 3-11Open Access
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    3. Variations of Curves

      • Ulrich Pinkall, Oliver Gross
      Pages 13-28Open Access
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    4. Curves in \(\mathbb {R}^2\)

      • Ulrich Pinkall, Oliver Gross
      Pages 29-46Open Access
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    5. Parallel Normal Fields

      • Ulrich Pinkall, Oliver Gross
      Pages 47-57Open Access
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    6. Curves in \(\mathbb {R}^3\)

      • Ulrich Pinkall, Oliver Gross
      Pages 59-84Open Access
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  3. Surfaces

    1. Front Matter

      Pages 85-85
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    2. Surfaces and Riemannian Geometry

      • Ulrich Pinkall, Oliver Gross
      Pages 87-103Open Access
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    3. Integration and Stokes’ Theorem

      • Ulrich Pinkall, Oliver Gross
      Pages 105-115Open Access
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    4. Curvature

      • Ulrich Pinkall, Oliver Gross
      Pages 117-129Open Access
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    5. Levi-Civita Connection

      • Ulrich Pinkall, Oliver Gross
      Pages 131-138Open Access
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    6. Total Gaussian Curvature

      • Ulrich Pinkall, Oliver Gross
      Pages 139-149Open Access
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    7. Closed Surfaces

      • Ulrich Pinkall, Oliver Gross
      Pages 151-160Open Access
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    8. Variations of Surfaces

      • Ulrich Pinkall, Oliver Gross
      Pages 161-179Open Access
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    9. Willmore Surfaces

      • Ulrich Pinkall, Oliver Gross
      Pages 181-191Open Access
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  4. Back Matter

    Pages 193-203
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Technical University of Berlin, Berlin, Germany

    Ulrich Pinkall, Oliver Gross

About the authors

Ulrich Pinkall was born in 1955. In 1979 he graduated in Mathematics and Physics at the University of Freiburg, where in 1981 he also got his PhD in Mathematics. From 1984 to 1986 he worked at the MPI fĂĽr Mathematik in Bonn. From 1986 to 2023 he taught Mathematics at the Technical University of Berlin. From 1995 to 1996 he was Five College Professor in Amherst. From 1992 to 2002 he was head of the Sonderforschungsbereich Differential Geometry and Quantum Physics.

Oliver Gross received his B.Sc. and M.Sc. in Mathematics at TU Berlin in 2019 and 2020 respectively. Currently he is a researcher in the collaborative research center SFB TRR 109 “Discretization in Geometry and Dynamics" at TU Berlin under the supervision of Prof. Ulrich Pinkall.


Bibliographic Information

  • Book Title: Differential Geometry

  • Book Subtitle: From Elastic Curves to Willmore Surfaces

  • Authors: Ulrich Pinkall, Oliver Gross

  • Series Title: Compact Textbooks in Mathematics

  • DOI: https://doi.org/10.1007/978-3-031-39838-4

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s) 2024

  • Softcover ISBN: 978-3-031-39837-7Published: 14 February 2024

  • eBook ISBN: 978-3-031-39838-4Published: 13 February 2024

  • Series ISSN: 2296-4568

  • Series E-ISSN: 2296-455X

  • Edition Number: 1

  • Number of Pages: XI, 203

  • Number of Illustrations: 14 b/w illustrations, 66 illustrations in colour

  • Topics: Differential Geometry

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