Differential geometry and its applications.
2nd ed.

*(English)*Zbl 1153.53001
Washington, DC: Mathematical Association of America (MAA) (ISBN 978-0-88385-748-9/hbk). xxi, 467 p. (2007).

This textbook covers standard topics of differential geometry when presented for undergraduates interested in mathematics or physics, i.e., it deals with curves and surfaces in Euclidean spaces, as well as with initial facts on geodesics and curvature, manifolds and Riemannian geometry. Among less common topics we should mention constant mean curvature surfaces and calculus of variations, incuding Pontryagin maximum principle.

The theoretical treatment is endowed with numerous exercises and examples. The concepts and constructions presented are systematically visualized through the use of computer algebra system Maple. Related procedures are included in the text with necessary explanations. The book is carefully written, and may be recommended to students approaching differential geometry for the first time.

Contents:

1. The geometry of curves (Pages 1–66).

2. Surfaces (67–106).

3. Curvatures (107–159).

4. Constant mean curvature surfaces (161–207).

5. Ggeodesics, metrics and isometries (208–273).

6. Holonomy and the Gauss–Bonnet theorem (275–310).

7. The calculus of variations and geometry (311–395).

8. A glimps to higher dimensions (397–435).

The theoretical treatment is endowed with numerous exercises and examples. The concepts and constructions presented are systematically visualized through the use of computer algebra system Maple. Related procedures are included in the text with necessary explanations. The book is carefully written, and may be recommended to students approaching differential geometry for the first time.

Contents:

1. The geometry of curves (Pages 1–66).

2. Surfaces (67–106).

3. Curvatures (107–159).

4. Constant mean curvature surfaces (161–207).

5. Ggeodesics, metrics and isometries (208–273).

6. Holonomy and the Gauss–Bonnet theorem (275–310).

7. The calculus of variations and geometry (311–395).

8. A glimps to higher dimensions (397–435).

Reviewer: Victor Alexandrov (Novosibirsk)

##### 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 |

53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

49Q05 | Minimal surfaces and optimization |

53-04 | Software, source code, etc. for problems pertaining to differential geometry |