Elementary differential geometry.

*(English)*Zbl 0959.53001
Springer Undergraduate Mathematics Series. London: Springer. x, 332 p. (2001).

This book is a very well organized, rigorous and modern introduction to differential geometry, through theory of curves and surfaces in three-dimensional Euclidean space. The only pre-requisites are a good working knowledge of calculus, vectors and linear algebra, (and some background in complex analysis for Chapter 9).

The first chapter deals with the intuitive notion of a curve in the plane and in the space. This is followed by a chapter on the curvature and the Frenet-Serret apparatus. In Chapter 3 the author proves the isoperimetric inequality and the four vertex theorem in the Plane. Chapter 4 is dedicated to several different ways to formulate mathematically the notion of a surface in three-dimensional Euclidean space. In Chapters 5 and 6, the author introduces the basics of local surface theory through their first and second fundamental forms, and gives some geometric interpretations of the principal curvatures. An investigation of the Gaussian and mean curvatures in given in Chapter 7. One proves that, on a compact surface, there is a point where the Gaussian curvature is positive. The chapter ends with a discussion of the Gauss map and a related geometric interpretation of the Gaussian curvature. The next chapter deals with the geodesics of a surface. One proves the famous Clairaut’s theorem about geodesics on surfaces of revolution. Chapter 9 contains a good introduction to minimal surfaces and it concludes with a simple approach to the Weierstrass representation for minimal surfaces. In Chapter 10, the author proves Gauss’ Theorema Egregium. The last section of this chapter is dedicated to compact surfaces of constant Gaussian curvature and one proves the following beautiful theorem: every compact surface whose Gaussian curvature is constant is a sphere. The final chapter contains the most beautiful and profound result in the theory of compact surfaces: the Gauss-Bonnet theorem.

Examples are interspersed throughout, not only for the definitions, but also for the theorems. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that has just been introduced. Full solutions to all exercises are given at the end of the book. The book is an ideal text for self-study or a one-semester course in differential geometry for undergraduates in the mathematical sciences.

The first chapter deals with the intuitive notion of a curve in the plane and in the space. This is followed by a chapter on the curvature and the Frenet-Serret apparatus. In Chapter 3 the author proves the isoperimetric inequality and the four vertex theorem in the Plane. Chapter 4 is dedicated to several different ways to formulate mathematically the notion of a surface in three-dimensional Euclidean space. In Chapters 5 and 6, the author introduces the basics of local surface theory through their first and second fundamental forms, and gives some geometric interpretations of the principal curvatures. An investigation of the Gaussian and mean curvatures in given in Chapter 7. One proves that, on a compact surface, there is a point where the Gaussian curvature is positive. The chapter ends with a discussion of the Gauss map and a related geometric interpretation of the Gaussian curvature. The next chapter deals with the geodesics of a surface. One proves the famous Clairaut’s theorem about geodesics on surfaces of revolution. Chapter 9 contains a good introduction to minimal surfaces and it concludes with a simple approach to the Weierstrass representation for minimal surfaces. In Chapter 10, the author proves Gauss’ Theorema Egregium. The last section of this chapter is dedicated to compact surfaces of constant Gaussian curvature and one proves the following beautiful theorem: every compact surface whose Gaussian curvature is constant is a sphere. The final chapter contains the most beautiful and profound result in the theory of compact surfaces: the Gauss-Bonnet theorem.

Examples are interspersed throughout, not only for the definitions, but also for the theorems. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that has just been introduced. Full solutions to all exercises are given at the end of the book. The book is an ideal text for self-study or a one-semester course in differential geometry for undergraduates in the mathematical sciences.

Reviewer: Stere Ianus (Bucureşti)

##### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53A04 | Curves in Euclidean and related spaces |

53A05 | Surfaces in Euclidean and related spaces |

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

53C22 | Geodesics in global differential geometry |

53C20 | Global Riemannian geometry, including pinching |

53C40 | Global submanifolds |