Differential Geometry of Curves and Surfaces by Masaaki Umehara

Differential Geometry of Curves and Surfaces

byMasaaki Umehara, Kotaro Yamada, Wayne Rossman

Kobo ebook | May 12, 2017

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This engrossing volume on curve and surface theories is the result of many years of experience the authors have had with teaching the most essential aspects of this subject. The first half of the text is suitable for a university-level course, without the need for referencing other texts, as it is completely self-contained. More advanced material in the second half of the book, including appendices, also serves more experienced students well.

Furthermore, this text is also suitable for a seminar for graduate students, and for self-study. It is written in a robust style that gives the student the opportunity to continue his study at a higher level beyond what a course would usually offer. Further material is included, for example, closed curves, enveloping curves, curves of constant width, the fundamental theorem of surface theory, constant mean curvature surfaces, and existence of curvature line coordinates.

Surface theory from the viewpoint of manifolds theory is explained, and encompasses higher level material that is useful for the more advanced student. This includes, but is not limited to, indices of umbilics, properties of cycloids, existence of conformal coordinates, and characterizing conditions for singularities.

In summary, this textbook succeeds in elucidating detailed explanations of fundamental material, where the most essential basic notions stand out clearly, but does not shy away from the more advanced topics needed for research in this field. It provides a large collection of mathematically rich supporting topics. Thus, it is an ideal first textbook in this field.


  • Curves:

    • What Exactly is a "Curve"?
    • Curvature and the Frenet Formula
    • Closed Curves*
    • Geometry of Spirals*
    • Space Curves
  • Surfaces:

    • What Exactly is a "Surface"?
    • The First Fundamental Form
    • The Second Fundamental Form
    • Principal and Asymptotic Directions
    • Geodesics and the Gauss-Bonnet Theorem
    • Proof of the Gauss-Bonnet Theorem*
  • Surfaces From the Viewpoint of Manifolds*:

    • Differential Forms
    • Levi-Civita Connections
    • The Gauss-Bonnet Formula for 2-Manifolds
    • Poincaré-Hopf Index Theorem
    • The Laplacian and Isothermal Coordinates
    • The Gauss and Codazzi Equations
    • Cycloids as Brachistochrones
    • Geodesic Triangulations of Compact Riemannian 2-Manifolds
  • Appendix A — Supplements:

    • A Review of Calculus
    • The Fundamental Theorems for Ordinary Differential Equations
    • Euclidean Spaces
  • Appendix B — Advanced Topics on Curves and Surfaces:

    • Evolutes and the Cycloid Pendulum
    • Convex Curves and Curves of Constant Width
    • Line Integrals and the Isoperimetric Inequality
    • First Fundamental Forms and Maps
    • Curvature Line Coordinates and Asymptotic Line Coordinates
    • Surfaces with K = 0
    • A Relationship Between Surfaces with Constant Gaussian Curvature and With Constant Mean Curvature
    • Surfaces of Revolution of Negative Constant Gaussian Curvature
    • Criteria of Typical Singularities
    • Proof of the Fundamental Theorem for Surfaces

Readership: Undergraduate and graduate students, and researchers interested in differential geometry of curve and surface theories.
Key Features:

  • Avoiding a dictionary-like presentation, the text reads as one clear, naturally flowing story
  • Many important topics, such as the fundamental theorems of both curves and surfaces, geodesics, the Gauss–Bonnet theorem, etc, are all explained succinctly
  • Unlike other textbooks, mathematically rich and special topics are presented, such as properties of spirals, classification of regular homotopy type of plane curves, the existence of isothermal coordinates and curvature line coordinates, and criteria for singularities
Title:Differential Geometry of Curves and SurfacesFormat:Kobo ebookPublished:May 12, 2017Publisher:World Scientific Publishing CompanyLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9814740268

ISBN - 13:9789814740265

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